Day: December 31, 2022

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202212311431 Problem 1.4

Let g:\Omega\to\mathbb{R} be such that \displaystyle{\frac{\partial g}{\partial x}} and \displaystyle{\frac{\partial g}{\partial y}} exist at (x_0,y_0)\in\Omega, and suppose that one of these partials exists in a neighbourhood of (x_0,y_0) and is continuous at (x_0,y_0). Show that g is real-differentiable at (x_0,y_0).

Extracted from R. B. Ash & W. P. Novinger. (2004). Complex Variables.

Roughwork.

Granted that

\begin{aligned} w=f(z)&=u(x,y)+iv(x,y) \\ \textrm{s.t. }A & =u_0+iv_0 \\ z_0 & =x_0+iy_0 \\ \end{aligned}

the concepts of existence of limit

\begin{aligned} &\qquad\enspace \lim_{z\to z_0}f(z) = A \\ & \Longleftrightarrow \begin{cases} \lim_{(x,y)\to (x_0,y_0)}u(x,y)=u_0 \\ \lim_{(x,y)\to (x_0,y_0)}v(x,y)=v_0 \\ \end{cases} \textrm{exist} \end{aligned}

and continuity at a point

\begin{aligned} &\qquad\enspace w=f(z)\textrm{ continuous at }z_0 \\ &\Longleftrightarrow \textrm{ both }u(x,y)\textrm{ and }v(x,y)\textrm{ continuous at }z_0 \\ \end{aligned}

are held by definition.

This problem is not to be attempted.

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202212311359 Problem 1.3

Let z_1 and z_2 be nonzero complex numbers, and let \theta (0\leqslant \theta\leqslant \pi) be the angle between them. Show that

(a) \mathrm{Re}z_1\overline{z}_2=|z_1||z_2|\cos\theta, \mathrm{Im}z_1\overline{z}_2=\pm |z_1||z_2|\sin\theta, and consequently
(b) The area of the triangle formed by z_1, z_2, and z_2-z_1 is |\mathrm{Im}z_1\overline{z}_2|/2.

Extracted from R. B. Ash & W. P. Novinger. (2004). Complex Variables.

Roughwork.

The area of a triangle \triangle ABC, constructed by any two sides \mathbf{AB} and \mathbf{AC} with an included angle \theta, is

\begin{aligned} \textrm{Area} & = \frac{1}{2}\mathbf{AB}\times \mathbf{AC} \\ \bigg( & =\frac{1}{2}(AB)(AC)\sin\theta\bigg) \\ \end{aligned}

i.e., half the area of the parallelogram spanned by these two vectors.

Hence, \mathtt{(a)} \Longrightarrow \mathtt{(b)}.

This problem is not to be attempted.

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202212311329 Problem 1.2

Show that |z_1+z_2|=|z_1|+|z_2| iff z_1 and z_2 lie on a common ray from 0 iff one of z_1 or z_2 is a nonnegative multiple of the other.

Extracted from R. B. Ash & W. P. Novinger. (2004). Complex Variables.

Roughwork.

Let z_1=r_1e^{i\theta_{1}} and z_2=r_2e^{i\theta_{2}}, rewrite

\begin{aligned} |r_1e^{i\theta_{1}}+r_2e^{i\theta_{2}}| & = |r_1e^{i\theta_{1}}|+|r_2e^{i\theta_{2}}| \\ ? = r_3 & = r_1 + r_2 \\ \end{aligned}

by Lemma.

\begin{aligned} r_3e^{i\theta_{3}} & = r_1e^{i\theta_{1}}+r_2e^{i\theta_{2}} \\ & = r_1(\cos\theta_1+i\sin\theta_1) + r_2(\cos\theta_2+i\sin\theta_2) \\ & = (r_1\cos\theta_1+r_2\cos\theta_2) + i(r_1\sin\theta_1+r_2\sin\theta_2) \\ r_3^2 & = r_1^2+r_2^2+2r_1r_2\cos (\theta_1-\theta_2) \\ r_3 & = \sqrt{r_1^2+r_2^2+2r_1r_2\cos (\theta_1-\theta_2)} \\ \end{aligned}

we have

\begin{aligned} \cos (\theta_1-\theta_2) & = 1 \\ \theta_1 - \theta_2 & = 0 \\ \theta_1 & = \theta_2 \\ \end{aligned}

The rest is left an exercise for the reader.