What are all the values of $\cosh(\log2)$? This is problem IV.9.1 in Complex Function Theory by Sarason.
I know that $$ \cosh(z)=\frac{\exp(z)+\exp(-z)}{2}. $$ I also know that $$ \log(2)=\ln2 + 2\pi n i $$ with $n$ an integer. Thus $$ \cosh(\log(2))=\frac{\exp(\ln2 + 2\pi n i)+\exp(-\ln2 - 2\pi n i)}{2}\\ =\frac{2 \exp(2\pi n i)+(1/2)\exp( -2\pi n i)}{2}\\ =\frac{2 + 1/2}{2}=\frac{5}{4} $$ I feel as though I am missing something because there should be a multiplicity of such values.