HW 3: Complex Numbers

Question-1¶

Given $z = x +4iy$ find:

• $Re(z)$ and $Im(z)$ (real and imainary components of z)

• $Re(z^{*})$ and $Im(z^{*})$ (real and imainary components of z cojugate)

• $Re(z^2)$ and $Im(z^2)$

• $Re(z \cdot z^{*})$ and $Im(z \cdot z^{*})$

Question-2¶

Convert the following complex numbers from polar $z=re^{i\phi}$ to cartesian representation $z=x+iy$

• $z=e^{i\pi/4}$

• $z=e^{-2\pi i}+e^{4\pi i}$

Question-3¶

Show that for functions $\Phi_m(\phi) = \frac{1}{\sqrt{2\pi}} e^{im\phi}$%2520%253D%2520%255Cfrac%257B1%257D%257B%255Csqrt%257B2%255Cpi%257D%257D%2520e%255E%257Bim%255Cphi%257D?scale=1) with an integer parameter (e.g. $m=0,1,2,3...$) one has the following relations

$\int^{2\pi}_0 \Phi^{*}_m (\phi) \Phi_n (\phi) d\phi = 0$%2520%255CPhi_n%2520(%255Cphi)%2520d%255Cphi%2520%253D%25200?scale=1) When $n\neq m$

$\int^{2\pi}_0 \Phi^{*}_m (\phi) \Phi_m (\phi) d\phi = 1$%2520%255CPhi_m%2520(%255Cphi)%2520d%255Cphi%2520%253D%25201?scale=1) When $n=m$

Question-4¶

Using Euler’s relation for complex numbers, show that the following equality holds when n≠m.

Question-5¶

Show that sine and cosine functions can be expressed in terms of complex exponentials as follows:

• $cos\phi=\frac{e^{i\phi}+e^{-i\phi}}{2}$

• $sin\phi=\frac{e^{i\phi}-e^{-i\phi}}{2i}$