# Complex Variables

## Contents

## Problems

solution Evaluate

solution If Find

solution

solution

solution Show that the four points in the Argand plane represented by the complex numbers are the vertices of a square

solution Find the equation of the straight line joining the points

in the Argand plane

solution Evaluate .

solution Evaluate

solution Evaluate

solution If , then what is ?

solution If , then what is ?

solution If , then what is ?

solution If find .

solution If find .

solution If find .

solution Evaluate

solution Evaluate

solution List all the cube roots of

solution List all the cube roots of

solution If find and check if it satisfies the Cauchy-Riemann equations.

## Differentiation

solution Show that is non-analytic everywhere.

solution Find .

solution Show that

solution Show that

solution Show that if is harmonic then is analytic.

## Polynomials

solution Find the partial fraction decomposition of .

solution Deflate

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 at .

solution Write the Taylor expansion of at .

solution Write the Taylor expansion of at .

## Trigonometric Functions

solution Verify the identity:

solution Verify the identity:

solution Verify the identity:

solution Verify the identity:

solution Verify the identity:

## Exponential and Log

This site uses and for the principal values.

solution Evaluate

solution Find the domain of analyticity for

solution Evaluate

solution Find where

solution Evaluate

solution Evaluate

solution Evaluate on the principal branch.

solution Evaluate on the principal branch.

solution Evaluate on the principal branch.

solution Solve

## Find harmonic functions on certain domains

solution Find a function that is harmonic on the washer-shaped region between the circles 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 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 1 to 2 and equals 20 and 30 at 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 . 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 that takes the values -50 and 10 on the lower an upper lines.

## Series

solution Find the Laurent series for about the singular point .

solution Find the Maclaurin series for .

solution Find the Laurent series for about all its singular points.

solution Find the Laurent series about for the function

## Residues

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

## Complex Integrals

solution Give an upper bound for

solution Compute along the directed line segment from to .

solution Evaluate where is the circle traversed twice in the clockwise direction.

solution Evaluate .

solution Evaluate .

solution Give an upper bound for is the line segment from to 1.

solution Evaluate starts at the origin, traverses the bottom half of a unit circle centered at and then the line from to .

## Contour Integrals

## Residue Calculus

solution Evaluate

solution Evaluate

solution Evaluate

solution Evaluate

solution Prove that .

solution A function is zero when , and is real when is real, and is analytic when . If is the imaginary part of prove that holds when .

solution Prove that . **Hint:** Integrate over the semicircle contour in the upper half plane, then put .

solution Show that , if are real, is positive and .

solution Evaluate

## Proofs

solution Show that .

solution Show that .

solution Show that .

solution Show that .

solution Show that

## Facts

- The roots of a complex number written in polar form are

De Moivre's Theorem

If and then:

Prove it by induction.

- For every complex number and any positive integer , it is true that

- 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 is continuous at , then it must be true that .

- The function is one-to-one and continuous everywhere on the complex plane except at .