Divide in real and imaginary part

Question

I am stuck at a question and the following function needs to be divided in a real part and an imaginary part like: $z = a + bi$. I tried to solve this problem by using partial fraction, but that's not working out.

The function is: $$z = \frac{1}{e-i \pi}$$

The result is also given: $$z = \frac{e}{e^2+ \pi^2} + \frac{i \pi}{e^2 + \pi^2}$$

Answer 1

What you do in these cases is multiply and divide by the complex conjugate, so that you get: $$z=\frac{e+i\pi}{e^2+\pi^2}$$ and now you can easily find the real and imaginary parts.

Answer 2

$$\frac{1}{e - i \pi} = \frac{1}{e - i \pi} \frac{e + i \pi}{e + i \pi} = \frac{ e + i \pi }{e^2 + \pi^2 } = \frac{e}{e^2+ \pi^2} + \frac{i \pi}{e^2 + \pi^2}$$

Answer 3

Multiply the top and the bottom of $$ \frac{1}{e-i \pi}$$ by ${e +i \pi}$ to get $$ \frac{e+i\pi }{e^2+ \pi ^2}$$

Now split the real part and the imaginary parts to get $$z= \frac{e+i\pi }{e^2+ \pi ^2}= \frac {e}{e^2+\pi ^2}+ i \frac {\pi }{ e^2+\pi ^2} $$