Example Problem Set 1

Many important problems in mathematical modeling and scientific computing require the use of complex variables. Assuming that the quantities a, b, c, d are all reals and $i = \sqrt{-1}$, resolve the following expressions, or explain the following operations, involving complex variables z₁ = a + i b and z₂ = c + i d:

1.

Let z₃ = z₁z₂. What is the real part of z₃, and what is its imaginary part?

2.

What is $\Vert{z}_{1}\Vert$, the modulus of z₁, and what is $\Vert{z}_{3}\Vert$, the modulus of z₃=z₁z₂?

3.

What is $\angle {z}_{2}$, the angle of complex variable z₂?

4.

Express z₁ in polar complex form, not using the quantities a or b but rather the modulus $\Vert{z}_{1}\Vert$and angle $\angle {z}_{1}$.

5.

Suppose that z₁ and z₂ both have a modulus of 1. Explain how their product z₃ = z₁z₂ amounts to a rotation in the complex plane, with the aid of a diagram. Why is the multiplication of these complex variables reduced now to addition? Without using the quantities a, b, c, d, what is the value of $\Vert{z}_{3}\Vert$?

6.

Suppose that in polar complex form, ${z} = e^{2\pi i/5}$. What do you get if z is multiplied by itself 5 times? Give the simplest possible answer that you can.

7.

Consider the complex exponential function $f(x) = e^{2\pi i \omega x}$. What function is its real part? What function is its imaginary part?

8.

If the above function f(x) passes through a linear system, i.e. is operated upon by any conceivable linear differential or integral operator, what is the most dramatic way in which f(x) can possibly be affected?