Complex Variables

From Exampleproblems

Jump to: navigation, search

Contents


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 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_1z_2 = r_1r_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 \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\,.

Main Page

Personal tools

Flash!
A Free Fun Game!
For Android 4.0

Get A Wifi Network
Switcher Widget for
Android