![Page 1: Complex Differentiation Mohammed Nasser Department of statistics RU](https://reader030.vdocument.in/reader030/viewer/2022032702/56649ce45503460f949b0e6b/html5/thumbnails/1.jpg)
Complex Differentiation
Mohammed Nasser
Department of statistics
RU
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Derivatives
• Differentiation of complex-valued functions is completely analogous to the real case:
• Definition. Derivative. Let f(z) be a complex-valued function defined in a neighborhood of z0. Then the derivative of f(z) at z0 is given by
Provided this limit exists. f(z) is said to be differentiable at z0.
zzfzzf
zfz
)()(lim)( 00
00
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Some Exercises
Show that
1) f(z)= is nowwhere differentiable.
2) g(z)=zn has derivative nzn-1.
3) h(z)=ez has derivative ez.
4) l(z)=|z|2 is nowhere differentiable except z=0
5) Every real-valued function of complex variable is either non-differentiable or differentiable with derivative equal to 0.
z
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Solutions
1.0 0
0 0
0 0
0 0
( ) ( )( ) lim
lim lim
h
h h
f z h f zf z
hz h z h
h h
h 0
X-axis
Y-axis
If we go along X-axis A tends to 1.
A
If we go along Y-axis A tends to -1.
That implies the limit does not exist.
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Properties of Derivatives
Rule. Chain''
.0if,''
'
'''
. constant anyfor ''
'''
000
020
00000
00000
00
000
zgzgfzgfdzd
zgzg
zgzfzfzgz
gf
zgzfzgzfzfg
czcfzcf
zgzfzgf
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Analytic. Holomorphic.
• Definition. A complex-valued function f (z) is said to be analytic, or equivalently, holomorphic, on an open set if it has a derivative at every point of . (The term “regular” is also used.)
• It is important that a function may be differentiable at a single point only. Analyticity implies differentiability within a neighborhood of the point. This permits expansion of the function by a Taylor series about the point.
• If f (z) is analytic on the whole complex plane, then it is said to be an entire function.
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Rational Function.
• Definition. If f and g are polynomials in z, then h (z) = f (z)/g(z), g(z) 0 is called a rational function.
• Remarks.– All polynomial functions of z are entire.– A rational function of z is analytic at every
point for which its denominator is nonzero.– If a function can be reduced to a polynomial
function which does not involve , then it is analytic.
z
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Example 1
1 2 2
1 2 2
1
1( )
( 1)
,2 2
12 2( )
12 2
1 1( )
1 1
x iyf z
x yz z z z
let x yi
z z z zi
if zz z z z
iz
f zzz z z z
Thus f1(z) is analytic at all points except z=1.
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Example 2
2 22
2 2
2
2
( ) 3 1 3
,2 2
( ) 3 1 32 2 2 2
( ) 3 1
f z x y x i y
z z z zlet x y
iz z z z z z z z
f z ii i
f z zz z
Thus f2(z) is nowhere analytic.
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Testing for Analyticity
Determining the analyticity of a function by searching for in its expression that cannot be removed is at best awkward. Observe:
z
5 4 3 5
53 2 5
1( )
z z z z z zf z
z z z z
It would be difficult and time consuming to try to reduce this expression to a form in which you could be sure that the could not be removed. The method cannot be used when anything but algebraic functions are used.
z
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Cauchy-Riemann Equations (1)
If the function f (z) = u(x,y) + iv(x,y) is differentiable at z0 = x0 + iy0, then the limit
zzfzzf
zfz
)()(lim)( 00
00
can be evaluated by allowing z to approach zero from any direction in the complex plane.
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Cauchy-Riemann Equations (2)If it approaches along the x-axis, then z = x, and we obtain
x
yxivyxuyxxivyxxuzf
x
),(),(),(),(lim)(' 00000000
00
xyxvyxxv
ix
yxuyxxuzf
xx
),(),(lim
),(),(lim)(' 0000
0
0000
00
But the limits of the bracketed expression are just the first partial derivatives of u and v with respect to x, so that:
).,(),()(' 00000 yxxv
iyxxu
zf
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Cauchy-Riemann Equations (3)If it approaches along the y-axis, then z = iy, and we obtain
0 0 0 00 0
( , ) ( , )'( ) lim
y
u x y y u x yf z
i y
And, therefore
).,(),()(' 00000 yxyv
yxyu
izf
0 0 0 0
0
( , ) ( , )limy
v x y y v x yi
i y
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Cauchy-Riemann Equations (4)
By definition, a limit exists only if it is unique. Therefore, these two expressions must be equivalent. Equating real and imaginary parts, we have that
xv
yu
yv
xu
and
must hold at z0 = x0 + iy0 . These equations are called the Cauchy-Riemann Equations. Their importance is made clear in the following theorem.
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Cauchy-Riemann Equations (5)
• Theorem. Let f (z) = u(x,y) + iv(x,y) be defined in some open set containing the point z0. If the first partial derivatives of u and v exist in , and are continuous at z0 , and satisfy the Cauchy-Riemann equations at z0, then f (z) is differentiable at z0. Consequently, if the first partial derivatives are continuous and satisfy the Cauchy-Riemann equations at all points of , then f (z) is analytic in .
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Example 1
1,1,2,2
)()()( 22
xv
yu
yyv
xxu
xyiyxzf
Hence, the Cauchy-Riemann equations are satisfied only on the line x = y, and therefore in no open disk. Thus, by the theorem, f (z) is nowhere analytic.
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Example 2Prove that f (z) is entire and find its derivative.
yexv
yeyu
yeyv
yexu
yieyezf
xxxx
xx
sin,sin,cos,cos
:Solution
sincos)(
The first partials are continuous and satisfy the Cauchy-Riemann equations at every point.
.sincos)(' yieyexv
ixu
zf xx
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Harmonic Functions
• Definition. Harmonic. A real-valued function (x,y) is said to be harmonic in a domain D if all of its second-order partial derivatives are continuous in D and if each point of D satisfies
.02
2
2
2
yx
Theorem. If f (z) = u(x,y) + iv(x,y) is analytic in a domain D, then each of the functions u(x,y) and v(x,y) is harmonic in D.
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Harmonic Conjugate
• Given a function u(x,y) harmonic in, say, an open disk, then we can find another harmonic function v(x,y) so that u + iv is an analytic function of z in the disk. Such a function v is called a harmonic conjugate of u.
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ExampleConstruct an analytic function whose real part is: .3),( 23 yxyxyxu
Solution: First verify that this function is harmonic.
.066
6 and16
6and33
2
2
2
2
2
2
2
222
xxyu
xu
and
xyu
xyyu
xxu
yxxu
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Example, Continued
16)2(
33)1( 22
xyyu
xv
andyxxu
yv
Integrate (1) with respect to y:
)(3),()3(
33
33
32
22
22
xyyxyxv
yyxv
yyxv
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Example, Continued
Now take the derivative of v(x,y) with respect to x:
).('6 xxyxv
According to equation (2), this equals 6xy – 1. Thus,
.3),(
.)(and,)(So
.1ly,Equivalent.1)('and
16)('6
32 CyyxyxvAnd
Cxxxx
xx
xyxxy
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Example, Continued
The desired analytic function f (z) = u + iv is:
Cxyyxiyxyxzf 3223 33)(
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Remember Complex Exponential
• We would like the complex exponential to be a natural extension of the real case, with f (z) = ez entire. We begin by examining ez = ex+iy = exeiy.
• eiy = cos y + i sin y by Euler’s and DeMoivre’s relations.
• Definition. Complex Exponential Function. If z = x + iy, then ez = ex(cos y + i sin y).
• That is, |ez|= ex and arg ez = y.
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More on Exponentials
• Recall that a function f is one-to-one on a set S if the equation f (z1) = f (z2), where z1, z2 S, implies that z1 = z2. The complex exponential function is not one-to-one on the whole plane.
• Theorem. A necessary and sufficient condition that ez = 1 is that z = 2ki, where k is an integer.
Also, a necessary and sufficient condition that is that z1 = z2 + 2ki, where k is an
integer. Thus ez is a periodic function.
21 ee zz
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How is the case with multi-valued functions like z1/n, logz etc??