How to separate out real and imaginary terms

Question

I have an equation like this: $$a+ib = \log(x+iy).$$

I need to separate the real and imaginary part in RHS so that I can equate the real part of LHS to real part of RHS and imaginary to imaginary part of RHS.

Answer 1

You could convert the complex number from algebraic to polar form $r\cdot e^{j\theta} $ where $r=\sqrt{x^2+y^2} $ and $\theta=\arctan\left(\frac{y}{x}\right) $

After that, you can split the $\log$ function into real and imaginary parts:

$$ \log(r\cdot e^{j\theta}) =\log(r) +j\theta\log(e) $$ $$ = \log(r) +j\theta $$ (assuming natural logarithm)

Answer 2

Note your equation is equivalent to

$$ e^{a+jb}=a+jb\\ e^ae^{jb}=a+jb\\ e^a(\cos(b)+j\sin(b))=a+jb $$ Then real and imaginary parts have been separated.