Complex Variables

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Problems

solution Evaluate i^{{243}}\,

solution If Failed to parse (unknown error):

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 Failed to parse (unknown error):

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 Failed to parse (unknown error):

solution Evaluate Failed to parse (unknown error):

solution If Failed to parse (unknown error): , then what is Failed to parse (unknown error): ?

solution If Failed to parse (unknown error): , then what is Failed to parse (unknown error): ?

solution If Failed to parse (unknown error): , then what is Failed to parse (unknown error): ?

solution If Failed to parse (unknown error):

find Failed to parse (unknown error): 

.

solution If Failed to parse (unknown error):

find Failed to parse (unknown error): 

.

solution If Failed to parse (unknown error):

find Failed to parse (unknown error): 

.

solution Evaluate Failed to parse (unknown error):

solution Evaluate Failed to parse (unknown error):

solution List all the cube roots of Failed to parse (unknown error):

solution List all the cube roots of Failed to parse (unknown error):

solution If Failed to parse (unknown error):

find Failed to parse (unknown error): 
and check if it satisfies the Cauchy-Riemann equations.

Differentiation

solution Show that Failed to parse (unknown error):

is non-analytic everywhere.

solution Find Failed to parse (unknown error): .

solution Show that Failed to parse (unknown error):

solution Show that Failed to parse (unknown error):

solution Show that if Failed to parse (unknown error):

is harmonic then Failed to parse (unknown error): 
is analytic.

Polynomials

solution Find the partial fraction decomposition of Failed to parse (unknown error): .

solution Deflate Failed to parse (unknown error):

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 Failed to parse (unknown error):

at Failed to parse (unknown error): 

.

solution Write the Taylor expansion of Failed to parse (unknown error):

at Failed to parse (unknown error): 

.

solution Write the Taylor expansion of Failed to parse (unknown error):

at Failed to parse (unknown error): 

.

Trigonometric Functions

solution Verify the identity: Failed to parse (unknown error):


solution Verify the identity: Failed to parse (unknown error):


solution Verify the identity: Failed to parse (unknown error):


solution Verify the identity: Failed to parse (unknown error):


solution Verify the identity: Failed to parse (unknown error):


Exponential and Log

This site uses Failed to parse (unknown error):

and Failed to parse (unknown error): 
for the principal values.

solution Evaluate Failed to parse (unknown error):


solution Find the domain of analyticity for Failed to parse (unknown error):


solution Evaluate Failed to parse (unknown error):


solution Find where Failed to parse (unknown error):


solution Evaluate Failed to parse (unknown error):


solution Evaluate Failed to parse (unknown error):


solution Evaluate Failed to parse (unknown error):

on the principal branch.

solution Evaluate Failed to parse (unknown error):

on the principal branch.

solution Evaluate Failed to parse (unknown error):

on the principal branch.

solution Solve Failed to parse (unknown error):


Find harmonic functions on certain domains

solution Find a function that is harmonic on the washer-shaped region between the circles Failed to parse (unknown error):

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 Failed to parse (unknown error):

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 Failed to parse (unknown error): 1 to 2 and equals 20 and 30 at Failed to parse (unknown error): 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 Failed to parse (unknown error): . 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 Failed to parse (unknown error):

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 Failed to parse (unknown error): .

solution Find the Maclaurin series for Failed to parse (unknown error): .

solution Find the Laurent series for Failed to parse (unknown error):

about all its singular points.

solution Find the Laurent series about Failed to parse (unknown error):

for the function Failed to parse (unknown error): 


Residues

Find the residues of Failed to parse (unknown error):

at all its isolated singular points and at infinity (if infinity is not a limit point of singular points), where Failed to parse (unknown error): 
is given by

solution Failed to parse (unknown error):


solution Failed to parse (unknown error):


solution Failed to parse (unknown error):


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 Failed to parse (unknown error):


solution Failed to parse (unknown error):


solution Failed to parse (unknown error):


solution Failed to parse (unknown error):


solution Failed to parse (unknown error):


solution Failed to parse (unknown error):


solution Failed to parse (unknown error):


solution Failed to parse (unknown error):


solution Failed to parse (unknown error):


solution Failed to parse (unknown error):


solution Failed to parse (unknown error):


solution Failed to parse (unknown error):


Complex Integrals

solution Give an upper bound for Failed to parse (unknown error):


solution Compute Failed to parse (unknown error):

along the directed line segment from Failed to parse (unknown error): 
to

Failed to parse (unknown error): .

solution Evaluate Failed to parse (unknown error):

where Failed to parse (unknown error): 
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 Failed to parse (unknown error):

to 1.

solution Evaluate Failed to parse (unknown error):

starts at the origin, traverses the bottom half of a unit circle centered at Failed to parse (unknown error): 
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 Failed to parse (unknown error):


solution Failed to parse (unknown error):


solution Failed to parse (unknown error):


solution Failed to parse (unknown error):


solution Failed to parse (unknown error):


solution Failed to parse (unknown error):


Residue Calculus

solution Evaluate Failed to parse (unknown error):


solution Evaluate Failed to parse (unknown error):


solution Evaluate Failed to parse (unknown error):


solution Evaluate Failed to parse (unknown error):


solution Prove that Failed to parse (unknown error): .


solution A function Failed to parse (unknown error):

is zero when Failed to parse (unknown error): 

, and is real when Failed to parse (unknown error):

is real, and is analytic when Failed to parse (unknown error): . If Failed to parse (unknown error):

is the imaginary part of Failed to parse (unknown error): 
prove that Failed to parse (unknown error): 
holds when Failed to parse (unknown error): 

.


solution Prove that Failed to parse (unknown error): .   Hint: Integrate Failed to parse (unknown error):

over the semicircle contour in the upper half plane, then put Failed to parse (unknown error): 

.


solution Show that Failed to parse (unknown error): ,   if Failed to parse (unknown error):

are real, Failed to parse (unknown error): 
is positive and Failed to parse (unknown error): 

.


solution Evaluate Failed to parse (unknown error):


Proofs

solution Show that Failed to parse (unknown error): .

solution Show that Failed to parse (unknown error): .

solution Show that Failed to parse (unknown error): .

solution Show that Failed to parse (unknown error): .

solution Show that Failed to parse (unknown error):

Facts

  • The Failed to parse (unknown error):
roots of a complex number written in polar form Failed to parse (unknown error): 
are

Failed to parse (unknown error):

De Moivre's Theorem
If Failed to parse (unknown error): and Failed to parse (unknown error): then:
Failed to parse (unknown error):


Failed to parse (unknown error): 



Prove it by induction.
  • Failed to parse (unknown error):


  • For every complex number Failed to parse (unknown error):
and any positive integer Failed to parse (unknown error): 

, it is true that
Failed to parse (unknown error):

  • 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 Failed to parse (unknown error):
is continuous at Failed to parse (unknown error): 

, then it must be true that Failed to parse (unknown error): .

  • The function Failed to parse (unknown error):
is one-to-one and continuous everywhere on the complex plane except at Failed to parse (unknown error): 

.

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