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Let us consider the function:

$$f(s)=1/(iπ)*(log(h(s)/g(s)))$$

where $h$ and $g$ are complex functions such that the $log$ function is well defined and $i²=-1$.

My question is: How one can find the real and the imaginary parts of $f$ in term of the the real and the imaginary parts of $g$ and $h$.

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Firstly use: $$ \log(h(s)/g(s))= $$ $$ \log|\frac{h(s)}{g(s)}|+i\arg(\frac{h(s)}{g(s)}) $$ By definition of the principle branch of the complex logarithm. $\frac{1}{\pi i}$ can be expressed as $\frac{-i}{\pi}$ by standard division of complex numbers.

Now $|\frac{h(s)}{g(s)}|$ and $\arg(\frac{h(s)}{g(s)})$ can be expressed in terms of the real and imaginary terms of both functions, provided g is nonzero.

Using all of these facts you should be able to express real and imaginary parts of $f$ in terms of real and imaginary parts of $g,h$.

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