Not sure if I have done this correctly, seems too straight forward, any help is very appreciated.
QUESTION:
Find the real and imaginary parts of $f(z) = \cos(z)$.
ATTEMPT:
$\cos(z) = \cos(x+iy) = \cos x\cos(iy) − \sin x\sin(iy) =
\cos x\cosh y − i\sin x\sinh y$
Is that correct?
Best Answer
By definition, $$ \cos z=\frac{e^{iz}+e^{-iz}}{2},\qquad \sin z=\frac{e^{iz}-e^{-iz}}{2i} $$ In particular, for real $y$, $$ \cos(iy)=\frac{e^{-y}+e^{y}}{2}=\cosh y $$ and $$ \sin(iy)=\frac{e^{-y}-e^{y}}{2i}=i\frac{e^{y}-e^{-y}}{2}=i\sinh y $$
So, yes, you're correct.