I am asked to find the singularities of the following complex-valued function:
$$f(z) = \frac{\sin(2x)}{\cosh(2y)-\cos(2x)}+i\frac{\sinh(2y)}{cos(2x)-cosh(2y)}$$
One idea I had was to somehow write everything in terms of $z$, but I could not figure out how to do that.
Best Answer
Write your function
$$f(z) = \frac{\sin (2x) - i\sinh (2y)}{\cosh (2y) - \cos (2x)} = \frac{\sin (2x) - \sin (2iy)}{\cosh (2y) - \cos (2x)} = \frac{\sin (2x) - \sin (2iy)}{\cos (2iy) - \cos (2x)}.$$
Now use the identities
$$\sin A - \sin B = 2 \cos \frac{A+B}{2}\sin\frac{A-B}{2}$$
and
$$\cos A - \cos B = -2\sin\frac{A+B}{2}\sin\frac{A-B}{2}$$
to simplify and obtain an expression of $f$ that is easier to analyse.