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Derivatives Differentiation Formulas
Derivatives of Complex Functions
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 2: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/2.jpg)
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Derivatives Differentiation Formulas
Introduction
1. The idea for the derivative lies in the desire to computeinstantaneous velocities or slopes of tangent lines.
2. In both cases, we want to know what happens when the
denominator in the difference quotientf (x+h)− f (x)
hgoes to
zero.3. Although the visualization is challenging, if not impossible, we
can investigate the same question for functions that map complexnumbers to complex numbers.
4. After all, the algebra and the idea of a limit translate to C.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 3: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/3.jpg)
logo1
Derivatives Differentiation Formulas
Introduction1. The idea for the derivative lies in the desire to compute
instantaneous velocities or slopes of tangent lines.
2. In both cases, we want to know what happens when the
denominator in the difference quotientf (x+h)− f (x)
hgoes to
zero.3. Although the visualization is challenging, if not impossible, we
can investigate the same question for functions that map complexnumbers to complex numbers.
4. After all, the algebra and the idea of a limit translate to C.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 4: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/4.jpg)
logo1
Derivatives Differentiation Formulas
Introduction1. The idea for the derivative lies in the desire to compute
instantaneous velocities or slopes of tangent lines.2. In both cases, we want to know what happens when the
denominator in the difference quotientf (x+h)− f (x)
hgoes to
zero.
3. Although the visualization is challenging, if not impossible, wecan investigate the same question for functions that map complexnumbers to complex numbers.
4. After all, the algebra and the idea of a limit translate to C.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 5: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/5.jpg)
logo1
Derivatives Differentiation Formulas
Introduction1. The idea for the derivative lies in the desire to compute
instantaneous velocities or slopes of tangent lines.2. In both cases, we want to know what happens when the
denominator in the difference quotientf (x+h)− f (x)
hgoes to
zero.3. Although the visualization is challenging
, if not impossible, wecan investigate the same question for functions that map complexnumbers to complex numbers.
4. After all, the algebra and the idea of a limit translate to C.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 6: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/6.jpg)
logo1
Derivatives Differentiation Formulas
Introduction1. The idea for the derivative lies in the desire to compute
instantaneous velocities or slopes of tangent lines.2. In both cases, we want to know what happens when the
denominator in the difference quotientf (x+h)− f (x)
hgoes to
zero.3. Although the visualization is challenging, if not impossible
, wecan investigate the same question for functions that map complexnumbers to complex numbers.
4. After all, the algebra and the idea of a limit translate to C.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 7: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/7.jpg)
logo1
Derivatives Differentiation Formulas
Introduction1. The idea for the derivative lies in the desire to compute
instantaneous velocities or slopes of tangent lines.2. In both cases, we want to know what happens when the
denominator in the difference quotientf (x+h)− f (x)
hgoes to
zero.3. Although the visualization is challenging, if not impossible, we
can investigate the same question for functions that map complexnumbers to complex numbers.
4. After all, the algebra and the idea of a limit translate to C.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 8: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/8.jpg)
logo1
Derivatives Differentiation Formulas
Introduction1. The idea for the derivative lies in the desire to compute
instantaneous velocities or slopes of tangent lines.2. In both cases, we want to know what happens when the
denominator in the difference quotientf (x+h)− f (x)
hgoes to
zero.3. Although the visualization is challenging, if not impossible, we
can investigate the same question for functions that map complexnumbers to complex numbers.
4. After all, the algebra and the idea of a limit translate to C.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 9: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/9.jpg)
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Derivatives Differentiation Formulas
Definition.
Let f be defined in a neighborhood of the point z0. Then fis called differentiable at z0 if and only if the limit
limz→z0
f (z)− f (z0)z− z0
exists.In this case we set
f ′(z0) := limz→z0
f (z)− f (z0)z− z0
and call it the derivative of f at z0. Other notations for the derivative
at z0 aredfdz
(z0) and Df (z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 10: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/10.jpg)
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Derivatives Differentiation Formulas
Definition. Let f be defined in a neighborhood of the point z0.
Then fis called differentiable at z0 if and only if the limit
limz→z0
f (z)− f (z0)z− z0
exists.In this case we set
f ′(z0) := limz→z0
f (z)− f (z0)z− z0
and call it the derivative of f at z0. Other notations for the derivative
at z0 aredfdz
(z0) and Df (z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 11: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/11.jpg)
logo1
Derivatives Differentiation Formulas
Definition. Let f be defined in a neighborhood of the point z0. Then fis called differentiable at z0
if and only if the limit
limz→z0
f (z)− f (z0)z− z0
exists.In this case we set
f ′(z0) := limz→z0
f (z)− f (z0)z− z0
and call it the derivative of f at z0. Other notations for the derivative
at z0 aredfdz
(z0) and Df (z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 12: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/12.jpg)
logo1
Derivatives Differentiation Formulas
Definition. Let f be defined in a neighborhood of the point z0. Then fis called differentiable at z0 if and only if the limit
limz→z0
f (z)− f (z0)z− z0
exists.
In this case we set
f ′(z0) := limz→z0
f (z)− f (z0)z− z0
and call it the derivative of f at z0. Other notations for the derivative
at z0 aredfdz
(z0) and Df (z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 13: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/13.jpg)
logo1
Derivatives Differentiation Formulas
Definition. Let f be defined in a neighborhood of the point z0. Then fis called differentiable at z0 if and only if the limit
limz→z0
f (z)− f (z0)z− z0
exists.In this case we set
f ′(z0) := limz→z0
f (z)− f (z0)z− z0
and call it the derivative of f at z0. Other notations for the derivative
at z0 aredfdz
(z0) and Df (z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 14: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/14.jpg)
logo1
Derivatives Differentiation Formulas
Definition. Let f be defined in a neighborhood of the point z0. Then fis called differentiable at z0 if and only if the limit
limz→z0
f (z)− f (z0)z− z0
exists.In this case we set
f ′(z0) := limz→z0
f (z)− f (z0)z− z0
and call it the derivative of f at z0.
Other notations for the derivative
at z0 aredfdz
(z0) and Df (z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 15: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/15.jpg)
logo1
Derivatives Differentiation Formulas
Definition. Let f be defined in a neighborhood of the point z0. Then fis called differentiable at z0 if and only if the limit
limz→z0
f (z)− f (z0)z− z0
exists.In this case we set
f ′(z0) := limz→z0
f (z)− f (z0)z− z0
and call it the derivative of f at z0. Other notations for the derivative
at z0 aredfdz
(z0)
and Df (z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 16: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/16.jpg)
logo1
Derivatives Differentiation Formulas
Definition. Let f be defined in a neighborhood of the point z0. Then fis called differentiable at z0 if and only if the limit
limz→z0
f (z)− f (z0)z− z0
exists.In this case we set
f ′(z0) := limz→z0
f (z)− f (z0)z− z0
and call it the derivative of f at z0. Other notations for the derivative
at z0 aredfdz
(z0) and Df (z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 17: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/17.jpg)
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Derivatives Differentiation Formulas
The function f with an open domain is called differentiable
(orholomorphic or analytic) if and only if it is differentiable at every z0in its domain. The function f is called entire if and only if it isdifferentiable at every z ∈ C. If f is not differentiable at z0, but forevery neighborhood of z0 there is a point so that f is analytic in aneighborhood around that point, then z0 is called a singularity.
It can be proved that f is differentiable if and only if
lim∆z→0
f (z0 +∆z)− f (z0)∆z
exists. In this case, the above limit is the
derivative.
With ∆w := f (z0 +∆z)− f (z0) we also writedwdz
= lim∆z→0
∆w∆z
.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 18: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/18.jpg)
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Derivatives Differentiation Formulas
The function f with an open domain is called differentiable (orholomorphic or analytic)
if and only if it is differentiable at every z0in its domain. The function f is called entire if and only if it isdifferentiable at every z ∈ C. If f is not differentiable at z0, but forevery neighborhood of z0 there is a point so that f is analytic in aneighborhood around that point, then z0 is called a singularity.
It can be proved that f is differentiable if and only if
lim∆z→0
f (z0 +∆z)− f (z0)∆z
exists. In this case, the above limit is the
derivative.
With ∆w := f (z0 +∆z)− f (z0) we also writedwdz
= lim∆z→0
∆w∆z
.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 19: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/19.jpg)
logo1
Derivatives Differentiation Formulas
The function f with an open domain is called differentiable (orholomorphic or analytic) if and only if it is differentiable at every z0in its domain.
The function f is called entire if and only if it isdifferentiable at every z ∈ C. If f is not differentiable at z0, but forevery neighborhood of z0 there is a point so that f is analytic in aneighborhood around that point, then z0 is called a singularity.
It can be proved that f is differentiable if and only if
lim∆z→0
f (z0 +∆z)− f (z0)∆z
exists. In this case, the above limit is the
derivative.
With ∆w := f (z0 +∆z)− f (z0) we also writedwdz
= lim∆z→0
∆w∆z
.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 20: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/20.jpg)
logo1
Derivatives Differentiation Formulas
The function f with an open domain is called differentiable (orholomorphic or analytic) if and only if it is differentiable at every z0in its domain. The function f is called entire if and only if it isdifferentiable at every z ∈ C.
If f is not differentiable at z0, but forevery neighborhood of z0 there is a point so that f is analytic in aneighborhood around that point, then z0 is called a singularity.
It can be proved that f is differentiable if and only if
lim∆z→0
f (z0 +∆z)− f (z0)∆z
exists. In this case, the above limit is the
derivative.
With ∆w := f (z0 +∆z)− f (z0) we also writedwdz
= lim∆z→0
∆w∆z
.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 21: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/21.jpg)
logo1
Derivatives Differentiation Formulas
The function f with an open domain is called differentiable (orholomorphic or analytic) if and only if it is differentiable at every z0in its domain. The function f is called entire if and only if it isdifferentiable at every z ∈ C. If f is not differentiable at z0, but forevery neighborhood of z0 there is a point so that f is analytic in aneighborhood around that point, then z0 is called a singularity.
It can be proved that f is differentiable if and only if
lim∆z→0
f (z0 +∆z)− f (z0)∆z
exists. In this case, the above limit is the
derivative.
With ∆w := f (z0 +∆z)− f (z0) we also writedwdz
= lim∆z→0
∆w∆z
.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 22: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/22.jpg)
logo1
Derivatives Differentiation Formulas
The function f with an open domain is called differentiable (orholomorphic or analytic) if and only if it is differentiable at every z0in its domain. The function f is called entire if and only if it isdifferentiable at every z ∈ C. If f is not differentiable at z0, but forevery neighborhood of z0 there is a point so that f is analytic in aneighborhood around that point, then z0 is called a singularity.
It can be proved that f is differentiable if and only if
lim∆z→0
f (z0 +∆z)− f (z0)∆z
exists.
In this case, the above limit is the
derivative.
With ∆w := f (z0 +∆z)− f (z0) we also writedwdz
= lim∆z→0
∆w∆z
.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 23: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/23.jpg)
logo1
Derivatives Differentiation Formulas
The function f with an open domain is called differentiable (orholomorphic or analytic) if and only if it is differentiable at every z0in its domain. The function f is called entire if and only if it isdifferentiable at every z ∈ C. If f is not differentiable at z0, but forevery neighborhood of z0 there is a point so that f is analytic in aneighborhood around that point, then z0 is called a singularity.
It can be proved that f is differentiable if and only if
lim∆z→0
f (z0 +∆z)− f (z0)∆z
exists. In this case, the above limit is the
derivative.
With ∆w := f (z0 +∆z)− f (z0) we also writedwdz
= lim∆z→0
∆w∆z
.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 24: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/24.jpg)
logo1
Derivatives Differentiation Formulas
The function f with an open domain is called differentiable (orholomorphic or analytic) if and only if it is differentiable at every z0in its domain. The function f is called entire if and only if it isdifferentiable at every z ∈ C. If f is not differentiable at z0, but forevery neighborhood of z0 there is a point so that f is analytic in aneighborhood around that point, then z0 is called a singularity.
It can be proved that f is differentiable if and only if
lim∆z→0
f (z0 +∆z)− f (z0)∆z
exists. In this case, the above limit is the
derivative.
With ∆w := f (z0 +∆z)− f (z0) we also writedwdz
= lim∆z→0
∆w∆z
.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 25: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/25.jpg)
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Derivatives Differentiation Formulas
Example.
Compute the derivative of f (z) = z3.
dz3
dz= lim
∆z→0
(z+∆z)3− z3
∆z
= lim∆z→0
z3 +3z2∆z+3z∆z2 +∆z3− z3
∆z
= lim∆z→0
3z2∆z+3z∆z2 +∆z3
∆z
= lim∆z→0
∆z(3z2 +3z∆z+∆z2
)∆z
= lim∆z→0
3z2 +3z∆z+∆z2
= 3z2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 26: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/26.jpg)
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Derivatives Differentiation Formulas
Example. Compute the derivative of f (z) = z3.
dz3
dz= lim
∆z→0
(z+∆z)3− z3
∆z
= lim∆z→0
z3 +3z2∆z+3z∆z2 +∆z3− z3
∆z
= lim∆z→0
3z2∆z+3z∆z2 +∆z3
∆z
= lim∆z→0
∆z(3z2 +3z∆z+∆z2
)∆z
= lim∆z→0
3z2 +3z∆z+∆z2
= 3z2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 27: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/27.jpg)
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Derivatives Differentiation Formulas
Example. Compute the derivative of f (z) = z3.
dz3
dz
= lim∆z→0
(z+∆z)3− z3
∆z
= lim∆z→0
z3 +3z2∆z+3z∆z2 +∆z3− z3
∆z
= lim∆z→0
3z2∆z+3z∆z2 +∆z3
∆z
= lim∆z→0
∆z(3z2 +3z∆z+∆z2
)∆z
= lim∆z→0
3z2 +3z∆z+∆z2
= 3z2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 28: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/28.jpg)
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Derivatives Differentiation Formulas
Example. Compute the derivative of f (z) = z3.
dz3
dz= lim
∆z→0
(z+∆z)3− z3
∆z
= lim∆z→0
z3 +3z2∆z+3z∆z2 +∆z3− z3
∆z
= lim∆z→0
3z2∆z+3z∆z2 +∆z3
∆z
= lim∆z→0
∆z(3z2 +3z∆z+∆z2
)∆z
= lim∆z→0
3z2 +3z∆z+∆z2
= 3z2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 29: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/29.jpg)
logo1
Derivatives Differentiation Formulas
Example. Compute the derivative of f (z) = z3.
dz3
dz= lim
∆z→0
(z+∆z)3− z3
∆z
= lim∆z→0
z3 +3z2∆z+3z∆z2 +∆z3− z3
∆z
= lim∆z→0
3z2∆z+3z∆z2 +∆z3
∆z
= lim∆z→0
∆z(3z2 +3z∆z+∆z2
)∆z
= lim∆z→0
3z2 +3z∆z+∆z2
= 3z2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 30: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/30.jpg)
logo1
Derivatives Differentiation Formulas
Example. Compute the derivative of f (z) = z3.
dz3
dz= lim
∆z→0
(z+∆z)3− z3
∆z
= lim∆z→0
z3 +3z2∆z+3z∆z2 +∆z3− z3
∆z
= lim∆z→0
3z2∆z+3z∆z2 +∆z3
∆z
= lim∆z→0
∆z(3z2 +3z∆z+∆z2
)∆z
= lim∆z→0
3z2 +3z∆z+∆z2
= 3z2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 31: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/31.jpg)
logo1
Derivatives Differentiation Formulas
Example. Compute the derivative of f (z) = z3.
dz3
dz= lim
∆z→0
(z+∆z)3− z3
∆z
= lim∆z→0
z3 +3z2∆z+3z∆z2 +∆z3− z3
∆z
= lim∆z→0
3z2∆z+3z∆z2 +∆z3
∆z
= lim∆z→0
∆z(3z2 +3z∆z+∆z2
)∆z
= lim∆z→0
3z2 +3z∆z+∆z2
= 3z2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 32: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/32.jpg)
logo1
Derivatives Differentiation Formulas
Example. Compute the derivative of f (z) = z3.
dz3
dz= lim
∆z→0
(z+∆z)3− z3
∆z
= lim∆z→0
z3 +3z2∆z+3z∆z2 +∆z3− z3
∆z
= lim∆z→0
3z2∆z+3z∆z2 +∆z3
∆z
= lim∆z→0
∆z(3z2 +3z∆z+∆z2
)∆z
= lim∆z→0
3z2 +3z∆z+∆z2
= 3z2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 33: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/33.jpg)
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Derivatives Differentiation Formulas
Example. Compute the derivative of f (z) = z3.
dz3
dz= lim
∆z→0
(z+∆z)3− z3
∆z
= lim∆z→0
z3 +3z2∆z+3z∆z2 +∆z3− z3
∆z
= lim∆z→0
3z2∆z+3z∆z2 +∆z3
∆z
= lim∆z→0
∆z(3z2 +3z∆z+∆z2
)∆z
= lim∆z→0
3z2 +3z∆z+∆z2
= 3z2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 34: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/34.jpg)
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Derivatives Differentiation Formulas
Example.
Show that f (z) = z is not differentiable at any z0 ∈ C.
lim∆z→0
z0 +∆z− z0
∆z= lim
∆z→0
z0 +∆z− z0
∆z
= lim∆z→0
∆z∆z
... which does not exist.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 35: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/35.jpg)
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Derivatives Differentiation Formulas
Example. Show that f (z) = z is not differentiable at any z0 ∈ C.
lim∆z→0
z0 +∆z− z0
∆z= lim
∆z→0
z0 +∆z− z0
∆z
= lim∆z→0
∆z∆z
... which does not exist.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 36: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/36.jpg)
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Derivatives Differentiation Formulas
Example. Show that f (z) = z is not differentiable at any z0 ∈ C.
lim∆z→0
z0 +∆z− z0
∆z
= lim∆z→0
z0 +∆z− z0
∆z
= lim∆z→0
∆z∆z
... which does not exist.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 37: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/37.jpg)
logo1
Derivatives Differentiation Formulas
Example. Show that f (z) = z is not differentiable at any z0 ∈ C.
lim∆z→0
z0 +∆z− z0
∆z= lim
∆z→0
z0 +∆z− z0
∆z
= lim∆z→0
∆z∆z
... which does not exist.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 38: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/38.jpg)
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Derivatives Differentiation Formulas
Example. Show that f (z) = z is not differentiable at any z0 ∈ C.
lim∆z→0
z0 +∆z− z0
∆z= lim
∆z→0
z0 +∆z− z0
∆z
= lim∆z→0
∆z∆z
... which does not exist.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 39: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/39.jpg)
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Derivatives Differentiation Formulas
Example. Show that f (z) = z is not differentiable at any z0 ∈ C.
lim∆z→0
z0 +∆z− z0
∆z= lim
∆z→0
z0 +∆z− z0
∆z
= lim∆z→0
∆z∆z
... which does not exist.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 40: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/40.jpg)
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Derivatives Differentiation Formulas
Theorem.
If the complex function f is differentiable at z0, then f iscontinuous at z0. Hence, every differentiable function is continuous.However, not every continuous function is differentiable.
Proof. For continuity at z0 note that
0 = limz→z0
(z− z0) limz→z0
f (z)− f (z0)z− z0
= limz→z0
(z− z0)f (z)− f (z0)
z− z0
= limz→z0
f (z)− f (z0).
For the last statement, consider f (z) = z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 41: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/41.jpg)
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Derivatives Differentiation Formulas
Theorem. If the complex function f is differentiable at z0, then f iscontinuous at z0.
Hence, every differentiable function is continuous.However, not every continuous function is differentiable.
Proof. For continuity at z0 note that
0 = limz→z0
(z− z0) limz→z0
f (z)− f (z0)z− z0
= limz→z0
(z− z0)f (z)− f (z0)
z− z0
= limz→z0
f (z)− f (z0).
For the last statement, consider f (z) = z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 42: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/42.jpg)
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Derivatives Differentiation Formulas
Theorem. If the complex function f is differentiable at z0, then f iscontinuous at z0. Hence, every differentiable function is continuous.
However, not every continuous function is differentiable.
Proof. For continuity at z0 note that
0 = limz→z0
(z− z0) limz→z0
f (z)− f (z0)z− z0
= limz→z0
(z− z0)f (z)− f (z0)
z− z0
= limz→z0
f (z)− f (z0).
For the last statement, consider f (z) = z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 43: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/43.jpg)
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Derivatives Differentiation Formulas
Theorem. If the complex function f is differentiable at z0, then f iscontinuous at z0. Hence, every differentiable function is continuous.However, not every continuous function is differentiable.
Proof. For continuity at z0 note that
0 = limz→z0
(z− z0) limz→z0
f (z)− f (z0)z− z0
= limz→z0
(z− z0)f (z)− f (z0)
z− z0
= limz→z0
f (z)− f (z0).
For the last statement, consider f (z) = z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 44: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/44.jpg)
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Derivatives Differentiation Formulas
Theorem. If the complex function f is differentiable at z0, then f iscontinuous at z0. Hence, every differentiable function is continuous.However, not every continuous function is differentiable.
Proof.
For continuity at z0 note that
0 = limz→z0
(z− z0) limz→z0
f (z)− f (z0)z− z0
= limz→z0
(z− z0)f (z)− f (z0)
z− z0
= limz→z0
f (z)− f (z0).
For the last statement, consider f (z) = z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 45: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/45.jpg)
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Derivatives Differentiation Formulas
Theorem. If the complex function f is differentiable at z0, then f iscontinuous at z0. Hence, every differentiable function is continuous.However, not every continuous function is differentiable.
Proof. For continuity at z0 note that
0
= limz→z0
(z− z0) limz→z0
f (z)− f (z0)z− z0
= limz→z0
(z− z0)f (z)− f (z0)
z− z0
= limz→z0
f (z)− f (z0).
For the last statement, consider f (z) = z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 46: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/46.jpg)
logo1
Derivatives Differentiation Formulas
Theorem. If the complex function f is differentiable at z0, then f iscontinuous at z0. Hence, every differentiable function is continuous.However, not every continuous function is differentiable.
Proof. For continuity at z0 note that
0 = limz→z0
(z− z0) limz→z0
f (z)− f (z0)z− z0
= limz→z0
(z− z0)f (z)− f (z0)
z− z0
= limz→z0
f (z)− f (z0).
For the last statement, consider f (z) = z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 47: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/47.jpg)
logo1
Derivatives Differentiation Formulas
Theorem. If the complex function f is differentiable at z0, then f iscontinuous at z0. Hence, every differentiable function is continuous.However, not every continuous function is differentiable.
Proof. For continuity at z0 note that
0 = limz→z0
(z− z0) limz→z0
f (z)− f (z0)z− z0
= limz→z0
(z− z0)f (z)− f (z0)
z− z0
= limz→z0
f (z)− f (z0).
For the last statement, consider f (z) = z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 48: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/48.jpg)
logo1
Derivatives Differentiation Formulas
Theorem. If the complex function f is differentiable at z0, then f iscontinuous at z0. Hence, every differentiable function is continuous.However, not every continuous function is differentiable.
Proof. For continuity at z0 note that
0 = limz→z0
(z− z0) limz→z0
f (z)− f (z0)z− z0
= limz→z0
(z− z0)f (z)− f (z0)
z− z0
= limz→z0
f (z)− f (z0).
For the last statement, consider f (z) = z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 49: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/49.jpg)
logo1
Derivatives Differentiation Formulas
Theorem. If the complex function f is differentiable at z0, then f iscontinuous at z0. Hence, every differentiable function is continuous.However, not every continuous function is differentiable.
Proof. For continuity at z0 note that
0 = limz→z0
(z− z0) limz→z0
f (z)− f (z0)z− z0
= limz→z0
(z− z0)f (z)− f (z0)
z− z0
= limz→z0
f (z)− f (z0).
For the last statement, consider f (z) = z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 50: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/50.jpg)
logo1
Derivatives Differentiation Formulas
Theorem. If the complex function f is differentiable at z0, then f iscontinuous at z0. Hence, every differentiable function is continuous.However, not every continuous function is differentiable.
Proof. For continuity at z0 note that
0 = limz→z0
(z− z0) limz→z0
f (z)− f (z0)z− z0
= limz→z0
(z− z0)f (z)− f (z0)
z− z0
= limz→z0
f (z)− f (z0).
For the last statement, consider f (z) = z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 51: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/51.jpg)
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Derivatives Differentiation Formulas
Theorem.
Let f and g be differentiable at z0 and let c ∈ C. Then thefunctions f +g, f −g and cf are all differentiable at z0 and
(f +g)′(z0) = f ′(z0)+g′(z0),(f −g)′(z0) = f ′(z0)−g′(z0),
(cf )′(z0) = cf ′(z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 52: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/52.jpg)
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Derivatives Differentiation Formulas
Theorem. Let f and g be differentiable at z0 and let c ∈ C.
Then thefunctions f +g, f −g and cf are all differentiable at z0 and
(f +g)′(z0) = f ′(z0)+g′(z0),(f −g)′(z0) = f ′(z0)−g′(z0),
(cf )′(z0) = cf ′(z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 53: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/53.jpg)
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Derivatives Differentiation Formulas
Theorem. Let f and g be differentiable at z0 and let c ∈ C. Then thefunctions f +g, f −g and cf are all differentiable at z0
and
(f +g)′(z0) = f ′(z0)+g′(z0),(f −g)′(z0) = f ′(z0)−g′(z0),
(cf )′(z0) = cf ′(z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 54: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/54.jpg)
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Derivatives Differentiation Formulas
Theorem. Let f and g be differentiable at z0 and let c ∈ C. Then thefunctions f +g, f −g and cf are all differentiable at z0 and
(f +g)′(z0) = f ′(z0)+g′(z0),
(f −g)′(z0) = f ′(z0)−g′(z0),(cf )′(z0) = cf ′(z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 55: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/55.jpg)
logo1
Derivatives Differentiation Formulas
Theorem. Let f and g be differentiable at z0 and let c ∈ C. Then thefunctions f +g, f −g and cf are all differentiable at z0 and
(f +g)′(z0) = f ′(z0)+g′(z0),(f −g)′(z0) = f ′(z0)−g′(z0),
(cf )′(z0) = cf ′(z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 56: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/56.jpg)
logo1
Derivatives Differentiation Formulas
Theorem. Let f and g be differentiable at z0 and let c ∈ C. Then thefunctions f +g, f −g and cf are all differentiable at z0 and
(f +g)′(z0) = f ′(z0)+g′(z0),(f −g)′(z0) = f ′(z0)−g′(z0),
(cf )′(z0) = cf ′(z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 57: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/57.jpg)
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Derivatives Differentiation Formulas
Proof (addition only).
(f +g)′(z0)
= limz→z0
(f +g)(z)− (f +g)(z0)z− z0
= limz→z0
f (z)+g(z)− f (z0)−g(z0)z− z0
= limz→z0
f (z)− f (z0)+g(z)−g(z0)z− z0
= limz→z0
f (z)− f (z0)z− z0
+ limz→z0
g(z)−g(z0)z− z0
= f ′(z0)+g′(z0)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 58: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/58.jpg)
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Derivatives Differentiation Formulas
Proof (addition only).
(f +g)′(z0)
= limz→z0
(f +g)(z)− (f +g)(z0)z− z0
= limz→z0
f (z)+g(z)− f (z0)−g(z0)z− z0
= limz→z0
f (z)− f (z0)+g(z)−g(z0)z− z0
= limz→z0
f (z)− f (z0)z− z0
+ limz→z0
g(z)−g(z0)z− z0
= f ′(z0)+g′(z0)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 59: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/59.jpg)
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Derivatives Differentiation Formulas
Proof (addition only).
(f +g)′(z0)
= limz→z0
(f +g)(z)− (f +g)(z0)z− z0
= limz→z0
f (z)+g(z)− f (z0)−g(z0)z− z0
= limz→z0
f (z)− f (z0)+g(z)−g(z0)z− z0
= limz→z0
f (z)− f (z0)z− z0
+ limz→z0
g(z)−g(z0)z− z0
= f ′(z0)+g′(z0)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 60: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/60.jpg)
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Derivatives Differentiation Formulas
Proof (addition only).
(f +g)′(z0)
= limz→z0
(f +g)(z)− (f +g)(z0)z− z0
= limz→z0
f (z)+g(z)− f (z0)−g(z0)z− z0
= limz→z0
f (z)− f (z0)+g(z)−g(z0)z− z0
= limz→z0
f (z)− f (z0)z− z0
+ limz→z0
g(z)−g(z0)z− z0
= f ′(z0)+g′(z0)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 61: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/61.jpg)
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Derivatives Differentiation Formulas
Proof (addition only).
(f +g)′(z0)
= limz→z0
(f +g)(z)− (f +g)(z0)z− z0
= limz→z0
f (z)+g(z)− f (z0)−g(z0)z− z0
= limz→z0
f (z)− f (z0)+g(z)−g(z0)z− z0
= limz→z0
f (z)− f (z0)z− z0
+ limz→z0
g(z)−g(z0)z− z0
= f ′(z0)+g′(z0)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 62: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/62.jpg)
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Derivatives Differentiation Formulas
Proof (addition only).
(f +g)′(z0)
= limz→z0
(f +g)(z)− (f +g)(z0)z− z0
= limz→z0
f (z)+g(z)− f (z0)−g(z0)z− z0
= limz→z0
f (z)− f (z0)+g(z)−g(z0)z− z0
= limz→z0
f (z)− f (z0)z− z0
+ limz→z0
g(z)−g(z0)z− z0
= f ′(z0)+g′(z0)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 63: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/63.jpg)
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Derivatives Differentiation Formulas
Proof (addition only).
(f +g)′(z0)
= limz→z0
(f +g)(z)− (f +g)(z0)z− z0
= limz→z0
f (z)+g(z)− f (z0)−g(z0)z− z0
= limz→z0
f (z)− f (z0)+g(z)−g(z0)z− z0
= limz→z0
f (z)− f (z0)z− z0
+ limz→z0
g(z)−g(z0)z− z0
= f ′(z0)+g′(z0)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 64: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/64.jpg)
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Derivatives Differentiation Formulas
Proof (addition only).
(f +g)′(z0)
= limz→z0
(f +g)(z)− (f +g)(z0)z− z0
= limz→z0
f (z)+g(z)− f (z0)−g(z0)z− z0
= limz→z0
f (z)− f (z0)+g(z)−g(z0)z− z0
= limz→z0
f (z)− f (z0)z− z0
+ limz→z0
g(z)−g(z0)z− z0
= f ′(z0)+g′(z0)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 65: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/65.jpg)
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Derivatives Differentiation Formulas
Theorem.
Product and Quotient Rule. Let f and g be differentiableat z0. Then fg is differentiable at x with
(fg)′(z0) = f ′(z0)g(z0)+g′(z0)f (z0).
Moreover, if g(z0) 6= 0, then the quotientfg
is differentiable at z0 with(fg
)′(z0) =
f ′(z0)g(z0)−g′(z0)f (z0)(g(z0)
)2 .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 66: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/66.jpg)
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Derivatives Differentiation Formulas
Theorem. Product and Quotient Rule. Let f and g be differentiableat z0.
Then fg is differentiable at x with
(fg)′(z0) = f ′(z0)g(z0)+g′(z0)f (z0).
Moreover, if g(z0) 6= 0, then the quotientfg
is differentiable at z0 with(fg
)′(z0) =
f ′(z0)g(z0)−g′(z0)f (z0)(g(z0)
)2 .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 67: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/67.jpg)
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Derivatives Differentiation Formulas
Theorem. Product and Quotient Rule. Let f and g be differentiableat z0. Then fg is differentiable at x
with
(fg)′(z0) = f ′(z0)g(z0)+g′(z0)f (z0).
Moreover, if g(z0) 6= 0, then the quotientfg
is differentiable at z0 with(fg
)′(z0) =
f ′(z0)g(z0)−g′(z0)f (z0)(g(z0)
)2 .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 68: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/68.jpg)
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Derivatives Differentiation Formulas
Theorem. Product and Quotient Rule. Let f and g be differentiableat z0. Then fg is differentiable at x with
(fg)′(z0) = f ′(z0)g(z0)+g′(z0)f (z0).
Moreover, if g(z0) 6= 0, then the quotientfg
is differentiable at z0 with(fg
)′(z0) =
f ′(z0)g(z0)−g′(z0)f (z0)(g(z0)
)2 .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 69: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/69.jpg)
logo1
Derivatives Differentiation Formulas
Theorem. Product and Quotient Rule. Let f and g be differentiableat z0. Then fg is differentiable at x with
(fg)′(z0) = f ′(z0)g(z0)+g′(z0)f (z0).
Moreover, if g(z0) 6= 0, then the quotientfg
is differentiable at z0
with(fg
)′(z0) =
f ′(z0)g(z0)−g′(z0)f (z0)(g(z0)
)2 .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 70: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/70.jpg)
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Derivatives Differentiation Formulas
Theorem. Product and Quotient Rule. Let f and g be differentiableat z0. Then fg is differentiable at x with
(fg)′(z0) = f ′(z0)g(z0)+g′(z0)f (z0).
Moreover, if g(z0) 6= 0, then the quotientfg
is differentiable at z0 with(fg
)′(z0) =
f ′(z0)g(z0)−g′(z0)f (z0)(g(z0)
)2 .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 71: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/71.jpg)
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Derivatives Differentiation Formulas
Proof (quotient rule only).
limz→z0
fg(z)− f
g(z0)
z− z0
= limz→z0
f (z)g(z) −
f (z0)g(z0)
z− z0= lim
z→z0
f (z)g(z0)−f (z0)g(z)g(z)g(z0)
z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z)z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z0)+ f (z0)g(z0)− f (z0)g(z)z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z0)−(f (z0)g(z)− f (z0)g(z0)
)z− z0
= limz→z0
1g(z)g(z0)
(f (z)g(z0)− f (z0)g(z0)
z− z0− f (z0)g(z)− f (z0)g(z0)
z− z0
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 72: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/72.jpg)
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Derivatives Differentiation Formulas
Proof (quotient rule only).
limz→z0
fg(z)− f
g(z0)
z− z0
= limz→z0
f (z)g(z) −
f (z0)g(z0)
z− z0= lim
z→z0
f (z)g(z0)−f (z0)g(z)g(z)g(z0)
z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z)z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z0)+ f (z0)g(z0)− f (z0)g(z)z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z0)−(f (z0)g(z)− f (z0)g(z0)
)z− z0
= limz→z0
1g(z)g(z0)
(f (z)g(z0)− f (z0)g(z0)
z− z0− f (z0)g(z)− f (z0)g(z0)
z− z0
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 73: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/73.jpg)
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Derivatives Differentiation Formulas
Proof (quotient rule only).
limz→z0
fg(z)− f
g(z0)
z− z0
= limz→z0
f (z)g(z) −
f (z0)g(z0)
z− z0
= limz→z0
f (z)g(z0)−f (z0)g(z)g(z)g(z0)
z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z)z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z0)+ f (z0)g(z0)− f (z0)g(z)z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z0)−(f (z0)g(z)− f (z0)g(z0)
)z− z0
= limz→z0
1g(z)g(z0)
(f (z)g(z0)− f (z0)g(z0)
z− z0− f (z0)g(z)− f (z0)g(z0)
z− z0
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 74: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/74.jpg)
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Derivatives Differentiation Formulas
Proof (quotient rule only).
limz→z0
fg(z)− f
g(z0)
z− z0
= limz→z0
f (z)g(z) −
f (z0)g(z0)
z− z0= lim
z→z0
f (z)g(z0)−f (z0)g(z)g(z)g(z0)
z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z)z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z0)+ f (z0)g(z0)− f (z0)g(z)z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z0)−(f (z0)g(z)− f (z0)g(z0)
)z− z0
= limz→z0
1g(z)g(z0)
(f (z)g(z0)− f (z0)g(z0)
z− z0− f (z0)g(z)− f (z0)g(z0)
z− z0
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 75: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/75.jpg)
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Derivatives Differentiation Formulas
Proof (quotient rule only).
limz→z0
fg(z)− f
g(z0)
z− z0
= limz→z0
f (z)g(z) −
f (z0)g(z0)
z− z0= lim
z→z0
f (z)g(z0)−f (z0)g(z)g(z)g(z0)
z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z)z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z0)+ f (z0)g(z0)− f (z0)g(z)z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z0)−(f (z0)g(z)− f (z0)g(z0)
)z− z0
= limz→z0
1g(z)g(z0)
(f (z)g(z0)− f (z0)g(z0)
z− z0− f (z0)g(z)− f (z0)g(z0)
z− z0
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 76: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/76.jpg)
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Derivatives Differentiation Formulas
Proof (quotient rule only).
limz→z0
fg(z)− f
g(z0)
z− z0
= limz→z0
f (z)g(z) −
f (z0)g(z0)
z− z0= lim
z→z0
f (z)g(z0)−f (z0)g(z)g(z)g(z0)
z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z)z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z0)+ f (z0)g(z0)− f (z0)g(z)z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z0)−(f (z0)g(z)− f (z0)g(z0)
)z− z0
= limz→z0
1g(z)g(z0)
(f (z)g(z0)− f (z0)g(z0)
z− z0− f (z0)g(z)− f (z0)g(z0)
z− z0
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 77: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/77.jpg)
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Derivatives Differentiation Formulas
Proof (quotient rule only).
limz→z0
fg(z)− f
g(z0)
z− z0
= limz→z0
f (z)g(z) −
f (z0)g(z0)
z− z0= lim
z→z0
f (z)g(z0)−f (z0)g(z)g(z)g(z0)
z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z)z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z0)+ f (z0)g(z0)− f (z0)g(z)z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z0)−(f (z0)g(z)− f (z0)g(z0)
)z− z0
= limz→z0
1g(z)g(z0)
(f (z)g(z0)− f (z0)g(z0)
z− z0− f (z0)g(z)− f (z0)g(z0)
z− z0
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 78: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/78.jpg)
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Derivatives Differentiation Formulas
Proof (quotient rule only).
limz→z0
fg(z)− f
g(z0)
z− z0
= limz→z0
f (z)g(z) −
f (z0)g(z0)
z− z0= lim
z→z0
f (z)g(z0)−f (z0)g(z)g(z)g(z0)
z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z)z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z0)+ f (z0)g(z0)− f (z0)g(z)z− z0
= limz→z0
1g(z)g(z0)
f (z)g(z0)− f (z0)g(z0)−(f (z0)g(z)− f (z0)g(z0)
)z− z0
= limz→z0
1g(z)g(z0)
(f (z)g(z0)− f (z0)g(z0)
z− z0− f (z0)g(z)− f (z0)g(z0)
z− z0
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 79: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/79.jpg)
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Derivatives Differentiation Formulas
Proof (quotient rule cont.).
limz→z0
1g(z)g(z0)
(f (z)g(z0)− f (z0)g(z0)
z− z0− f (z0)g(z)− f (z0)g(z0)
z− z0
)= lim
z→z0
1g(z)g(z0)
(f (z)− f (z0)
z− z0g(z0)−
g(z)−g(z0)z− z0
f (z0))
=1(
g(z0))2
(f ′(z0)g(z0)−g′(z0)f (z0)
)=
f ′(z0)g(z0)−g′(z0)f (z0)(g(z0)
)2 .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 80: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/80.jpg)
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Derivatives Differentiation Formulas
Proof (quotient rule cont.).
limz→z0
1g(z)g(z0)
(f (z)g(z0)− f (z0)g(z0)
z− z0− f (z0)g(z)− f (z0)g(z0)
z− z0
)
= limz→z0
1g(z)g(z0)
(f (z)− f (z0)
z− z0g(z0)−
g(z)−g(z0)z− z0
f (z0))
=1(
g(z0))2
(f ′(z0)g(z0)−g′(z0)f (z0)
)=
f ′(z0)g(z0)−g′(z0)f (z0)(g(z0)
)2 .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 81: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/81.jpg)
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Derivatives Differentiation Formulas
Proof (quotient rule cont.).
limz→z0
1g(z)g(z0)
(f (z)g(z0)− f (z0)g(z0)
z− z0− f (z0)g(z)− f (z0)g(z0)
z− z0
)= lim
z→z0
1g(z)g(z0)
(f (z)− f (z0)
z− z0g(z0)−
g(z)−g(z0)z− z0
f (z0))
=1(
g(z0))2
(f ′(z0)g(z0)−g′(z0)f (z0)
)=
f ′(z0)g(z0)−g′(z0)f (z0)(g(z0)
)2 .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 82: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/82.jpg)
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Derivatives Differentiation Formulas
Proof (quotient rule cont.).
limz→z0
1g(z)g(z0)
(f (z)g(z0)− f (z0)g(z0)
z− z0− f (z0)g(z)− f (z0)g(z0)
z− z0
)= lim
z→z0
1g(z)g(z0)
(f (z)− f (z0)
z− z0g(z0)−
g(z)−g(z0)z− z0
f (z0))
=1(
g(z0))2
(f ′(z0)g(z0)−g′(z0)f (z0)
)
=f ′(z0)g(z0)−g′(z0)f (z0)(
g(z0))2 .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 83: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/83.jpg)
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Derivatives Differentiation Formulas
Proof (quotient rule cont.).
limz→z0
1g(z)g(z0)
(f (z)g(z0)− f (z0)g(z0)
z− z0− f (z0)g(z)− f (z0)g(z0)
z− z0
)= lim
z→z0
1g(z)g(z0)
(f (z)− f (z0)
z− z0g(z0)−
g(z)−g(z0)z− z0
f (z0))
=1(
g(z0))2
(f ′(z0)g(z0)−g′(z0)f (z0)
)=
f ′(z0)g(z0)−g′(z0)f (z0)(g(z0)
)2 .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 84: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/84.jpg)
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Derivatives Differentiation Formulas
Proof (quotient rule cont.).
limz→z0
1g(z)g(z0)
(f (z)g(z0)− f (z0)g(z0)
z− z0− f (z0)g(z)− f (z0)g(z0)
z− z0
)= lim
z→z0
1g(z)g(z0)
(f (z)− f (z0)
z− z0g(z0)−
g(z)−g(z0)z− z0
f (z0))
=1(
g(z0))2
(f ′(z0)g(z0)−g′(z0)f (z0)
)=
f ′(z0)g(z0)−g′(z0)f (z0)(g(z0)
)2 .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 85: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/85.jpg)
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Derivatives Differentiation Formulas
Theorem.
Chain Rule. Let f ,g be complex functions and let z0 besuch that g is differentiable at z0 and f is differentiable at g(z0). Thenf ◦g is differentiable at z0 and the derivative is(f ◦g)′(z0) = f ′
(g(z0)
)g′(z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 86: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/86.jpg)
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Derivatives Differentiation Formulas
Theorem. Chain Rule.
Let f ,g be complex functions and let z0 besuch that g is differentiable at z0 and f is differentiable at g(z0). Thenf ◦g is differentiable at z0 and the derivative is(f ◦g)′(z0) = f ′
(g(z0)
)g′(z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 87: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/87.jpg)
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Derivatives Differentiation Formulas
Theorem. Chain Rule. Let f ,g be complex functions and let z0 besuch that g is differentiable at z0 and f is differentiable at g(z0).
Thenf ◦g is differentiable at z0 and the derivative is(f ◦g)′(z0) = f ′
(g(z0)
)g′(z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 88: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/88.jpg)
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Derivatives Differentiation Formulas
Theorem. Chain Rule. Let f ,g be complex functions and let z0 besuch that g is differentiable at z0 and f is differentiable at g(z0). Thenf ◦g is differentiable at z0
and the derivative is(f ◦g)′(z0) = f ′
(g(z0)
)g′(z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 89: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/89.jpg)
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Derivatives Differentiation Formulas
Theorem. Chain Rule. Let f ,g be complex functions and let z0 besuch that g is differentiable at z0 and f is differentiable at g(z0). Thenf ◦g is differentiable at z0 and the derivative is(f ◦g)′(z0) = f ′
(g(z0)
)g′(z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 90: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/90.jpg)
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Derivatives Differentiation Formulas
Proof (small technicality ignored).
limz→z0
f ◦g(z)− f ◦g(z0)z− z0
= limz→z0
f(g(z)
)− f
(g(z0)
)z− z0
· g(z)−g(z0)g(z)−g(z0)
= limz→z0
f(g(z)
)− f
(g(z0)
)g(z)−g(z0)
· g(z)−g(z0)z− z0
= limz→z0
f(g(z)
)− f
(g(z0)
)g(z)−g(z0)
· limz→z0
g(z)−g(z0)z− z0
= limu→g(z0)
f (u)− f(g(z0)
)u−g(z0)
· limz→z0
g(z)−g(z0)z− z0
= f ′(g(z0)
)g′(z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 91: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/91.jpg)
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Derivatives Differentiation Formulas
Proof (small technicality ignored).
limz→z0
f ◦g(z)− f ◦g(z0)z− z0
= limz→z0
f(g(z)
)− f
(g(z0)
)z− z0
· g(z)−g(z0)g(z)−g(z0)
= limz→z0
f(g(z)
)− f
(g(z0)
)g(z)−g(z0)
· g(z)−g(z0)z− z0
= limz→z0
f(g(z)
)− f
(g(z0)
)g(z)−g(z0)
· limz→z0
g(z)−g(z0)z− z0
= limu→g(z0)
f (u)− f(g(z0)
)u−g(z0)
· limz→z0
g(z)−g(z0)z− z0
= f ′(g(z0)
)g′(z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 92: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/92.jpg)
logo1
Derivatives Differentiation Formulas
Proof (small technicality ignored).
limz→z0
f ◦g(z)− f ◦g(z0)z− z0
= limz→z0
f(g(z)
)− f
(g(z0)
)z− z0
· g(z)−g(z0)g(z)−g(z0)
= limz→z0
f(g(z)
)− f
(g(z0)
)g(z)−g(z0)
· g(z)−g(z0)z− z0
= limz→z0
f(g(z)
)− f
(g(z0)
)g(z)−g(z0)
· limz→z0
g(z)−g(z0)z− z0
= limu→g(z0)
f (u)− f(g(z0)
)u−g(z0)
· limz→z0
g(z)−g(z0)z− z0
= f ′(g(z0)
)g′(z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 93: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/93.jpg)
logo1
Derivatives Differentiation Formulas
Proof (small technicality ignored).
limz→z0
f ◦g(z)− f ◦g(z0)z− z0
= limz→z0
f(g(z)
)− f
(g(z0)
)z− z0
· g(z)−g(z0)g(z)−g(z0)
= limz→z0
f(g(z)
)− f
(g(z0)
)g(z)−g(z0)
· g(z)−g(z0)z− z0
= limz→z0
f(g(z)
)− f
(g(z0)
)g(z)−g(z0)
· limz→z0
g(z)−g(z0)z− z0
= limu→g(z0)
f (u)− f(g(z0)
)u−g(z0)
· limz→z0
g(z)−g(z0)z− z0
= f ′(g(z0)
)g′(z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 94: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/94.jpg)
logo1
Derivatives Differentiation Formulas
Proof (small technicality ignored).
limz→z0
f ◦g(z)− f ◦g(z0)z− z0
= limz→z0
f(g(z)
)− f
(g(z0)
)z− z0
· g(z)−g(z0)g(z)−g(z0)
= limz→z0
f(g(z)
)− f
(g(z0)
)g(z)−g(z0)
· g(z)−g(z0)z− z0
= limz→z0
f(g(z)
)− f
(g(z0)
)g(z)−g(z0)
· limz→z0
g(z)−g(z0)z− z0
= limu→g(z0)
f (u)− f(g(z0)
)u−g(z0)
· limz→z0
g(z)−g(z0)z− z0
= f ′(g(z0)
)g′(z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 95: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/95.jpg)
logo1
Derivatives Differentiation Formulas
Proof (small technicality ignored).
limz→z0
f ◦g(z)− f ◦g(z0)z− z0
= limz→z0
f(g(z)
)− f
(g(z0)
)z− z0
· g(z)−g(z0)g(z)−g(z0)
= limz→z0
f(g(z)
)− f
(g(z0)
)g(z)−g(z0)
· g(z)−g(z0)z− z0
= limz→z0
f(g(z)
)− f
(g(z0)
)g(z)−g(z0)
· limz→z0
g(z)−g(z0)z− z0
= limu→g(z0)
f (u)− f(g(z0)
)u−g(z0)
· limz→z0
g(z)−g(z0)z− z0
= f ′(g(z0)
)g′(z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 96: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/96.jpg)
logo1
Derivatives Differentiation Formulas
Proof (small technicality ignored).
limz→z0
f ◦g(z)− f ◦g(z0)z− z0
= limz→z0
f(g(z)
)− f
(g(z0)
)z− z0
· g(z)−g(z0)g(z)−g(z0)
= limz→z0
f(g(z)
)− f
(g(z0)
)g(z)−g(z0)
· g(z)−g(z0)z− z0
= limz→z0
f(g(z)
)− f
(g(z0)
)g(z)−g(z0)
· limz→z0
g(z)−g(z0)z− z0
= limu→g(z0)
f (u)− f(g(z0)
)u−g(z0)
· limz→z0
g(z)−g(z0)z− z0
= f ′(g(z0)
)g′(z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 97: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/97.jpg)
logo1
Derivatives Differentiation Formulas
Proof (small technicality ignored).
limz→z0
f ◦g(z)− f ◦g(z0)z− z0
= limz→z0
f(g(z)
)− f
(g(z0)
)z− z0
· g(z)−g(z0)g(z)−g(z0)
= limz→z0
f(g(z)
)− f
(g(z0)
)g(z)−g(z0)
· g(z)−g(z0)z− z0
= limz→z0
f(g(z)
)− f
(g(z0)
)g(z)−g(z0)
· limz→z0
g(z)−g(z0)z− z0
= limu→g(z0)
f (u)− f(g(z0)
)u−g(z0)
· limz→z0
g(z)−g(z0)z− z0
= f ′(g(z0)
)g′(z0).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 98: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/98.jpg)
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Derivatives Differentiation Formulas
Example
(derivatives work just like they did in calculus). Find thederivative of f (x) = (iz+4)3 (
z2 +1)2.
ddz
(iz+4)3 (z2 +1
)2
= 3(iz+4)2 i(z2 +1
)2+(iz+4)3 2
(z2 +1
)2z
= 3i(iz+4)2 (z2 +1
)2+4z(iz+4)3 (
z2 +1)
= (iz+4)2 (z2 +1
)(3iz2 +3i+4iz2 +16z
)= (iz+4)2 (
z2 +1)(
7iz2 +16z+3i)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 99: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/99.jpg)
logo1
Derivatives Differentiation Formulas
Example (derivatives work just like they did in calculus).
Find thederivative of f (x) = (iz+4)3 (
z2 +1)2.
ddz
(iz+4)3 (z2 +1
)2
= 3(iz+4)2 i(z2 +1
)2+(iz+4)3 2
(z2 +1
)2z
= 3i(iz+4)2 (z2 +1
)2+4z(iz+4)3 (
z2 +1)
= (iz+4)2 (z2 +1
)(3iz2 +3i+4iz2 +16z
)= (iz+4)2 (
z2 +1)(
7iz2 +16z+3i)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 100: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/100.jpg)
logo1
Derivatives Differentiation Formulas
Example (derivatives work just like they did in calculus). Find thederivative of f (x) = (iz+4)3 (
z2 +1)2.
ddz
(iz+4)3 (z2 +1
)2
= 3(iz+4)2 i(z2 +1
)2+(iz+4)3 2
(z2 +1
)2z
= 3i(iz+4)2 (z2 +1
)2+4z(iz+4)3 (
z2 +1)
= (iz+4)2 (z2 +1
)(3iz2 +3i+4iz2 +16z
)= (iz+4)2 (
z2 +1)(
7iz2 +16z+3i)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 101: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/101.jpg)
logo1
Derivatives Differentiation Formulas
Example (derivatives work just like they did in calculus). Find thederivative of f (x) = (iz+4)3 (
z2 +1)2.
ddz
(iz+4)3 (z2 +1
)2
= 3(iz+4)2 i(z2 +1
)2+(iz+4)3 2
(z2 +1
)2z
= 3i(iz+4)2 (z2 +1
)2+4z(iz+4)3 (
z2 +1)
= (iz+4)2 (z2 +1
)(3iz2 +3i+4iz2 +16z
)= (iz+4)2 (
z2 +1)(
7iz2 +16z+3i)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 102: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/102.jpg)
logo1
Derivatives Differentiation Formulas
Example (derivatives work just like they did in calculus). Find thederivative of f (x) = (iz+4)3 (
z2 +1)2.
ddz
(iz+4)3 (z2 +1
)2
= 3(iz+4)2 i(z2 +1
)2
+(iz+4)3 2(z2 +1
)2z
= 3i(iz+4)2 (z2 +1
)2+4z(iz+4)3 (
z2 +1)
= (iz+4)2 (z2 +1
)(3iz2 +3i+4iz2 +16z
)= (iz+4)2 (
z2 +1)(
7iz2 +16z+3i)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 103: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/103.jpg)
logo1
Derivatives Differentiation Formulas
Example (derivatives work just like they did in calculus). Find thederivative of f (x) = (iz+4)3 (
z2 +1)2.
ddz
(iz+4)3 (z2 +1
)2
= 3(iz+4)2 i(z2 +1
)2+(iz+4)3 2
(z2 +1
)2z
= 3i(iz+4)2 (z2 +1
)2+4z(iz+4)3 (
z2 +1)
= (iz+4)2 (z2 +1
)(3iz2 +3i+4iz2 +16z
)= (iz+4)2 (
z2 +1)(
7iz2 +16z+3i)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 104: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/104.jpg)
logo1
Derivatives Differentiation Formulas
Example (derivatives work just like they did in calculus). Find thederivative of f (x) = (iz+4)3 (
z2 +1)2.
ddz
(iz+4)3 (z2 +1
)2
= 3(iz+4)2 i(z2 +1
)2+(iz+4)3 2
(z2 +1
)2z
= 3i(iz+4)2 (z2 +1
)2+4z(iz+4)3 (
z2 +1)
= (iz+4)2 (z2 +1
)(3iz2 +3i+4iz2 +16z
)= (iz+4)2 (
z2 +1)(
7iz2 +16z+3i)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 105: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/105.jpg)
logo1
Derivatives Differentiation Formulas
Example (derivatives work just like they did in calculus). Find thederivative of f (x) = (iz+4)3 (
z2 +1)2.
ddz
(iz+4)3 (z2 +1
)2
= 3(iz+4)2 i(z2 +1
)2+(iz+4)3 2
(z2 +1
)2z
= 3i(iz+4)2 (z2 +1
)2+4z(iz+4)3 (
z2 +1)
= (iz+4)2 (z2 +1
)(3iz2 +3i+4iz2 +16z
)
= (iz+4)2 (z2 +1
)(7iz2 +16z+3i
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions
![Page 106: Derivatives of Complex Functions - USM](https://reader031.vdocument.in/reader031/viewer/2022011723/61d2da4e48231b56fb55fce3/html5/thumbnails/106.jpg)
logo1
Derivatives Differentiation Formulas
Example (derivatives work just like they did in calculus). Find thederivative of f (x) = (iz+4)3 (
z2 +1)2.
ddz
(iz+4)3 (z2 +1
)2
= 3(iz+4)2 i(z2 +1
)2+(iz+4)3 2
(z2 +1
)2z
= 3i(iz+4)2 (z2 +1
)2+4z(iz+4)3 (
z2 +1)
= (iz+4)2 (z2 +1
)(3iz2 +3i+4iz2 +16z
)= (iz+4)2 (
z2 +1)(
7iz2 +16z+3i)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Derivatives of Complex Functions