Differentiating a purely imaginary function

Question

If a function is defined as being the imaginary part of some expression, how do I take the derivative of the function? Do I:

(a) Take the imaginary part of the expression, and then differentiate?

(b) Differentiate the whole expression, and then take the imaginary part?

Or does it not make a difference?

Concrete example: For the function

$$ \Phi(x,y) = \mathrm{Im}\Big\{\frac{2}{\pi}\mathrm{ln} \space \mathrm{tanh}(x+iy)\Big\} $$

how do I evaluate $\nabla^2 \Phi(x,y)$?

Thanks!

Answer 1

Any complex function can be written $f(z) = p(z) + iq(z)$ for real-valued functions $p,q$ (not necessarily in a way you can actually write down, just theoretically). So $\mathrm{Im} (f'(z)) = \mathrm{Im}(p'(z) + iq'(z)) = q'(z)$ and ${d \over {dz}} \mathrm{Im}(f(z)) = {d\over dz} q(z) = q'(z)$

So yeah, either works.