Using the Residue Theorem to Evaluate Contour Integrals

Question

I am at an impass when it comes to this problem, where I must take the following function:

$$f(z) = \frac{e^{2z}}{4\cosh(z)-5}$$

and evaluate the contour integral of it around a circle of radius $\frac{3}{2}$ (centered at the origin) using the residue theorem. I am struggling mainly to find any other poles or singular points that are within the circle, as it is already given that $\ln(2)$ is a pole. I thought maybe I need to convert this into complex form, such that

$$\ln(2) = \ln(2) +i(2n\pi)$$

But any pole here other than $\ln(2)$ would be outside of the circle, yet the solution in the book suggests that there are in fact more singular points inside the circle that should be taken account of.

Please help, I am a physics major and numbers without diagrams or real world analogies are downright terrifying to me. Thanks!

Answer 1

Hint: The solutions to $$4\cosh z -5=0$$ are $$z=\pm\ln 2 +2k\pi i$$ for integral $k$, if that helps.