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topic entry on complex analysis∗
pahio†
2013-03-21 22:11:45
Introduction
Complex analysis may be defined as the study of analytic functions of a complex
variable. The origins of this subject lie in the observation that, given a function
which has a convergent Taylor series, one can substitute complex numbers for
the variable and obtain a convergent series which defines a function of a complex
variable. For instance, putting imaginary numbers into the power series for the
exponential function, we find
x2
x3
x4
x5
x6
x7
−i +
+i −
− i + ···
2
3!
4!
5!
6!
7!
2
3
4
5
6
7
x
x
x
x
x
x
e−ix = 1 − ix −
+i +
−i −
+ i + ···
2
3!
4!
5!
6!
7!
Adding and subtracting these series, we find
eix
=
1 + ix −
x2
x4
x6
1 ix
(e + e−ix ) = 1 −
+
−
+ −···
2
2!
4!
6!
1 ix
x3
x5
x7
(e − e−ix ) = x −
+
−
+ −···
2i
3!
5!
7!
We recognize these series as the Taylor-Madhava series for the sine and the
cosine functions respectively. We hence have
1 ix
(e − e−ix )
2i
1 ix
cos x =
(e + e−ix )
2
eix = cos x + i sin x.
sin x =
These equations let us re-express trigonometric functions in terms of complex
exponentials. Using them, deriving and verifying trigonometric identities becomes a straightforward exercise in algebra using the laws of exponents.
∗ hTopicEntryOnComplexAnalysisi
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1
We call functions of a complex variable which can be expressed in terms of
a power series as complex analytic. More precisely, if D is an open subset of C,
we say that a function f : D → C is complex analytic if, for every point w in
D, there exists a positive number δ and a sequence of complex numbers ck such
that the series
∞
X
ck (z − w)k
k=0
converges to f (z) when z ∈ D and |z − w| < δ.
An important feature of this definition is that it is not required that a single
series works for all points of D. For instance, suppose we define the function
f : Cr{1} → C as
1
.
f (z) =
1−z
While it it turns out that f is analytic, no single series will give us the values of
f for all allowed values of z. For instance, we have the familiar geometric series:
f (z) =
∞
X
zk
k=0
However, this series diverges when |z| > 1. For such values of z, we need to
use other series. For instance, when z is near 2, we have the following series:
f (z) =
∞
X
(−1)k+1 (z − 2)k
k=0
This series, however, diverges when |z − 2| > 1. While, for every allowed value
of z we can find some power series which will converge to f (z), no single power
series will converge to f (z) for all permissible values of z.
It is possible to define the operations of differentiation and integration for
complex functions. These operations are well-defined for analytic functions and
have the usual properties familiar from real analysis.
The class of analytic functions is interesting to study for at least two main
reasons. Firstly, many functions which arise in pure and applied mathematics, such as polynomials, rational functions, exponential functions. logarithms,
trigonometric functions, and solutions of differential equations are analytic. Second, the class of analytic functions enjoys many remarkable properties which do
not hold for other classes of functions, such as the following:
Closure The class of complex analytic functions is closed under the usual algebraic operations, taking derivative and integrals, composition, and taking
uniform limits.
Rigidity Given a complex analytic function f : D → C, where D is an open
subset of C, if we know the values of f at an infinite number of points of D
which have a limit point in D, then we know the value of f at all points of
D. For instance, given a complex analytic function on some neighborhood
2
of the real axis, the values of that function in the whole neighborhood will
be determined by its values on the real axis.
Cauchy and Morera theorems The integral of a complex analytic function
along any contractible closed loop equals zero. Conversely, if the integral
of a complex function about every contractible loop happens to be zero,
then that function is analytic.
Complex differentiability If a complex function is differentiable, then it has
derivatives of all orders. This contrasts sharply with the case of real
analysis, where a function may be differentiable only a fixed number of
times.
Harmonicity The real and imaginary parts of a complex analytic function are
harmonic, i.e. satisfy Laplace’s equation. Conversely, given a harmonic
function on the plane, there exists a complex analytic function of which it
is the real part.
Conformal mapping A complex function is analytic if and only if it preserves
maps pairs of intersecting curves into pairs which intersect at the same
angle.
As one can see, there are many ways to characterize complex analytic functions,
many of which have nothing to do with power series. This suggests that analytic
functions are somehow a naturally occurring subset of complex functions. This
variety of distinct ways of characterizing analytic functions means that one
has a variety of methods which may be used to study them and prove deep and
surprising results by bringing insights and techniques from geometry, differential
equations, and functional analysis to bear on problems of complex analysis. This
also works the other way — one can use complex analysis to prove results in
other branches of mathmatics which have nothing to do with complex numbers.
For instance, the problem of minimal surfaces can be solved by using complex
analysis.
0.1
Complex numbers
1. complex plane, equality of complex numbers
2. topology of the complex plane
3. triangle inequality of complex numbers
4. argument of product and quotient
5. unit disc, annulus, closed complex plane
6. nth root
7. taking square root algebraically
3
8. quadratic equation in C
9. complex function
0.2
Complex functions
1. de Moivre identity
2. addition formula
3. complex exponential function
4. periodicity of exponential function
5. complex sine and cosine
6. values of complex cosine
7. complex tangent and cotangent
8. example of summation by parts
9. Euler’s formulas (see also this)
10. complex logarithm
11. general power
12. fundamental theorems in complex analysis
13. index of special functions
0.3
Analytic function
1. holomorphic
2. meromorphic
3. periodic functions
4. isolated singularity
5. complex derivative
6. Cauchy-Riemann equations
7. power series
8. Bohr’s theorem
9. identity theorem of holomorphic functions
10. Weierstrass double series theorem
4
11. entire functions
12. properties of entire functions
13. pole of function
14. zeros and poles of rational function
15. when all singularities are poles
16. Casorati-Weierstrass theorem
17. Picard’s theorem
18. Laurent series
19. coefficients of Laurent series
20. residue
21. regular at infinity
22. Nevanlinna theory
0.4
Complex integration
1. contour integral
2. estimating theorem of contour integral
3. theorems on complex function series
4. holomorphic function associated with continuous function
5. Cauchy integral theorem
6. Cauchy integral formula; variant of Cauchy integral formula
7. residue theorem
8. example of using residue theorem
9. argument principle
10. complex antiderivative
5
0.5
Analytic continuation
1. analytic continuation
2. meromorphic continuation
3. analytic continuation by power series
4. monodromy theorem
5. Schwarz’ reflection principle
6. example of analytic continuation
7. analytic continuation of gamma function
8. analytic continuation of Riemann zeta to critical strip
9. analytic continuation of Riemann zeta (using integral)
0.6
Riemann zeta function
1. Riemann zeta function
2. Euler product formula
3. Riemann functional equation
4. critical strip
5. value of the Riemann zeta function at 0, at 2, at 4
6. formulae for zeta in the critical strip
0.7
Conformal mapping
1. conformal mapping
2. conformal mapping theorem
3. simple example of composed conformal mapping
4. example of conformal mapping
5. Schwarz–Christoffel transformation
6