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<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_{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\,.

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