HW 3: Complex Numbers

Question-1¶

Given $z = x + 4iy$ find:

• $\operatorname{Re}(z)$ and $\operatorname{Im}(z)$ (real and imaginary parts of $z$)

• $\operatorname{Re}(z^{*})$ and $\operatorname{Im}(z^{*})$ (real and imaginary parts of the conjugate $z^{*}$)

• $\operatorname{Re}(z^2)$ and $\operatorname{Im}(z^2)$

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

Question-2¶

Convert the following complex numbers from polar form $z = r e^{i\phi}$ to Cartesian form $z = x + iy$:

• $z = e^{i\pi/4}$

• $z = e^{-2\pi i} + e^{4\pi i}$

Question-3¶

Show that the functions $\Phi_m(\phi) = \frac{1}{\sqrt{2\pi}}\, e^{im\phi}$ with an integer parameter (e.g. $m = 0, 1, 2, 3, \ldots$) satisfy the following relations:

Question-4¶

Using Euler’s relation for complex numbers, show that the following equality holds when $n \neq m$:

Question-5¶

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

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

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