How to find partial fraction of a complex expression

Question

I am learning complex analysis, and i am asked to find the series for $$f(z)=\frac{1}{(z-1)(z-2i)}$$ , it says that it can be done at the point z=1+i . It then follows in the book to write $$\frac{1}{(z-1)(z-2i)}=\frac{1+2i}{5}(\frac{1}{z-1}-\frac{1}{z-2i})$$ Can someone explain me how to do partial fractions with complex arguments like here? Thank you very much!

Answer 1

The same way as you would do it with real numbers:\begin{align}\frac1{(z-1)(z-2i)}&=\frac A{z-1}+\frac B{z-2i}\\&=\frac{A(z-2i)+B(z-1)}{(z-1)(z-2i)}\\&=\frac{(A+B)z-2iA-B}{(z-1)(z-2i)},\end{align}And now you solve the system$$\left\{\begin{array}{l}A+B=0\\-2iA-B=1.\end{array}\right.$$