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Complex Analysis

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} ~.

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