Examples and References

### Examples

1. Let $z_1 = 2 e^{j\pi/4}$ and $z_2 = 8 e^{j\pi/3}$. Find

a) $2z_1-z_2$
b) $\frac{1}{z_1}$
c) $\frac{z_1}{z_2^2}$
d) $\sqrt[3]{z_2}$

2. What is $j^j$?

3. Let $z$ be any complex number. Is it true that $(e^z)^\star= e^{z^\star}$?

4. Plot the magnitude and phase of the function $X(f) = e^{j\pi f}+e^{j 3 \pi f}$, for $-1 \leq f \leq 1$.

5. Prove that

$\int e^{ax} \ \cos(bx) \ dx = \frac{e^{ax}}{a^2+b^2} \left(a \cos(bx) + b \sin(bx) \right)$

### References

A good online reference for complex numbers is the wiki page http://en.wikipedia.org/wiki/Complex_number.