Complex Analysis

作者: 周思益 | 来源:发表于2021-04-11 03:20 被阅读0次

This note is based on a book called A first course in complex analysis with applications.

Complex Numbers and Complex Plane

Principal Argument The symbol arg( $z$ ) actually represents a set of values, but the argument $\theta$ of a complex number that lies in the interval $-\pi<\theta\leq\pi$ is called the principal value of arg( $z$ ) or the principal argument of $z$ .

In general, arg( $z$ ) and Arg( $z$ ) are related by
${\rm arg}(z) ={\rm Arg}(z) +2\pi n, \quad n=0,\pm 1,\pm 2, \ldots$

de Moivre's Formula When $z=\cos\theta+i\sin\theta$ , we have $|z|=r=1$ then
$(\cos\theta+i\sin\theta)^n = \cos n\theta+i \sin n \theta$
This formula is known as de Moivre's formula.

If we take arg( $z$ ) from the interval $(-\pi,\pi)$ , the relationship between a complex number $z$ and its argument is single-valued; that is, every nonzero complex number has precisely one angle in $(-\pi,\pi)$ . But there is nothing special about the interval $(-\pi,\pi)$ ; we also establish a single-valued relationship by using the interval $(0,2\pi)$ to dfin the principal value of the argument of $z$ . For the interval $(-\pi,\pi)$ , the negative real axis in analogous to a barrier that we agree not to cross; the technical name for this barrier is a branch cut.

Annulus The set $S_1$ of points satisfying the inequality $\rho_1<|z-z_0|$ lie exterior to the circle of radius $\rho_1$ centered at $z_0$ , whereas the set $S_2$ of points satisfying $|z-z_0|<\rho_2$ lie interior to the circle of radius $\rho_2$ centered at $z_0$ . Thus if $0<\rho_1<\rho_2$ , the set of points satisfying the simultaneous inequality
$\rho_1<|z-z_0|<\rho_2$
is the intersection of the sets $S_1$ and $S_2$ . This intersection is an open circular ring centered at $z_0$ . Figure 1.19(d) illustrates such a ring centered at the origin. The set defined by $(2)$ is called an open circular annulus. By allowing $\rho_1=0$ , we obtain a deleted neighborhood of $z_0$ .

Application of complex analysis in the context of electrical engineering

Consider the problem of finding the steady-state current $i_p(t)$ in an LRC-series circuit in which the charge $q(t)$ on the capacitor for time $t>0$ is described by the differential equation
$L\frac{d^2q}{dt^2}+R\frac{dq}{dt}+\frac{1}{C}q=E_0\sin \gamma t$
Assuming a solution of the form $q_p(t)=A\sin\gamma t+B\cos\gamma t$ . Then we are left with
$A= E_0X/(-\gamma Z^2),\quad B=E_0 R/(-\gamma Z^2)$
where the quantities
$X = L\gamma-1/C\gamma, \quad Z = \sqrt{X^2+R^2}$
They are called the reactance and impedance of the circuit, respectively. Thus the solution is
$q_p(t) = - \frac{-E_0 X}{\gamma Z^2}\sin\gamma t- \frac{E_0R}{\gamma Z^2} \cos \gamma t$
From this solution and $i_p = q'_p(t)$ , we obtain the steady-state current
$i_p(t) = \frac{E_0}{Z} \bigg( \frac{R}{Z} \sin \gamma t- \frac{X}{Z}\cos \gamma t \bigg)$
To avoid confusion with the current i, an electrical engineer will denote the imaginary unit $i$ by the symbol $j$ . Then replace $E_0\sin \gamma t$ with ${\rm Im}(E_0 e^{j\gamma t})$ . Then we try the solution $i_p(t)={\rm Im}(A e^{j\gamma t})$ . Then we substitute it into the original equation and obtain the result
$(jL\gamma+R+1/jC\gamma)A = E_0$
From it, we obtain
$A = \frac{E_0}{R+j\bigg( L\gamma - \frac{1}{C\gamma} \bigg)} = \frac{E_0}{R+jX}$
The denominator of the last expression is called the complex impedance of the circuit $Z_c = R+jX$ .

Then the steady-state current is given by
$i_p(t) = {\rm Im} \bigg( \frac{E_0}{Z} e^{-j\theta}e^{j\gamma t} \bigg)$

Complex Functions and Mappings

Example: Image of a Vertical Line under $w=z^2$ .

Image of a Vertical Line under $w=z^2$

Example: Image of a Vertical Line under $w=z^2$ .

Image of a Horizontal Line under $w=z^2$.

Principal Square Root Function $z^{1/2}$
The function $z^{1/2}$ defined by
$z^{1/2} = \sqrt{|z|} e^{i {\rm Arg}(z)/2}$
is called the Principal square root function.

Reciprocal FunctionThe function $1/z$ , whose domain is the set of all nonzero complex numbers, is called the reciprocal function.

Example: Mapping of a semi-infinite strip. Find the image of the semi-infinite horizontal strip defined by $1\leq y\leq2$ . $x\geq 0$ under $w=1/2$ .

Reciprical map

Analytic Functions

Analytic Functions Even though the requirement of differentiability is a stringent demand, there is a class of functions that is of great importance whose members satisfy even more severe requirements. These functions are called analytic functions.

Analyticity at a Point A complex function $w=f(z)$ is said to be analytic at a point $z_0$ if $f$ is differentiable at $z_0$ and at every point in some neighborhood of $z_0$ .

A function $f$ is analytic in a domain D if it is analytic at every point in $D$ . A function $f$ that is analytic throughout a domain $D$ is called holomorphic or regular.

Analyticity at a point is not the same as differentiability at a point. Analyticity at a point is a neighborhood property. Analyticity is a property that is defined over an open set. For example $f(z) = |z|^2$ is differentiable at $z=0$ but is not differentiable anywhere else. Even though $f(z)=|z|^2$ is differentiable at $z=0$ , it is not analytic at that point because there exists no neighborhood of $z=0$ throughout which $f$ is differentiable. Hence the function $f(z) = |z|^2$ is nowhere analytic.

Entire Functions A function that is analytic at every point $z$ in the complex plane is said to be an entire function.

Cauchy-Riemann Equations Suppose $f(z)=u(x,y)+iv(x,y)$ is differentiable at a point $z=x+iy$ . Then at $z$ the first-order partial derivatives of $u$ and $v$ exist and satisfy the Cauchy-Riemann equations
$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}~, \quad \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x} ~.$