A function of a complex number is a function f:\mathbb{C} \rightarrow \mathbb{C}, that is
f(z) = w where z = x + yi, w = u(x,y) + v(x,y)i. A complex
function is a pair of real functions put together (analogous to a
complex number being comprised of real numbers). For example, f(z) = z^2 = (x^2 - y^2) + 2xyi = u(x,y) +
v(x,y)i.
Holomorphic function
f(z) is holomorphic (analytic, regular)
in a region R of the complex plane if
it has a unique derivative at every point in R.
Derivative of a complex function
This has the usual definition, f'(z) =
\frac{df}{dz} = \lim_{h \rightarrow \infty}
\frac{f(z+h)-f(z)}{h}. However, for the derivative to exist it
has to meet more stringent requirements. Recall that for a real
function, the derivative exists iff the right and left derivatives exist
and are equal. However, since complex functions exist in the complex
plane (which is two dimensional), we need the left, right, up, down and
diagonal limits to exist and to be equal.
If f(z) = u(x,y) + v(x,y)i is
holomorphic in a region R, then u and v
satisfy the Cauchy-Riemann relations. The relations are thus a
necessary, but not sufficient condition for f to be holomorphic.
Theorem
If \frac{\partial u}{\partial x},
\frac{\partial u}{\partial u}, \frac{\partial v}{\partial x},
\frac{\partial v}{\partial y} exist, are continuous and satisfy
the Cauchy-Riemann relations in a region R
\subseteq \mathbb{C}, then f(z) =
u(x,y) + v(x,y)i is holomorphic in R.
Theorem
If f(z) is holomorphic in a region
R of the complex plane, then it has
derivatives of all orders at all points in R i.e. it will be infinitely differentiable.
Theorem
If f(z) is holomorphic in R, then it can be expanded as a Taylor series
around any point z_0 in R. Radius of convergence of Taylor series
around z_0 will be: R_\text{conv} = \min|z_0 - z_s|, where
z_s is a singular point of f(z) (i.e. a point where f is not holomorphic, its complex derivative
is undefined)
Theorem
If f(z) = u(x,y) + v(x,y)i is a
holomorphic in a region R, then u and v
satisfy Laplace’s equation in R: \nabla^2 u = 0, \quad \nabla^2v = 0. If
f is holomorphic then u and v are
considered harmonic functions (since they satisfy Laplace’s equation).
Thus, if g and h satisfy Laplace’s equation, then they can
serve as the real or imaginary parts of a holomorphic complex function.
Complex Integrals and
Cauchy’s Theorem
To integrate f(z) from point a to point b
where a,b \in \mathbb{C}, we have to
consider what is essentially the line integral of the function between
a and b.
Suppose f(z) = u + vi where z = x + yi. The path C from a to
b can be parametrized as x = x(t), y = y(t) \qquad t \in [a,b].
Relates to contour integration. Suppose f(z) is a complex function and C is a closed curve in the complex plane. If
the following conditions are satisfied:
f(z) is holomorphic on and
everywhere inside C
C is a simple curve (doesn’t cross
itself)
C has a finite number of corners
(e.g. is not a Weierstrass function, or Brownian motion),
then, \oint_C f(z)dz = 0.
Cauchy Integral Formula
If f(z) is holomorphic on or inside a
simple closed curve C, then: f(a) = \frac{1}{2\pi i} \oint_C
\frac{f(z)}{z-a}dz, where a is
inside C. If a is outside C, the integral will evaluate to 0.
Laurent Series
Generalised Taylor series for complex functions.
Laurent’s Theorem
Suppose that R is a region enclosed by
two concentric circles (of any size). Suppose further that f(z) is holomorphic in R. Then f(z) =
\sum_{k=0}^{\infty} a_k (z - z_0)^k + \sum_{j=1}^{\infty}b_j
(z-z_0)^{-j}\\
= a_0 + a_1(z-z_0) + \ldots + \frac{b_1}{z-z_0} + \frac{b_2}{z-z_0} +
\ldots, which is the sum of the analytic part (a terms) and the principal part (b terms).
Now to find the coefficients of a Taylor series, you find the
derivatives of f(x) at the point about
which expansion occurs. For a Laurent series, you have to use contour
integrals to calculate the coefficients.
For the analytic part, a_n = \frac{1}{2\pi
i} \oint_C \frac{f(z)}{(z-z_0)=^{n+1}} dz. For the principal
part, b_n = \frac{1}{2\pi i} \oint_C
\frac{f(z)}{(z-z_0)=^{-n+1}} dz. Here, C is a closed curve inside R that surrounds z_0.
Notes/definitions about poles/singularities/residues…
Residue Theorem
Suppose f(z) is a complex function
with many singular points, z_1, z_2, ...,
z_n. If C is a curve that
encloses all these points, then \oint_{C}f(z)dz = 2\pi i
\sum_{j=0}^{n}b_{ij}.