Complex Variables

From Exampleproblems

1 Problems
2 Differentiation
3 Polynomials
4 Trigonometric Functions
5 Exponential and Log
6 Find harmonic functions on certain domains
7 Series
8 Residues
9 Complex Integrals
10 Contour Integrals
11 Residue Calculus
12 Proofs
13 Facts

Problems

solution Evaluate $i^{243} \,$

solution If $z=5+(3\sqrt{2})i\,$ Find $z\bar{z} \,$

solution $\sqrt{7+24i} \,$

solution $\sqrt{-8-6i} \,$

solution Show that the four points in the Argand plane represented by the complex numbers are the vertices of a square $2+i,4+3i,2+5i,3i\,$

solution Find the equation of the straight line joining the points $(-9+6i),(11-4i)\,$

in the Argand plane

solution Evaluate $i^{23} \,$ .

solution Evaluate $(1+i) \cdot (7-i)\,$

solution Evaluate $\frac{1+i}{7-i}\,$

solution If $arg(z)=\frac{\pi}{20}\,$ , then what is $arg(-z)\,$ ?

solution If $arg(z)=\frac{\pi}{20}\,$ , then what is $arg(\bar{z})\,$ ?

solution If $arg(z)=\frac{\pi}{20}\,$ , then what is $arg(z^3)\,$ ?

solution If $z_1=2+i, z_2=3-2i\,$ find $|3z_1-4z_2|\,$ .

solution If $z_1=2+i\,$ find $z_1^3-3z_1^2+4z_1-8\,$ .

solution If $z_3=-\frac{1}{2}+\frac{\sqrt{3}}{2}i\,$ find $\left(\bar{z_3}\right)^4\,$ .

solution Evaluate $\int_0^{2\pi}\cos^8\theta\,d\theta\,$

solution Evaluate $\mathrm{Re}\left[(a+bi)^p\right]\,$

solution List all the cube roots of $\sqrt{2}+i\sqrt{2}\,$

solution List all the cube roots of $-1+i\,$

solution If $u(x,y) = e^x\sin y\,$ find $f(x,y) = u(x,y) + i v(x,y)\,$ and check if it satisfies the Cauchy-Riemann equations.

Differentiation

solution Show that $\frac{d}{dz}\overline{z}\,$ is non-analytic everywhere.

solution Find $\frac{d}{dz}\left[\frac{1+z}{1-z}\right]\,$ .

solution Show that $\lim_{z\to 0} \frac{\sin z}{z} = 1\,$

solution Show that $\lim_{z\to 0} \frac{\cos z - 1}{z} = 0\,$

solution Show that if $\phi(x,y)\,$ is harmonic then $\phi_x - i \phi_y\,$ is analytic.

Polynomials

solution Find the partial fraction decomposition of $R(z)=\frac{4z+4}{z(z-1)(z-2)^2}\,$ .

solution Deflate $z^3+(2-i)z^2-2iz\,$

solution Show that a polynomial with real coefficients can always be expressed as a product of linear and quadratic factors with real coefficients.

solution Write the Taylor expansion of $f(z)=z^5+3z+4\,$ at $z=2\,$ .

solution Write the Taylor expansion of $f(z)=z^{10}\,$ at $z=2\,$ .

solution Write the Taylor expansion of $f(z)=(z-1)(z-2)^3\,$ at $z=2\,$ .

Trigonometric Functions

solution Verify the identity: $\sin(a \pm b) = \sin(a)\cos(b)\pm\cos(a)\sin(b)\,$

solution Verify the identity: $\sin(iz) = i\sinh(z)\,$

solution Verify the identity: $\sinh(iz) = i\sin(z)\,$

solution Verify the identity: $\cos(iz) = \cosh(z)\,$

solution Verify the identity: $\cosh(iz) = \cos(z)\,$

Exponential and Log

This site uses $\mathrm{Log}\,$ and $\mathrm{Arg}\,$ for the principal values.

solution Evaluate $\sum_{k=0}^{100} e^{kz}\,$

solution Find the domain of analyticity for $\mathrm{Log}(3z-i)\,$

solution Evaluate $\mathrm{Log}(z^2-1)\,$

solution Find where $\sin z = \cos z\,$

solution Evaluate $\mathrm{Log}(i)\,$

solution Evaluate $\mathrm{Log}(-1-i)\,$

solution Evaluate $(-1)^{2i}\,$ on the principal branch.

solution Evaluate $(-i)^i\,$ on the principal branch.

solution Evaluate $(1+i)^i\,$ on the principal branch.

solution Solve $e^z = (1+i)/\sqrt{2}\,$

Find harmonic functions on certain domains

solution Find a function that is harmonic on the washer-shaped region between the circles $|z|=1, |z|=2\,$ and takes the values 20 and 30 on the inner and outer circles.

solution Find a function that is harmonic on the wedge-shaped region between the rays in the complex plane with principal argument $3\pi/4, 5\pi/4\,$ and takes the values 20 and 30 on rays with the smaller and larger angles.

solution Find a function that is harmonic on the vertical strip from $x =$ 1 to 2 and equals 20 and 30 at $x =$ 1 and 2.

solution Find a function that is harmonic on the washer-shaped region between the circles with radii 1 and 2 and center $(1, i)$ . It should be 0 and 10 on the inner and outer circle.

solution Find a function that is harmonic on the strip between the lines $y=-x+3, y=-x-3\,$ that takes the values -50 and 10 on the lower an upper lines.

Series

solution Find the Laurent series for $f(z) = z^2 e^{1/z}\,$ about the singular point $z=0\,$ .

solution Find the Maclaurin series for $\tan^{-1}z\,$ .

solution Find the Laurent series for $\frac{1}{z^2(z-1)}\,$ about all its singular points.

solution Find the Laurent series about $z_0=1\,$ for the function $\frac{e^{2z}}{(z-1)^3}\,$

Residues

Find the residues of $f(z)\,$ at all its isolated singular points and at infinity (if infinity is not a limit point of singular points), where $f(z)\,$ is given by

solution $\frac{1}{z^3-z^5}\,$

solution $\frac{1}{z(1-z^2)}\,$

solution $\frac{z^{2n}}{(1+z)^n}, n \isin \mathbb{Z^+}\,$

solution $\frac{z^2}{(z^2+1)^2}\,$

solution $\frac{\sin 2z}{(z+1)^3}\,$

solution $\frac{e^z}{z^2(z^2+9)}\,$

solution $\cot^2z\,$

solution $\cot^3z\,$

solution $\cos\frac{1}{z-2}\,$

solution $z^3\cos\frac{1}{z-2}\,$

solution $e^{z+\frac{1}{z}}\,$

solution $\sin z \sin \frac{1}{z}\,$

solution $\frac{\cos z}{z^2(z-\pi)^3}\,$

solution $\frac{z^2+z-1}{z^2(z-1)}\,$

solution $\frac{\sin z}{z^2+4}\,$

solution $\frac{e^{i z}}{\sqrt{z}}\,$

solution $\frac{(\mathrm{Log } (z))^2}{1+z^2}\,$

solution $\frac{e^{tz}}{(z+2)^2}, t \isin \mathbb{R^+}\,$

solution $\frac{e^{tz}}{z \sinh (az)}, a \isin \mathbb{R}, t \isin \mathbb{R^+}\,$

solution $\frac{e^{tz} \sinh (a \sqrt{z})}{z\sinh \sqrt{z}}, a \isin \mathbb{R}, t \isin \mathbb{R^+}\,$

Complex Integrals

solution Give an upper bound for $\Bigg| \int_{|z|=3} \frac{dz}{z^2-i} \Bigg| \,$

solution Compute $\int_\Gamma \mbox{Re}\ z \ dz$ along the directed line segment from $z = 0\,$ to $z = 1 + 2i\,$ .

solution Evaluate $\int_\Gamma \frac{z}{(z+2)(z-1)}dz\,$ where $\Gamma\,$ is the circle $|z|=3\,$ traversed twice in the clockwise direction.

solution Evaluate $\int_C (xy+ix^2)dz, C=z(t)=t+it, 0\le t\le 1\,$ .

solution Evaluate $\int_C (xy+ix^2)dz, C=z(t)=t+it^2, 0\le t\le 1\,$ .

solution Give an upper bound for $\int_C \frac{dz}{z^4}, C\,$ is the line segment from $i\,$ to 1.

solution Evaluate $\int_C \sin z dz, C\,$ starts at the origin, traverses the bottom half of a unit circle centered at $z_0=1/2\,$ and then the line from $z=1\,$ to $z=i\pi\,$ .

Contour Integrals

solution $\oint_{|z|=2} \frac{1-2z}{z(z-1)(z-3)} dz\,$

solution $\oint_{|z|=5} ze^{3/z} dz \,$

solution $\oint_{|z|=5} \frac{\cos z}{z^2(z-\pi)^3}\,$

solution $\oint_{|z|=2} \frac{\sin z}{z-0}\,dz\,$

solution $\oint_{|z|=2} \frac{\cos z}{z-0}\,dz\,$

solution $\oint_{|z|=1} e^{1/z}\sin\frac{1}{z} dz\,$

solution $\oint_{|z|=2} \frac{dz}{z(z+1)^2(z+3)} dz\,$

Residue Calculus

solution Evaluate $\int_0^{2\pi} \frac{\sin^2\theta}{5+4\cos\theta}\,d\theta\,$

solution Evaluate $\int_0^{2\pi} \frac{d\theta}{5+4\sin\theta}\,$

solution Evaluate $\int_0^{2\pi} \frac{d\theta}{a^2\sin^2\theta + b^2\cos^2\theta}\,$

solution Evaluate $\int_0^\infty \frac{\sin x}{x}dx\,$

solution Prove that $\int_0^\infty \frac{x^\alpha}{1 + 2x\cos\phi + x^2}\ dx = \frac{\pi}{\sin(\pi\alpha)}\frac{\sin(\phi\alpha)}{\sin\phi}, -1 < \alpha < 1, \alpha\phi \ne 0, -\pi < \phi < \pi$ .

solution A function $\phi(z)\,$ is zero when $z = 0\,$ , and is real when $x\,$ is real, and is analytic when $|z| \leq 1\,$ . If $f(x, y)\,$ is the imaginary part of $\phi(x + iy)\,$ prove that $\int_0^{2\pi} \frac{x\sin\theta}{1 - 2x\cos\theta + x^2}f(\cos\theta, \sin\theta)\ d\theta = \pi\phi(x)$ holds when $-1 < x < 1\,$ .

solution Prove that $\int_0^{\pi/2} \frac{r\theta\sin(2\theta)}{1 - 2r\cos(2\theta) + r^2}\ d\theta = \begin{cases} \frac{\pi}{4}\ln(1 + r) & \mathrm{if}\ r^2 < 1\\ \frac{\pi}{4}\ln(1 + 1/r) & \mathrm{if}\ r^2 > 1 \end{cases}$ . Hint: Integrate $\frac{2zr}{z^2(1 + r)^2 + (1 - r)^2}\frac{\mathrm{Log}(1 - iz)}{1 + z^2}$ over the semicircle contour in the upper half plane, then put $x = \tan\theta\,$ .

solution Show that $\int_0^\infty \left\{\prod_{k = 1}^n \frac{\sin(\phi_k x)}{x}\right\}\left\{\prod_{j = 1}^m \cos(a_jx)\right\}\frac{\sin(ax)}{x}\ dx = \frac{\pi}{2}\phi_1\phi_2\ldots\phi_n$ , if $\phi_1, \phi_2, \ldots, \phi_n, a_1, a_2, \ldots, a_m$ are real, $a\,$ is positive and $a > \sum_{k = 1}^n |\phi_k| + \sum_{j = 1}^m |a_j|$ .

solution Evaluate $\int_0^\infty \frac{dx}{x^3+1} \,$

Proofs

solution Show that $\overline{z_1+z_2}=\overline{z_1}+\overline{z_2}\,$ .

solution Show that $|z_1z_2|=|z_1||z_2|\,$ .

solution Show that $|z_1+z_2|\le|z_1|+|z_2|\,$ .

solution Show that $|z_1+z_2+z_3|\le |z_1| + |z_2| + |z_3|\,$ .

solution Show that $\overline{z_1z_2}=\overline{z_1}\,\overline{z_2}\,$

Facts

The $\mathrm{m^{th}}\,$ roots of a complex number written in polar form $z=|z|e^{i\theta}\,$ are

$z^{\frac{1}{m}} = |z|^\frac{1}{m}\exp\left(\frac{i(\theta+2k\pi)}{m}\right), k=0,1,2,...,m-1\,$

De Moivre's Theorem

If  and  then:






Prove it by induction.

$\ln(re^{i\varphi})=\ln r + i (\varphi + 2k\pi)$

For every complex number $z \ne 0\,$ and any positive integer $n$ , it is true that
$|\bar{z}| = |z|, |(\bar{z})^n| = |z|^n, arg(z^n) = n\,arg(z)\,$

Every subset of the complex plane is compact if and only if it is closed and bounded.

The complement of an open set is closed and vice versa.

If $f(z)\,$ is continuous at $z=z_0\,$ , then it must be true that $\lim_{z\rightarrow z_0} f(z)=f(z_0)\,$ .

The function $w = \frac{1}{z}\,$ is one-to-one and continuous everywhere on the complex plane except at $z=0\,$ .

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