contour integrals of functions of a complex variable · introduction 1.complex functions of a...
TRANSCRIPT
![Page 1: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/1.jpg)
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Contours Contour Integrals Examples
Contour Integrals of Functions of a ComplexVariable
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 2: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/2.jpg)
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Contours Contour Integrals Examples
Introduction
1. Complex functions of a complex variable are usually integratedalong parametric curves.
2. The integrals are ultimately reduced to integrals of complexfunctions of a real variable as introduced in the previouspresentation.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 3: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/3.jpg)
logo1
Contours Contour Integrals Examples
Introduction1. Complex functions of a complex variable are usually integrated
along parametric curves.
2. The integrals are ultimately reduced to integrals of complexfunctions of a real variable as introduced in the previouspresentation.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 4: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/4.jpg)
logo1
Contours Contour Integrals Examples
Introduction1. Complex functions of a complex variable are usually integrated
along parametric curves.2. The integrals are ultimately reduced to integrals of complex
functions of a real variable as introduced in the previouspresentation.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 5: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/5.jpg)
logo1
Contours Contour Integrals Examples
Definitions
A set C of points (x,y) in the complex plane is called an arc if andonly if there are continuous functions x(t) and y(t) with a≤ t ≤ b sothat for every point (x,y) in C there is a t so that x = x(t) and y = y(t).
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
C
r(x(a),y(a)
)
r(x(b),y(b)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 6: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/6.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsA set C of points (x,y) in the complex plane is called an arc if andonly if there are continuous functions x(t) and y(t) with a≤ t ≤ b sothat for every point (x,y) in C there is a t so that x = x(t) and y = y(t).
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
C
r(x(a),y(a)
)
r(x(b),y(b)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 7: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/7.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsA set C of points (x,y) in the complex plane is called an arc if andonly if there are continuous functions x(t) and y(t) with a≤ t ≤ b sothat for every point (x,y) in C there is a t so that x = x(t) and y = y(t).
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
C
r(x(a),y(a)
)
r(x(b),y(b)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 8: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/8.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsA set C of points (x,y) in the complex plane is called an arc if andonly if there are continuous functions x(t) and y(t) with a≤ t ≤ b sothat for every point (x,y) in C there is a t so that x = x(t) and y = y(t).
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
C
r(x(a),y(a)
)
r(x(b),y(b)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 9: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/9.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsA set C of points (x,y) in the complex plane is called an arc if andonly if there are continuous functions x(t) and y(t) with a≤ t ≤ b sothat for every point (x,y) in C there is a t so that x = x(t) and y = y(t).
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
C
r(x(a),y(a)
)
r(x(b),y(b)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 10: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/10.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsA set C of points (x,y) in the complex plane is called an arc if andonly if there are continuous functions x(t) and y(t) with a≤ t ≤ b sothat for every point (x,y) in C there is a t so that x = x(t) and y = y(t).
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
C
r(x(a),y(a)
)
r(x(b),y(b)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 11: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/11.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsA set C of points (x,y) in the complex plane is called an arc if andonly if there are continuous functions x(t) and y(t) with a≤ t ≤ b sothat for every point (x,y) in C there is a t so that x = x(t) and y = y(t).
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
C
r(x(a),y(a)
)
r(x(b),y(b)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 12: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/12.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsA set C of points (x,y) in the complex plane is called an arc if andonly if there are continuous functions x(t) and y(t) with a≤ t ≤ b sothat for every point (x,y) in C there is a t so that x = x(t) and y = y(t).
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
C
r(x(a),y(a)
)
r(x(b),y(b)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 13: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/13.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsA set C of points (x,y) in the complex plane is called an arc if andonly if there are continuous functions x(t) and y(t) with a≤ t ≤ b sothat for every point (x,y) in C there is a t so that x = x(t) and y = y(t).
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
C
r(x(a),y(a)
)
r(x(b),y(b)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 14: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/14.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsA set C of points (x,y) in the complex plane is called an arc if andonly if there are continuous functions x(t) and y(t) with a≤ t ≤ b sothat for every point (x,y) in C there is a t so that x = x(t) and y = y(t).
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>
U
C
r(x(a),y(a)
)
r(x(b),y(b)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 15: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/15.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsA set C of points (x,y) in the complex plane is called an arc if andonly if there are continuous functions x(t) and y(t) with a≤ t ≤ b sothat for every point (x,y) in C there is a t so that x = x(t) and y = y(t).
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
C
r(x(a),y(a)
)
r(x(b),y(b)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 16: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/16.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsA set C of points (x,y) in the complex plane is called an arc if andonly if there are continuous functions x(t) and y(t) with a≤ t ≤ b sothat for every point (x,y) in C there is a t so that x = x(t) and y = y(t).
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
C
r(x(a),y(a)
)
r(x(b),y(b)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 17: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/17.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsA set C of points (x,y) in the complex plane is called an arc if andonly if there are continuous functions x(t) and y(t) with a≤ t ≤ b sothat for every point (x,y) in C there is a t so that x = x(t) and y = y(t).
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
C
r
(x(a),y(a)
)
r(x(b),y(b)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 18: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/18.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsA set C of points (x,y) in the complex plane is called an arc if andonly if there are continuous functions x(t) and y(t) with a≤ t ≤ b sothat for every point (x,y) in C there is a t so that x = x(t) and y = y(t).
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
C
r(x(a),y(a)
)
r(x(b),y(b)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 19: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/19.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsA set C of points (x,y) in the complex plane is called an arc if andonly if there are continuous functions x(t) and y(t) with a≤ t ≤ b sothat for every point (x,y) in C there is a t so that x = x(t) and y = y(t).
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
C
r(x(a),y(a)
)
r
(x(b),y(b)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 20: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/20.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsA set C of points (x,y) in the complex plane is called an arc if andonly if there are continuous functions x(t) and y(t) with a≤ t ≤ b sothat for every point (x,y) in C there is a t so that x = x(t) and y = y(t).
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
C
r(x(a),y(a)
)
r(x(b),y(b)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 21: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/21.jpg)
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Contours Contour Integrals Examples
Definitions
With the usual switching between two-dimensional notation andcomplex notation, we also write the continuous function asz(t) = x(t)+ iy(t).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 22: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/22.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsWith the usual switching between two-dimensional notation andcomplex notation, we also write the continuous function asz(t) = x(t)+ iy(t).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 23: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/23.jpg)
logo1
Contours Contour Integrals Examples
Definitions
An arc C is a Jordan arc or a simple arc if and only if for t1 6= t2 wehave z(t1) 6= z(t2).
-
6ℑ(z)
ℜ(z)
q
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 24: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/24.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsAn arc C is a Jordan arc or a simple arc if and only if for t1 6= t2 wehave z(t1) 6= z(t2).
-
6ℑ(z)
ℜ(z)
q
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 25: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/25.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsAn arc C is a Jordan arc or a simple arc if and only if for t1 6= t2 wehave z(t1) 6= z(t2).
-
6ℑ(z)
ℜ(z)
q
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 26: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/26.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsAn arc C is a Jordan arc or a simple arc if and only if for t1 6= t2 wehave z(t1) 6= z(t2).
-
6ℑ(z)
ℜ(z)
q
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 27: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/27.jpg)
logo1
Contours Contour Integrals Examples
Definitions
An arc C is called a simple closed curve if, except for z(a) = z(b) wehave that t1 6= t2 implies z(t1) 6= z(t2).
-
6ℑ(z)
ℜ(z)
q
�I
Ii
y
�6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 28: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/28.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsAn arc C is called a simple closed curve if, except for z(a) = z(b) wehave that t1 6= t2 implies z(t1) 6= z(t2).
-
6ℑ(z)
ℜ(z)
q
�I
Ii
y
�6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 29: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/29.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsAn arc C is called a simple closed curve if, except for z(a) = z(b) wehave that t1 6= t2 implies z(t1) 6= z(t2).
-
6ℑ(z)
ℜ(z)
q
�I
Ii
y
�6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 30: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/30.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsAn arc C is called a simple closed curve if, except for z(a) = z(b) wehave that t1 6= t2 implies z(t1) 6= z(t2).
-
6ℑ(z)
ℜ(z)
q
�I
Ii
y
�6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 31: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/31.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsAn arc C is called a simple closed curve if, except for z(a) = z(b) wehave that t1 6= t2 implies z(t1) 6= z(t2).
-
6ℑ(z)
ℜ(z)
q
�
I
Ii
y
�6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 32: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/32.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsAn arc C is called a simple closed curve if, except for z(a) = z(b) wehave that t1 6= t2 implies z(t1) 6= z(t2).
-
6ℑ(z)
ℜ(z)
q
�I
Ii
y
�6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 33: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/33.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsAn arc C is called a simple closed curve if, except for z(a) = z(b) wehave that t1 6= t2 implies z(t1) 6= z(t2).
-
6ℑ(z)
ℜ(z)
q
�I
I
i
y
�6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 34: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/34.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsAn arc C is called a simple closed curve if, except for z(a) = z(b) wehave that t1 6= t2 implies z(t1) 6= z(t2).
-
6ℑ(z)
ℜ(z)
q
�I
Ii
y
�6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 35: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/35.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsAn arc C is called a simple closed curve if, except for z(a) = z(b) wehave that t1 6= t2 implies z(t1) 6= z(t2).
-
6ℑ(z)
ℜ(z)
q
�I
Ii
y
�6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 36: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/36.jpg)
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Contours Contour Integrals Examples
DefinitionsAn arc C is called a simple closed curve if, except for z(a) = z(b) wehave that t1 6= t2 implies z(t1) 6= z(t2).
-
6ℑ(z)
ℜ(z)
q
�I
Ii
y
�
6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 37: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/37.jpg)
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Contours Contour Integrals Examples
DefinitionsAn arc C is called a simple closed curve if, except for z(a) = z(b) wehave that t1 6= t2 implies z(t1) 6= z(t2).
-
6ℑ(z)
ℜ(z)
q
�I
Ii
y
�6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 38: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/38.jpg)
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Contours Contour Integrals Examples
Definitions
A simple closed curve is called positively oriented if and only if it istraversed in the counterclockwise (mathematically positive) direction.
-
6ℑ(z)
ℜ(z)
K
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 39: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/39.jpg)
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Contours Contour Integrals Examples
DefinitionsA simple closed curve is called positively oriented if and only if it istraversed in the counterclockwise (mathematically positive) direction.
-
6ℑ(z)
ℜ(z)
K
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 40: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/40.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsA simple closed curve is called positively oriented if and only if it istraversed in the counterclockwise (mathematically positive) direction.
-
6ℑ(z)
ℜ(z)
K
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 41: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/41.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsA simple closed curve is called positively oriented if and only if it istraversed in the counterclockwise (mathematically positive) direction.
-
6ℑ(z)
ℜ(z)
K
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 42: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/42.jpg)
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Contours Contour Integrals Examples
Definitions
An arc C is called smooth if and only if the function z(t) that traversesC is differentiable with continuous derivative and z′(t) 6= 0 for all t.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
C
r(x(a),y(a)
)
r(x(b),y(b)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 43: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/43.jpg)
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Contours Contour Integrals Examples
DefinitionsAn arc C is called smooth if and only if the function z(t) that traversesC is differentiable with continuous derivative and z′(t) 6= 0 for all t.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
C
r(x(a),y(a)
)
r(x(b),y(b)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 44: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/44.jpg)
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Contours Contour Integrals Examples
DefinitionsAn arc C is called smooth if and only if the function z(t) that traversesC is differentiable with continuous derivative and z′(t) 6= 0 for all t.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
C
r(x(a),y(a)
)
r(x(b),y(b)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 45: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/45.jpg)
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Contours Contour Integrals Examples
DefinitionsAn arc C is called smooth if and only if the function z(t) that traversesC is differentiable with continuous derivative and z′(t) 6= 0 for all t.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U>U
C
(x(a),y(a)
) r
r(x(b),y(b)
)
No!
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 46: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/46.jpg)
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Contours Contour Integrals Examples
DefinitionsAn arc C is called smooth if and only if the function z(t) that traversesC is differentiable with continuous derivative and z′(t) 6= 0 for all t.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U>U
C
(x(a),y(a)
) r
r(x(b),y(b)
)No!
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 47: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/47.jpg)
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Contours Contour Integrals Examples
Definitions
An arc is called a contour or a piecewise smooth arc if and only if itconsists of smooth arcs joined end-to-end. It is called a simple closedcontour if and only if there is no self-intersection except that theinitial point equals the final point.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U>U
C
(x(a),y(a)
) r
r(x(b),y(b)
)O.k. for contours.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 48: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/48.jpg)
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Contours Contour Integrals Examples
DefinitionsAn arc is called a contour or a piecewise smooth arc if and only if itconsists of smooth arcs joined end-to-end. It is called a simple closedcontour if and only if there is no self-intersection except that theinitial point equals the final point.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U>U
C
(x(a),y(a)
) r
r(x(b),y(b)
)O.k. for contours.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 49: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/49.jpg)
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Contours Contour Integrals Examples
DefinitionsAn arc is called a contour or a piecewise smooth arc if and only if itconsists of smooth arcs joined end-to-end. It is called a simple closedcontour if and only if there is no self-intersection except that theinitial point equals the final point.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U>U
C
(x(a),y(a)
) r
r(x(b),y(b)
)
O.k. for contours.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 50: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/50.jpg)
logo1
Contours Contour Integrals Examples
DefinitionsAn arc is called a contour or a piecewise smooth arc if and only if itconsists of smooth arcs joined end-to-end. It is called a simple closedcontour if and only if there is no self-intersection except that theinitial point equals the final point.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U>U
C
(x(a),y(a)
) r
r(x(b),y(b)
)O.k. for contours.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 51: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/51.jpg)
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Contours Contour Integrals Examples
Example.
With z(θ) = eiθ and 0≤ θ ≤ 2π , the unit circleC = {z ∈ C : |z|= 1} is a positively oriented simple closed curve.
-
6ℑ(z)
ℜ(z)
K
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 52: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/52.jpg)
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Contours Contour Integrals Examples
Example. With z(θ) = eiθ and 0≤ θ ≤ 2π , the unit circleC = {z ∈ C : |z|= 1} is a positively oriented simple closed curve.
-
6ℑ(z)
ℜ(z)
K
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 53: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/53.jpg)
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Contours Contour Integrals Examples
Example. With z(θ) = eiθ and 0≤ θ ≤ 2π , the unit circleC = {z ∈ C : |z|= 1} is a positively oriented simple closed curve.
-
6ℑ(z)
ℜ(z)
K
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 54: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/54.jpg)
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Contours Contour Integrals Examples
Example. With z(θ) = eiθ and 0≤ θ ≤ 2π , the unit circleC = {z ∈ C : |z|= 1} is a positively oriented simple closed curve.
-
6ℑ(z)
ℜ(z)
K
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 55: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/55.jpg)
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Contours Contour Integrals Examples
Example.
With z(θ) = e−iθ and 0≤ θ ≤ 2π , the unit circleC = {z ∈ C : |z|= 1} is a simple closed curve, but it is not positivelyoriented.
-
6ℑ(z)
ℜ(z)
U
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 56: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/56.jpg)
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Contours Contour Integrals Examples
Example. With z(θ) = e−iθ and 0≤ θ ≤ 2π , the unit circleC = {z ∈ C : |z|= 1} is a simple closed curve, but it is not positivelyoriented.
-
6ℑ(z)
ℜ(z)
U
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 57: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/57.jpg)
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Contours Contour Integrals Examples
Example. With z(θ) = e−iθ and 0≤ θ ≤ 2π , the unit circleC = {z ∈ C : |z|= 1} is a simple closed curve, but it is not positivelyoriented.
-
6ℑ(z)
ℜ(z)
U
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 58: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/58.jpg)
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Contours Contour Integrals Examples
Example. With z(θ) = e−iθ and 0≤ θ ≤ 2π , the unit circleC = {z ∈ C : |z|= 1} is a simple closed curve, but it is not positivelyoriented.
-
6ℑ(z)
ℜ(z)
U
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 59: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/59.jpg)
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Contours Contour Integrals Examples
Definition.
The length of a contour C parametrized by z(t) is
L :=∫ b
a
∣∣z′(t)∣∣ dt.
Discussion.
L =∫ b
a
∣∣z′(t)∣∣ dt
=∫ b
a
√(x′(t)
)2 +(y′(t)
)2 dt
which is the length formula from multivariable calculus.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 60: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/60.jpg)
logo1
Contours Contour Integrals Examples
Definition. The length of a contour C parametrized by z(t) is
L :=∫ b
a
∣∣z′(t)∣∣ dt.
Discussion.
L =∫ b
a
∣∣z′(t)∣∣ dt
=∫ b
a
√(x′(t)
)2 +(y′(t)
)2 dt
which is the length formula from multivariable calculus.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 61: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/61.jpg)
logo1
Contours Contour Integrals Examples
Definition. The length of a contour C parametrized by z(t) is
L :=∫ b
a
∣∣z′(t)∣∣ dt.
Discussion.
L =∫ b
a
∣∣z′(t)∣∣ dt
=∫ b
a
√(x′(t)
)2 +(y′(t)
)2 dt
which is the length formula from multivariable calculus.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 62: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/62.jpg)
logo1
Contours Contour Integrals Examples
Definition. The length of a contour C parametrized by z(t) is
L :=∫ b
a
∣∣z′(t)∣∣ dt.
Discussion.
L =∫ b
a
∣∣z′(t)∣∣ dt
=∫ b
a
√(x′(t)
)2 +(y′(t)
)2 dt
which is the length formula from multivariable calculus.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 63: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/63.jpg)
logo1
Contours Contour Integrals Examples
Definition. The length of a contour C parametrized by z(t) is
L :=∫ b
a
∣∣z′(t)∣∣ dt.
Discussion.
L =∫ b
a
∣∣z′(t)∣∣ dt
=∫ b
a
√(x′(t)
)2 +(y′(t)
)2 dt
which is the length formula from multivariable calculus.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 64: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/64.jpg)
logo1
Contours Contour Integrals Examples
Definition. The length of a contour C parametrized by z(t) is
L :=∫ b
a
∣∣z′(t)∣∣ dt.
Discussion.
L =∫ b
a
∣∣z′(t)∣∣ dt
=∫ b
a
√(x′(t)
)2 +(y′(t)
)2 dt
which is the length formula from multivariable calculus.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 65: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/65.jpg)
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Contours Contour Integrals Examples
Definition.
Let f be a continuous function of a complex variable andlet C be a contour with parametrization z(t). Then we define thecontour integral of f over C as∫
Cf (z) dz :=
∫ b
af (z(t))z′(t) dt.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 66: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/66.jpg)
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Contours Contour Integrals Examples
Definition. Let f be a continuous function of a complex variable andlet C be a contour with parametrization z(t).
Then we define thecontour integral of f over C as∫
Cf (z) dz :=
∫ b
af (z(t))z′(t) dt.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 67: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/67.jpg)
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Contours Contour Integrals Examples
Definition. Let f be a continuous function of a complex variable andlet C be a contour with parametrization z(t). Then we define thecontour integral of f over C as∫
Cf (z) dz
:=∫ b
af (z(t))z′(t) dt.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 68: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/68.jpg)
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Contours Contour Integrals Examples
Definition. Let f be a continuous function of a complex variable andlet C be a contour with parametrization z(t). Then we define thecontour integral of f over C as∫
Cf (z) dz :=
∫ b
af (z(t))z′(t) dt.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 69: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/69.jpg)
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Contours Contour Integrals Examples
Note.
If φ is a differentiable function with continuous, nonzeroderivative that maps the interval [α,β ] to the interval [a,b], thenz(φ(τ)) is another parametrization of C. It just uses the parameter τ in[α,β ] rather than the parameter t in [a,b]. But the integral of f over Cis not affected by interchanging parametrizations in this fashion,because with ξ (τ) := z(φ(τ)) we have∫
β
α
f (ξ (τ))ξ ′(τ) dτ =∫
β
α
f (z(φ(τ)))z′(φ(τ))φ ′(τ) dτ
=∫ b
af (z(t))z′(t) dt
Therefore, the definition of the contour integral is sensible, as it onlydepends on the shape of the contour, not on the way we parametrizeit. (Omitted proof that any two parametrizations “differ” by a φ asabove.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 70: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/70.jpg)
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Contours Contour Integrals Examples
Note. If φ is a differentiable function with continuous, nonzeroderivative that maps the interval [α,β ] to the interval [a,b], thenz(φ(τ)) is another parametrization of C.
It just uses the parameter τ in[α,β ] rather than the parameter t in [a,b]. But the integral of f over Cis not affected by interchanging parametrizations in this fashion,because with ξ (τ) := z(φ(τ)) we have∫
β
α
f (ξ (τ))ξ ′(τ) dτ =∫
β
α
f (z(φ(τ)))z′(φ(τ))φ ′(τ) dτ
=∫ b
af (z(t))z′(t) dt
Therefore, the definition of the contour integral is sensible, as it onlydepends on the shape of the contour, not on the way we parametrizeit. (Omitted proof that any two parametrizations “differ” by a φ asabove.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 71: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/71.jpg)
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Contours Contour Integrals Examples
Note. If φ is a differentiable function with continuous, nonzeroderivative that maps the interval [α,β ] to the interval [a,b], thenz(φ(τ)) is another parametrization of C. It just uses the parameter τ in[α,β ] rather than the parameter t in [a,b].
But the integral of f over Cis not affected by interchanging parametrizations in this fashion,because with ξ (τ) := z(φ(τ)) we have∫
β
α
f (ξ (τ))ξ ′(τ) dτ =∫
β
α
f (z(φ(τ)))z′(φ(τ))φ ′(τ) dτ
=∫ b
af (z(t))z′(t) dt
Therefore, the definition of the contour integral is sensible, as it onlydepends on the shape of the contour, not on the way we parametrizeit. (Omitted proof that any two parametrizations “differ” by a φ asabove.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 72: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/72.jpg)
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Contours Contour Integrals Examples
Note. If φ is a differentiable function with continuous, nonzeroderivative that maps the interval [α,β ] to the interval [a,b], thenz(φ(τ)) is another parametrization of C. It just uses the parameter τ in[α,β ] rather than the parameter t in [a,b]. But the integral of f over Cis not affected by interchanging parametrizations in this fashion,because with ξ (τ) := z(φ(τ)) we have
∫β
α
f (ξ (τ))ξ ′(τ) dτ =∫
β
α
f (z(φ(τ)))z′(φ(τ))φ ′(τ) dτ
=∫ b
af (z(t))z′(t) dt
Therefore, the definition of the contour integral is sensible, as it onlydepends on the shape of the contour, not on the way we parametrizeit. (Omitted proof that any two parametrizations “differ” by a φ asabove.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 73: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/73.jpg)
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Contours Contour Integrals Examples
Note. If φ is a differentiable function with continuous, nonzeroderivative that maps the interval [α,β ] to the interval [a,b], thenz(φ(τ)) is another parametrization of C. It just uses the parameter τ in[α,β ] rather than the parameter t in [a,b]. But the integral of f over Cis not affected by interchanging parametrizations in this fashion,because with ξ (τ) := z(φ(τ)) we have∫
β
α
f (ξ (τ))ξ ′(τ) dτ
=∫
β
α
f (z(φ(τ)))z′(φ(τ))φ ′(τ) dτ
=∫ b
af (z(t))z′(t) dt
Therefore, the definition of the contour integral is sensible, as it onlydepends on the shape of the contour, not on the way we parametrizeit. (Omitted proof that any two parametrizations “differ” by a φ asabove.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 74: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/74.jpg)
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Contours Contour Integrals Examples
Note. If φ is a differentiable function with continuous, nonzeroderivative that maps the interval [α,β ] to the interval [a,b], thenz(φ(τ)) is another parametrization of C. It just uses the parameter τ in[α,β ] rather than the parameter t in [a,b]. But the integral of f over Cis not affected by interchanging parametrizations in this fashion,because with ξ (τ) := z(φ(τ)) we have∫
β
α
f (ξ (τ))ξ ′(τ) dτ =∫
β
α
f (z(φ(τ)))z′(φ(τ))φ ′(τ) dτ
=∫ b
af (z(t))z′(t) dt
Therefore, the definition of the contour integral is sensible, as it onlydepends on the shape of the contour, not on the way we parametrizeit. (Omitted proof that any two parametrizations “differ” by a φ asabove.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 75: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/75.jpg)
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Contours Contour Integrals Examples
Note. If φ is a differentiable function with continuous, nonzeroderivative that maps the interval [α,β ] to the interval [a,b], thenz(φ(τ)) is another parametrization of C. It just uses the parameter τ in[α,β ] rather than the parameter t in [a,b]. But the integral of f over Cis not affected by interchanging parametrizations in this fashion,because with ξ (τ) := z(φ(τ)) we have∫
β
α
f (ξ (τ))ξ ′(τ) dτ =∫
β
α
f (z(φ(τ)))z′(φ(τ))φ ′(τ) dτ
=∫ b
af (z(t))z′(t) dt
Therefore, the definition of the contour integral is sensible, as it onlydepends on the shape of the contour, not on the way we parametrizeit. (Omitted proof that any two parametrizations “differ” by a φ asabove.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 76: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/76.jpg)
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Contours Contour Integrals Examples
Note. If φ is a differentiable function with continuous, nonzeroderivative that maps the interval [α,β ] to the interval [a,b], thenz(φ(τ)) is another parametrization of C. It just uses the parameter τ in[α,β ] rather than the parameter t in [a,b]. But the integral of f over Cis not affected by interchanging parametrizations in this fashion,because with ξ (τ) := z(φ(τ)) we have∫
β
α
f (ξ (τ))ξ ′(τ) dτ =∫
β
α
f (z(φ(τ)))z′(φ(τ))φ ′(τ) dτ
=∫ b
af (z(t))z′(t) dt
Therefore, the definition of the contour integral is sensible, as it onlydepends on the shape of the contour, not on the way we parametrizeit.
(Omitted proof that any two parametrizations “differ” by a φ asabove.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 77: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/77.jpg)
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Contours Contour Integrals Examples
Note. If φ is a differentiable function with continuous, nonzeroderivative that maps the interval [α,β ] to the interval [a,b], thenz(φ(τ)) is another parametrization of C. It just uses the parameter τ in[α,β ] rather than the parameter t in [a,b]. But the integral of f over Cis not affected by interchanging parametrizations in this fashion,because with ξ (τ) := z(φ(τ)) we have∫
β
α
f (ξ (τ))ξ ′(τ) dτ =∫
β
α
f (z(φ(τ)))z′(φ(τ))φ ′(τ) dτ
=∫ b
af (z(t))z′(t) dt
Therefore, the definition of the contour integral is sensible, as it onlydepends on the shape of the contour, not on the way we parametrizeit. (Omitted proof that any two parametrizations “differ” by a φ asabove.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 78: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/78.jpg)
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Contours Contour Integrals Examples
Rules.
1. If w is a complex number, then∫
Cwf (z) dz = w
∫C
f (z) dz.
2.∫
C(f +g)(z) dz =
∫C
f (z) dz+∫
Cg(z) dz.
3. Let −C denote the same contour as C, only traversed in theopposite direction. Then z(−τ) with −b≤ τ ≤−a is aparametrization and∫
−Cf (z) dz =
∫ −a
−bf (z(−τ))
ddτ
z(−τ) dτ =∫ −a
−bf (z(−τ))z′(−τ)(−1) dτ
=∫ a
bf (z(t))z′(t) dt =−
∫C
f (z) dz
4. If the endpoint of C1 is the starting point of C2, then the union ofthe two contours in denoted C := C1 +C2 and we have∫
Cf (z) dz =
∫C1
f (z) dz+∫
C2
g(z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 79: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/79.jpg)
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Contours Contour Integrals Examples
Rules.1. If w is a complex number, then
∫C
wf (z) dz = w∫
Cf (z) dz.
2.∫
C(f +g)(z) dz =
∫C
f (z) dz+∫
Cg(z) dz.
3. Let −C denote the same contour as C, only traversed in theopposite direction. Then z(−τ) with −b≤ τ ≤−a is aparametrization and∫
−Cf (z) dz =
∫ −a
−bf (z(−τ))
ddτ
z(−τ) dτ =∫ −a
−bf (z(−τ))z′(−τ)(−1) dτ
=∫ a
bf (z(t))z′(t) dt =−
∫C
f (z) dz
4. If the endpoint of C1 is the starting point of C2, then the union ofthe two contours in denoted C := C1 +C2 and we have∫
Cf (z) dz =
∫C1
f (z) dz+∫
C2
g(z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 80: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/80.jpg)
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Contours Contour Integrals Examples
Rules.1. If w is a complex number, then
∫C
wf (z) dz = w∫
Cf (z) dz.
2.∫
C(f +g)(z) dz =
∫C
f (z) dz+∫
Cg(z) dz.
3. Let −C denote the same contour as C, only traversed in theopposite direction. Then z(−τ) with −b≤ τ ≤−a is aparametrization and∫
−Cf (z) dz =
∫ −a
−bf (z(−τ))
ddτ
z(−τ) dτ =∫ −a
−bf (z(−τ))z′(−τ)(−1) dτ
=∫ a
bf (z(t))z′(t) dt =−
∫C
f (z) dz
4. If the endpoint of C1 is the starting point of C2, then the union ofthe two contours in denoted C := C1 +C2 and we have∫
Cf (z) dz =
∫C1
f (z) dz+∫
C2
g(z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 81: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/81.jpg)
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Contours Contour Integrals Examples
Rules.1. If w is a complex number, then
∫C
wf (z) dz = w∫
Cf (z) dz.
2.∫
C(f +g)(z) dz =
∫C
f (z) dz+∫
Cg(z) dz.
3. Let −C denote the same contour as C, only traversed in theopposite direction.
Then z(−τ) with −b≤ τ ≤−a is aparametrization and∫
−Cf (z) dz =
∫ −a
−bf (z(−τ))
ddτ
z(−τ) dτ =∫ −a
−bf (z(−τ))z′(−τ)(−1) dτ
=∫ a
bf (z(t))z′(t) dt =−
∫C
f (z) dz
4. If the endpoint of C1 is the starting point of C2, then the union ofthe two contours in denoted C := C1 +C2 and we have∫
Cf (z) dz =
∫C1
f (z) dz+∫
C2
g(z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 82: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/82.jpg)
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Contours Contour Integrals Examples
Rules.1. If w is a complex number, then
∫C
wf (z) dz = w∫
Cf (z) dz.
2.∫
C(f +g)(z) dz =
∫C
f (z) dz+∫
Cg(z) dz.
3. Let −C denote the same contour as C, only traversed in theopposite direction. Then z(−τ) with −b≤ τ ≤−a is aparametrization and
∫−C
f (z) dz =∫ −a
−bf (z(−τ))
ddτ
z(−τ) dτ =∫ −a
−bf (z(−τ))z′(−τ)(−1) dτ
=∫ a
bf (z(t))z′(t) dt =−
∫C
f (z) dz
4. If the endpoint of C1 is the starting point of C2, then the union ofthe two contours in denoted C := C1 +C2 and we have∫
Cf (z) dz =
∫C1
f (z) dz+∫
C2
g(z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 83: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/83.jpg)
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Contours Contour Integrals Examples
Rules.1. If w is a complex number, then
∫C
wf (z) dz = w∫
Cf (z) dz.
2.∫
C(f +g)(z) dz =
∫C
f (z) dz+∫
Cg(z) dz.
3. Let −C denote the same contour as C, only traversed in theopposite direction. Then z(−τ) with −b≤ τ ≤−a is aparametrization and∫
−Cf (z) dz
=∫ −a
−bf (z(−τ))
ddτ
z(−τ) dτ =∫ −a
−bf (z(−τ))z′(−τ)(−1) dτ
=∫ a
bf (z(t))z′(t) dt =−
∫C
f (z) dz
4. If the endpoint of C1 is the starting point of C2, then the union ofthe two contours in denoted C := C1 +C2 and we have∫
Cf (z) dz =
∫C1
f (z) dz+∫
C2
g(z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 84: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/84.jpg)
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Contours Contour Integrals Examples
Rules.1. If w is a complex number, then
∫C
wf (z) dz = w∫
Cf (z) dz.
2.∫
C(f +g)(z) dz =
∫C
f (z) dz+∫
Cg(z) dz.
3. Let −C denote the same contour as C, only traversed in theopposite direction. Then z(−τ) with −b≤ τ ≤−a is aparametrization and∫
−Cf (z) dz =
∫ −a
−bf (z(−τ))
ddτ
z(−τ) dτ
=∫ −a
−bf (z(−τ))z′(−τ)(−1) dτ
=∫ a
bf (z(t))z′(t) dt =−
∫C
f (z) dz
4. If the endpoint of C1 is the starting point of C2, then the union ofthe two contours in denoted C := C1 +C2 and we have∫
Cf (z) dz =
∫C1
f (z) dz+∫
C2
g(z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 85: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/85.jpg)
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Contours Contour Integrals Examples
Rules.1. If w is a complex number, then
∫C
wf (z) dz = w∫
Cf (z) dz.
2.∫
C(f +g)(z) dz =
∫C
f (z) dz+∫
Cg(z) dz.
3. Let −C denote the same contour as C, only traversed in theopposite direction. Then z(−τ) with −b≤ τ ≤−a is aparametrization and∫
−Cf (z) dz =
∫ −a
−bf (z(−τ))
ddτ
z(−τ) dτ =∫ −a
−bf (z(−τ))z′(−τ)(−1) dτ
=∫ a
bf (z(t))z′(t) dt =−
∫C
f (z) dz
4. If the endpoint of C1 is the starting point of C2, then the union ofthe two contours in denoted C := C1 +C2 and we have∫
Cf (z) dz =
∫C1
f (z) dz+∫
C2
g(z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 86: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/86.jpg)
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Contours Contour Integrals Examples
Rules.1. If w is a complex number, then
∫C
wf (z) dz = w∫
Cf (z) dz.
2.∫
C(f +g)(z) dz =
∫C
f (z) dz+∫
Cg(z) dz.
3. Let −C denote the same contour as C, only traversed in theopposite direction. Then z(−τ) with −b≤ τ ≤−a is aparametrization and∫
−Cf (z) dz =
∫ −a
−bf (z(−τ))
ddτ
z(−τ) dτ =∫ −a
−bf (z(−τ))z′(−τ)(−1) dτ
=∫ a
bf (z(t))z′(t) dt
=−∫
Cf (z) dz
4. If the endpoint of C1 is the starting point of C2, then the union ofthe two contours in denoted C := C1 +C2 and we have∫
Cf (z) dz =
∫C1
f (z) dz+∫
C2
g(z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 87: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/87.jpg)
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Contours Contour Integrals Examples
Rules.1. If w is a complex number, then
∫C
wf (z) dz = w∫
Cf (z) dz.
2.∫
C(f +g)(z) dz =
∫C
f (z) dz+∫
Cg(z) dz.
3. Let −C denote the same contour as C, only traversed in theopposite direction. Then z(−τ) with −b≤ τ ≤−a is aparametrization and∫
−Cf (z) dz =
∫ −a
−bf (z(−τ))
ddτ
z(−τ) dτ =∫ −a
−bf (z(−τ))z′(−τ)(−1) dτ
=∫ a
bf (z(t))z′(t) dt =−
∫C
f (z) dz
4. If the endpoint of C1 is the starting point of C2, then the union ofthe two contours in denoted C := C1 +C2 and we have∫
Cf (z) dz =
∫C1
f (z) dz+∫
C2
g(z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 88: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/88.jpg)
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Contours Contour Integrals Examples
Rules.1. If w is a complex number, then
∫C
wf (z) dz = w∫
Cf (z) dz.
2.∫
C(f +g)(z) dz =
∫C
f (z) dz+∫
Cg(z) dz.
3. Let −C denote the same contour as C, only traversed in theopposite direction. Then z(−τ) with −b≤ τ ≤−a is aparametrization and∫
−Cf (z) dz =
∫ −a
−bf (z(−τ))
ddτ
z(−τ) dτ =∫ −a
−bf (z(−τ))z′(−τ)(−1) dτ
=∫ a
bf (z(t))z′(t) dt =−
∫C
f (z) dz
4. If the endpoint of C1 is the starting point of C2, then the union ofthe two contours in denoted C := C1 +C2 and we have∫
Cf (z) dz =
∫C1
f (z) dz+∫
C2
g(z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 89: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/89.jpg)
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Contours Contour Integrals Examples
Example.
Compute the contour integral of f (z) = |z| around theupper half of the positively oriented unit circle.∫
Cf (z) dz =
∫π
0
∣∣eit∣∣ ieit dt
=∫
π
0ieit dt
= eit∣∣∣π0
= eiπ − e0 =−2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 90: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/90.jpg)
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Contours Contour Integrals Examples
Example. Compute the contour integral of f (z) = |z| around theupper half of the positively oriented unit circle.
∫C
f (z) dz =∫
π
0
∣∣eit∣∣ ieit dt
=∫
π
0ieit dt
= eit∣∣∣π0
= eiπ − e0 =−2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 91: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/91.jpg)
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Contours Contour Integrals Examples
Example. Compute the contour integral of f (z) = |z| around theupper half of the positively oriented unit circle.∫
Cf (z) dz
=∫
π
0
∣∣eit∣∣ ieit dt
=∫
π
0ieit dt
= eit∣∣∣π0
= eiπ − e0 =−2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 92: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/92.jpg)
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Contours Contour Integrals Examples
Example. Compute the contour integral of f (z) = |z| around theupper half of the positively oriented unit circle.∫
Cf (z) dz =
∫π
0
∣∣eit∣∣
ieit dt
=∫
π
0ieit dt
= eit∣∣∣π0
= eiπ − e0 =−2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 93: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/93.jpg)
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Contours Contour Integrals Examples
Example. Compute the contour integral of f (z) = |z| around theupper half of the positively oriented unit circle.∫
Cf (z) dz =
∫π
0
∣∣eit∣∣ ieit dt
=∫
π
0ieit dt
= eit∣∣∣π0
= eiπ − e0 =−2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 94: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/94.jpg)
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Contours Contour Integrals Examples
Example. Compute the contour integral of f (z) = |z| around theupper half of the positively oriented unit circle.∫
Cf (z) dz =
∫π
0
∣∣eit∣∣ ieit dt
=∫
π
0ieit dt
= eit∣∣∣π0
= eiπ − e0 =−2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 95: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/95.jpg)
logo1
Contours Contour Integrals Examples
Example. Compute the contour integral of f (z) = |z| around theupper half of the positively oriented unit circle.∫
Cf (z) dz =
∫π
0
∣∣eit∣∣ ieit dt
=∫
π
0ieit dt
= eit∣∣∣π0
= eiπ − e0 =−2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 96: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/96.jpg)
logo1
Contours Contour Integrals Examples
Example. Compute the contour integral of f (z) = |z| around theupper half of the positively oriented unit circle.∫
Cf (z) dz =
∫π
0
∣∣eit∣∣ ieit dt
=∫
π
0ieit dt
= eit∣∣∣π0
= eiπ − e0
=−2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 97: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/97.jpg)
logo1
Contours Contour Integrals Examples
Example. Compute the contour integral of f (z) = |z| around theupper half of the positively oriented unit circle.∫
Cf (z) dz =
∫π
0
∣∣eit∣∣ ieit dt
=∫
π
0ieit dt
= eit∣∣∣π0
= eiπ − e0 =−2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 98: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/98.jpg)
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Contours Contour Integrals Examples
Theorem.
If the complex function f (z) has an antiderivative F(z),then the integral of f over the contour C parametrized with z(t),a≤ t ≤ b is equal to∫
Cf (z) dz = F(z(b))−F(z(a)).
Proof. ∫C
f (z) dz =∫ b
af(z(t))z′(t) dt
= F(z(t))∣∣∣b
a= F(z(b))−F(z(a))
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 99: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/99.jpg)
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Contours Contour Integrals Examples
Theorem. If the complex function f (z) has an antiderivative F(z),then the integral of f over the contour C parametrized with z(t),a≤ t ≤ b is equal to∫
Cf (z) dz = F(z(b))−F(z(a)).
Proof. ∫C
f (z) dz =∫ b
af(z(t))z′(t) dt
= F(z(t))∣∣∣b
a= F(z(b))−F(z(a))
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 100: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/100.jpg)
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Contours Contour Integrals Examples
Theorem. If the complex function f (z) has an antiderivative F(z),then the integral of f over the contour C parametrized with z(t),a≤ t ≤ b is equal to∫
Cf (z) dz = F(z(b))−F(z(a)).
Proof.
∫C
f (z) dz =∫ b
af(z(t))z′(t) dt
= F(z(t))∣∣∣b
a= F(z(b))−F(z(a))
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 101: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/101.jpg)
logo1
Contours Contour Integrals Examples
Theorem. If the complex function f (z) has an antiderivative F(z),then the integral of f over the contour C parametrized with z(t),a≤ t ≤ b is equal to∫
Cf (z) dz = F(z(b))−F(z(a)).
Proof. ∫C
f (z) dz =∫ b
af(z(t))z′(t) dt
= F(z(t))∣∣∣b
a= F(z(b))−F(z(a))
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 102: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/102.jpg)
logo1
Contours Contour Integrals Examples
Theorem. If the complex function f (z) has an antiderivative F(z),then the integral of f over the contour C parametrized with z(t),a≤ t ≤ b is equal to∫
Cf (z) dz = F(z(b))−F(z(a)).
Proof. ∫C
f (z) dz =∫ b
af(z(t))z′(t) dt
= F(z(t))∣∣∣b
a
= F(z(b))−F(z(a))
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 103: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/103.jpg)
logo1
Contours Contour Integrals Examples
Theorem. If the complex function f (z) has an antiderivative F(z),then the integral of f over the contour C parametrized with z(t),a≤ t ≤ b is equal to∫
Cf (z) dz = F(z(b))−F(z(a)).
Proof. ∫C
f (z) dz =∫ b
af(z(t))z′(t) dt
= F(z(t))∣∣∣b
a= F(z(b))−F(z(a))
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 104: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/104.jpg)
logo1
Contours Contour Integrals Examples
Theorem. If the complex function f (z) has an antiderivative F(z),then the integral of f over the contour C parametrized with z(t),a≤ t ≤ b is equal to∫
Cf (z) dz = F(z(b))−F(z(a)).
Proof. ∫C
f (z) dz =∫ b
af(z(t))z′(t) dt
= F(z(t))∣∣∣b
a= F(z(b))−F(z(a))
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 105: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/105.jpg)
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Contours Contour Integrals Examples
Example.
Let n ∈ C(!) Integrate the function f (z) = zn around thepositively oriented unit circle.∫
Czn dz =
∫ 2π
0
(eit)n
ieit dt =∫ 2π
0iei(n+1)t dt
=1
n+1ei(n+1)t
∣∣∣2π
0(n 6=−1)∫
Cz−1 dz =
∫ 2π
0i dt = 2πi
In particular, for n an integer not equal to −1, the integral is zero.Note that the computation is not affected by the fact that the contourcrosses a branch cut. We choose a branch and stay consistent with it.A value at a single point (where the power function is discontinuous)does not affect an integral.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 106: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/106.jpg)
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Contours Contour Integrals Examples
Example. Let n ∈ C
(!) Integrate the function f (z) = zn around thepositively oriented unit circle.∫
Czn dz =
∫ 2π
0
(eit)n
ieit dt =∫ 2π
0iei(n+1)t dt
=1
n+1ei(n+1)t
∣∣∣2π
0(n 6=−1)∫
Cz−1 dz =
∫ 2π
0i dt = 2πi
In particular, for n an integer not equal to −1, the integral is zero.Note that the computation is not affected by the fact that the contourcrosses a branch cut. We choose a branch and stay consistent with it.A value at a single point (where the power function is discontinuous)does not affect an integral.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 107: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/107.jpg)
logo1
Contours Contour Integrals Examples
Example. Let n ∈ C(!)
Integrate the function f (z) = zn around thepositively oriented unit circle.∫
Czn dz =
∫ 2π
0
(eit)n
ieit dt =∫ 2π
0iei(n+1)t dt
=1
n+1ei(n+1)t
∣∣∣2π
0(n 6=−1)∫
Cz−1 dz =
∫ 2π
0i dt = 2πi
In particular, for n an integer not equal to −1, the integral is zero.Note that the computation is not affected by the fact that the contourcrosses a branch cut. We choose a branch and stay consistent with it.A value at a single point (where the power function is discontinuous)does not affect an integral.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 108: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/108.jpg)
logo1
Contours Contour Integrals Examples
Example. Let n ∈ C(!) Integrate the function f (z) = zn around thepositively oriented unit circle.
∫C
zn dz =∫ 2π
0
(eit)n
ieit dt =∫ 2π
0iei(n+1)t dt
=1
n+1ei(n+1)t
∣∣∣2π
0(n 6=−1)∫
Cz−1 dz =
∫ 2π
0i dt = 2πi
In particular, for n an integer not equal to −1, the integral is zero.Note that the computation is not affected by the fact that the contourcrosses a branch cut. We choose a branch and stay consistent with it.A value at a single point (where the power function is discontinuous)does not affect an integral.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 109: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/109.jpg)
logo1
Contours Contour Integrals Examples
Example. Let n ∈ C(!) Integrate the function f (z) = zn around thepositively oriented unit circle.∫
Czn dz
=∫ 2π
0
(eit)n
ieit dt =∫ 2π
0iei(n+1)t dt
=1
n+1ei(n+1)t
∣∣∣2π
0(n 6=−1)∫
Cz−1 dz =
∫ 2π
0i dt = 2πi
In particular, for n an integer not equal to −1, the integral is zero.Note that the computation is not affected by the fact that the contourcrosses a branch cut. We choose a branch and stay consistent with it.A value at a single point (where the power function is discontinuous)does not affect an integral.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 110: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/110.jpg)
logo1
Contours Contour Integrals Examples
Example. Let n ∈ C(!) Integrate the function f (z) = zn around thepositively oriented unit circle.∫
Czn dz =
∫ 2π
0
(eit)n
ieit dt
=∫ 2π
0iei(n+1)t dt
=1
n+1ei(n+1)t
∣∣∣2π
0(n 6=−1)∫
Cz−1 dz =
∫ 2π
0i dt = 2πi
In particular, for n an integer not equal to −1, the integral is zero.Note that the computation is not affected by the fact that the contourcrosses a branch cut. We choose a branch and stay consistent with it.A value at a single point (where the power function is discontinuous)does not affect an integral.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 111: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/111.jpg)
logo1
Contours Contour Integrals Examples
Example. Let n ∈ C(!) Integrate the function f (z) = zn around thepositively oriented unit circle.∫
Czn dz =
∫ 2π
0
(eit)n
ieit dt =∫ 2π
0iei(n+1)t dt
=1
n+1ei(n+1)t
∣∣∣2π
0(n 6=−1)∫
Cz−1 dz =
∫ 2π
0i dt = 2πi
In particular, for n an integer not equal to −1, the integral is zero.Note that the computation is not affected by the fact that the contourcrosses a branch cut. We choose a branch and stay consistent with it.A value at a single point (where the power function is discontinuous)does not affect an integral.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 112: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/112.jpg)
logo1
Contours Contour Integrals Examples
Example. Let n ∈ C(!) Integrate the function f (z) = zn around thepositively oriented unit circle.∫
Czn dz =
∫ 2π
0
(eit)n
ieit dt =∫ 2π
0iei(n+1)t dt
=1
n+1ei(n+1)t
∣∣∣2π
0
(n 6=−1)∫C
z−1 dz =∫ 2π
0i dt = 2πi
In particular, for n an integer not equal to −1, the integral is zero.Note that the computation is not affected by the fact that the contourcrosses a branch cut. We choose a branch and stay consistent with it.A value at a single point (where the power function is discontinuous)does not affect an integral.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 113: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/113.jpg)
logo1
Contours Contour Integrals Examples
Example. Let n ∈ C(!) Integrate the function f (z) = zn around thepositively oriented unit circle.∫
Czn dz =
∫ 2π
0
(eit)n
ieit dt =∫ 2π
0iei(n+1)t dt
=1
n+1ei(n+1)t
∣∣∣2π
0(n 6=−1)
∫C
z−1 dz =∫ 2π
0i dt = 2πi
In particular, for n an integer not equal to −1, the integral is zero.Note that the computation is not affected by the fact that the contourcrosses a branch cut. We choose a branch and stay consistent with it.A value at a single point (where the power function is discontinuous)does not affect an integral.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 114: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/114.jpg)
logo1
Contours Contour Integrals Examples
Example. Let n ∈ C(!) Integrate the function f (z) = zn around thepositively oriented unit circle.∫
Czn dz =
∫ 2π
0
(eit)n
ieit dt =∫ 2π
0iei(n+1)t dt
=1
n+1ei(n+1)t
∣∣∣2π
0(n 6=−1)∫
Cz−1 dz
=∫ 2π
0i dt = 2πi
In particular, for n an integer not equal to −1, the integral is zero.Note that the computation is not affected by the fact that the contourcrosses a branch cut. We choose a branch and stay consistent with it.A value at a single point (where the power function is discontinuous)does not affect an integral.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 115: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/115.jpg)
logo1
Contours Contour Integrals Examples
Example. Let n ∈ C(!) Integrate the function f (z) = zn around thepositively oriented unit circle.∫
Czn dz =
∫ 2π
0
(eit)n
ieit dt =∫ 2π
0iei(n+1)t dt
=1
n+1ei(n+1)t
∣∣∣2π
0(n 6=−1)∫
Cz−1 dz =
∫ 2π
0i dt
= 2πi
In particular, for n an integer not equal to −1, the integral is zero.Note that the computation is not affected by the fact that the contourcrosses a branch cut. We choose a branch and stay consistent with it.A value at a single point (where the power function is discontinuous)does not affect an integral.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 116: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/116.jpg)
logo1
Contours Contour Integrals Examples
Example. Let n ∈ C(!) Integrate the function f (z) = zn around thepositively oriented unit circle.∫
Czn dz =
∫ 2π
0
(eit)n
ieit dt =∫ 2π
0iei(n+1)t dt
=1
n+1ei(n+1)t
∣∣∣2π
0(n 6=−1)∫
Cz−1 dz =
∫ 2π
0i dt = 2πi
In particular, for n an integer not equal to −1, the integral is zero.Note that the computation is not affected by the fact that the contourcrosses a branch cut. We choose a branch and stay consistent with it.A value at a single point (where the power function is discontinuous)does not affect an integral.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 117: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/117.jpg)
logo1
Contours Contour Integrals Examples
Example. Let n ∈ C(!) Integrate the function f (z) = zn around thepositively oriented unit circle.∫
Czn dz =
∫ 2π
0
(eit)n
ieit dt =∫ 2π
0iei(n+1)t dt
=1
n+1ei(n+1)t
∣∣∣2π
0(n 6=−1)∫
Cz−1 dz =
∫ 2π
0i dt = 2πi
In particular, for n an integer not equal to −1, the integral is zero.
Note that the computation is not affected by the fact that the contourcrosses a branch cut. We choose a branch and stay consistent with it.A value at a single point (where the power function is discontinuous)does not affect an integral.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 118: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/118.jpg)
logo1
Contours Contour Integrals Examples
Example. Let n ∈ C(!) Integrate the function f (z) = zn around thepositively oriented unit circle.∫
Czn dz =
∫ 2π
0
(eit)n
ieit dt =∫ 2π
0iei(n+1)t dt
=1
n+1ei(n+1)t
∣∣∣2π
0(n 6=−1)∫
Cz−1 dz =
∫ 2π
0i dt = 2πi
In particular, for n an integer not equal to −1, the integral is zero.Note that the computation is not affected by the fact that the contourcrosses a branch cut.
We choose a branch and stay consistent with it.A value at a single point (where the power function is discontinuous)does not affect an integral.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 119: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/119.jpg)
logo1
Contours Contour Integrals Examples
Example. Let n ∈ C(!) Integrate the function f (z) = zn around thepositively oriented unit circle.∫
Czn dz =
∫ 2π
0
(eit)n
ieit dt =∫ 2π
0iei(n+1)t dt
=1
n+1ei(n+1)t
∣∣∣2π
0(n 6=−1)∫
Cz−1 dz =
∫ 2π
0i dt = 2πi
In particular, for n an integer not equal to −1, the integral is zero.Note that the computation is not affected by the fact that the contourcrosses a branch cut. We choose a branch and stay consistent with it.
A value at a single point (where the power function is discontinuous)does not affect an integral.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable
![Page 120: Contour Integrals of Functions of a Complex Variable · Introduction 1.Complex functions of a complex variable are usually integrated along parametric curves. 2.The integrals are](https://reader033.vdocument.in/reader033/viewer/2022042810/5f9eeb36d767e527a734266d/html5/thumbnails/120.jpg)
logo1
Contours Contour Integrals Examples
Example. Let n ∈ C(!) Integrate the function f (z) = zn around thepositively oriented unit circle.∫
Czn dz =
∫ 2π
0
(eit)n
ieit dt =∫ 2π
0iei(n+1)t dt
=1
n+1ei(n+1)t
∣∣∣2π
0(n 6=−1)∫
Cz−1 dz =
∫ 2π
0i dt = 2πi
In particular, for n an integer not equal to −1, the integral is zero.Note that the computation is not affected by the fact that the contourcrosses a branch cut. We choose a branch and stay consistent with it.A value at a single point (where the power function is discontinuous)does not affect an integral.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Contour Integrals of Functions of a Complex Variable