Chapter 14 - Functions of Complex Variables

Reference "Mathematical Methods in the Physical Sciences" 3e by Mary L. Boas.

We can represent a complex function (those including $i=\sqrt{-1}$) by $z$: $$\begin{array}{rl}z& =x+iy\end{array}$$ Or, alternatively, with $\theta =\arctan (y/x)$ and $r=\sqrt{x^2+y^2}$: $$z\equiv r{e}^{i\theta }$$ Complex functions are those that are a function of $z$: $$f(z)=f(x+iy)=u(x,y)+iv(x,y)$$ and are composed of their real and imaginary parts, represented by $u$ and $v$: $$\begin{array}{rl}\text{Re: }& u(x,y)\\ \text{Im: }& v(x,y)\end{array}$$

Note: $f(z)$ is customarily single-valued (one unique value per $z$). To handle functions that aren't single-valued ($\ln (z),\arctan (z)$), define a range such that $\theta$ or $x$ or $y$ are permitted.

Analytic Functions

A function $f(z)$ is analytic in a region of the complex plane if it has a unique derivative at every point of the region.

Similarly, the statement "$f(z)$ is analytic at some point $z=a$" means there's a unique derivative in some small circle around $z=a$.

For some analytic function $f(z)$, the derivative is defined as $$f^{\prime }(z)=\frac{df}{dz}=\underset{\Delta z\to 0}{\lim }\frac{\Delta f}{\Delta z}=\underset{\Delta z\to 0}{\lim }\frac{f(z+\Delta z)-f(z)}{\Delta z}$$ where $\Delta z=\Delta x+i\Delta y$. Note also some definitions:

• A regular point of $f(z)$ is a point at which $f(z)$ is analytic.
• A singular point / singularity of $f(z)$ is a point at which $f(z)$ is not analytic.
  • This is an isolated singular point if analytic everywhere else in a small circle around $f(z)$.

Ex. derivative of an analytic function: $\frac{d}{dz}{z}^2$. $$\begin{array}{rl}\frac{d}{dz}(z^2)& =\underset{\Delta z\to 0}{\lim }\frac{(z+\Delta z{)}^2-z^2}{\Delta z}\\ & =\underset{\Delta z\to 0}{\lim }\frac{z^2+2z\Delta z+(\Delta z{)}^2-z^2}{\Delta z}\\ & =\underset{\Delta z\to 0}{\lim }(2z+\Delta z)\\ & =2z\end{array}$$

Ex. derivative of a non-analytic function: $$\frac{d}{dz}|z{|}^2={\sqrt{z}}^2=\sqrt{x^2+y^2}$$ Using our limit idea above, $$\begin{array}{rl}\frac{d}{dz}(z^2)& =\underset{\Delta z\to 0}{\lim }\frac{|z+\Delta z^2|-|z{|}^2}{\Delta z}\end{array}$$ the numerator is always real (abs. values always real), but the denominator has different values depending on the approach. It's neither always real nor purely imaginary.

Thus, $f(z)=|z{|}^2$ is not analytic.

Some relevant theorems about analytic functions:

Theorem I: if $f(z)=u(x,y)+iv(x,y)$ is analytic in a region, then in the region $$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}$$ these are the Cauchy-Riemann conditions.

Theorem II: if $u(x,y)$ and $v(x,y)$ (and their partial derivatives w.r.t. $x$ and $y$) are both:

1. Continuous.
2. Satisfy the Cauchy-Riemann conditions.

Then $f(z)$ is analytic at all points in the region (though not necessarily the boundary).

Theorem III: If $f(z)$ analytic in a region, then it has derivatives of all orders at points inside the region, and can be expanded in a Taylor series around any point $z=z_0$ inside the region.

The power series converges inside the circle around $z_0$ which has a radius until the nearest singular point.

Theorem IV: If $f(z)=u+iv$ is analytic in a region, then $u$ and $v$ satisfy Laplace's equation in the region (i.e. $u$ and $v$ are harmonic functions).

Also, if $u$ or $v$ satisfy Laplace's equation in a simply-connected region, they are respectfully the real or imaginary parts of an analytic function $f(z)$.

Note: this means we can find solutions of Laplace's equation just by taking the real or imaginary parts of an analytic function of $z$.

Contour Integrals

Some additional theorems of analytic functions involve contour integrals - or path integrals in the complex plane.

Theorem V (Cauchy's theorem): If $C$ is some closed, smooth curve (corners allowed) in the complex plane, and $f(z)$ is analytic on and inside $C$, then $${\oint }_{\text{around C}}f(z)dz=0$$ Note: a finite number of corners are allowed.

Theorem VI (Cauchy integrals): if $f(z)$ is analytic on and inside some closed curve $C$, then $f(z=a)$ inside $C$ is given by the path integral along $C$ below: $$f(a)=\frac{1}{2\pi i}{\oint }_C\frac{f(z)}{z-a}dz$$ If $f(z)$ is given on the boundary of a region ($C$), then this integral gives the value of $f(z=a)$ at any point $a$ within $C$.

Laurent Series

Laurent series are able to represent complex functions $f(z)$ as power series, which include terms of negative degrees.

Theorem VII (Laurent's theorem): Let $C_1$ and $C_2$ be two circles centered at $z_0$, and $f(z)$ analytic in the region $R$ between the circles. $f(z)$ can be expanded as $$f(z)=a_0+a_1(z-z_0)+a_2(z-z_0{)}^2+\cdots +\frac{b_1}{z-z_0}+\frac{b_2}{(z-z_0{)}^2}+\cdots$$ convergent in $R$. The $b_n$ terms are called the principal part.

Some properties of $b_n$:

1. If all $b_n=0$, $f(z)$ is analytic at $z=z_0$ and $z_0$ is a regular point (see Analytic Functions).
2. If $b_n\ne 0$ but all terms after $b_{n+m}=0$, $f(z)$ has a "pole of order" $n$ at $z=z_0$. If $n=1$ (i.e. $b_1\ne 0$, $b_{n>1}=0$), $f(z)$ has a "simple pole".
3. If all $b_n\ne 0$, $f(z)$ has an "essential singularity" at $z=z_0$.
4. The coefficient $b_1$ is the "residue" of $f(z)$ at $z=z_0$.

Residue Theorem