Practice problems in Complex Numbers

 Compute the magnitude and phase of $(1-j) \bigg(\frac{1}{2} + j \frac{\sqrt{3}}{2}\bigg)$ Compute the magnitude and phase of $e^{j\pi/2} (1+j) (1+3j)$ Compute the magnitude and phase of $j e^{j \pi/3}$ Compute the magnitude and phase of $e^{j \pi/4} + e^{j 3 \pi/4}$ Compute the magnitude and phase of $(1+3j)^2$ Compute the magnitude and phase of $(1-3j)/(1+3j)^2$ Compute the magnitude and phase of $e^{j \pi/5} \times e^{j 2 \pi/5} \times e^{j 3 \pi/5} \ldots e^{j 9 \pi/5}$ Compute the magnitude and phase of $e^{j \pi/5} \times e^{j 2 \pi/5} \times e^{j 3 \pi/5} \ldots e^{j 9 \pi/5} \times e^{j 10 \pi/5}$ Let $z_1 = 1, z_2 = -\frac{1}{2} + j \frac{\sqrt{3}}{2}, z_3 = -\frac{1}{2} - j \frac{\sqrt{3}}{2}$ [a)] What are $z_1^3, z_2^3$ and $z_3^3$? [b)] Show that $z_3 = z_2^2$ [c)] Show that $z_1 + z_2 + z_3 = 0$ Can you now see why $z_1,z_2,z_3$ can be called the cube roots of unity. They are usually expressed as $1, \omega, \omega^2$. Part c shows that the sum of the cube roots of unity is zero. In one of the homework problems, we will show that this true for $n$th roots of unity for any $n$. Let $z_1 = 2 e^{j\pi/4}$ and $z_2 = 8 e^{j\pi/3}$. Find and express your answer in Cartesian and polar form [a)] $2z_1-z_2$ [b)] $\frac{1}{z_1}$ [c)] $\frac{z_1}{z_2^2}$ [d)] $\sqrt{z_2}$ Prove that $\int e^{ax} \ \cos(bx) \ dx = \frac{e^{ax}}{a^2+b^2} \left(a \cos(bx) + b \sin(bx) \right)$ You can use integration tricks you learned in your calculus class to solve this problem. That is not the point of the exercise. Try using Euler's identity and then using integration of exponentials to see if you can solve the problem.
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