# Complex Variables

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## 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 $f(x,y)=u(x,y)+iv(x,y)\,$ 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 ${\frac {z^{{2n}}}{(1+z)^{n}}},n\in {\mathbb {Z^{+}}}\,$

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 $\cot ^{2}z\,$

solution $\cot ^{3}z\,$

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 ${\frac {e^{{iz}}}{{\sqrt {z}}}}\,$

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

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

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

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

## 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 $\int _{C}(xy+ix^{2})dz,C=z(t)=t+it,0\leq t\leq 1\,$.

solution Evaluate $\int _{C}(xy+ix^{2})dz,C=z(t)=t+it^{2},0\leq t\leq 1\,$.

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

solution Evaluate $\int _{C}\sin zdz,C\,$ 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 $\int _{0}^{\infty }{\frac {x^{\alpha }}{1+2x\cos \phi +x^{2}}}\ dx={\frac {\pi }{\sin(\pi \alpha )}}{\frac {\sin(\phi \alpha )}{\sin \phi }},-1<\alpha <1,\alpha \phi \neq 0,-\pi <\phi <\pi$.

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.

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 $\int _{0}^{\infty }\left\{\prod _{{k=1}}^{n}{\frac {\sin(\phi _{k}x)}{x}}\right\}\left\{\prod _{{j=1}}^{m}\cos(a_{j}x)\right\}{\frac {\sin(ax)}{x}}\ dx={\frac {\pi }{2}}\phi _{1}\phi _{2}\ldots \phi _{n}$,   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 $|z_{1}z_{2}|=|z_{1}||z_{2}|\,$.

solution Show that $|z_{1}+z_{2}|\leq |z_{1}|+|z_{2}|\,$.

solution Show that $|z_{1}+z_{2}+z_{3}|\leq |z_{1}|+|z_{2}|+|z_{3}|\,$.

solution Show that $\overline {z_{1}z_{2}}=\overline {z_{1}}\,\overline {z_{2}}\,$

## 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 $z_{1}=r_{1}(\cos \theta _{1}+i\sin \theta _{1})\,$ and $z_{2}=r_{2}(\cos \theta _{2}+i\sin \theta _{2})\,$ then:
$z_{1}z_{2}=r_{1}r_{2}\left[\cos(\theta _{1}+\theta _{2})+i\sin(\theta _{1}+\theta _{2})\right]\,$
${\frac {z_{1}}{z_{2}}}={\frac {r_{1}}{r_{2}}}\left[\cos(\theta _{1}-\theta _{2})+i\sin(\theta _{1}-\theta _{2})\right]\,$
Prove it by induction.

• $\ln(re^{{i\varphi }})=\ln r+i(\varphi +2k\pi )$
• For every complex number $z\neq 0\,$ 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 $z=0\,$.