Solve f^iv(x)=cosh(x)

$$f ^ { i v } ( x ) = \cosh ( x )$$

Solve for f

$$f=\left(\frac{|\cosh(x)|}{|x|}\right)^{-\frac{Im(v)+iRe(v)}{\left(Re(v)\right)^{2}+\left(Im(v)\right)^{2}}}e^{\frac{Re(v)arg(\frac{\cosh(x)}{x})-iIm(v)arg(\frac{\cosh(x)}{x})}{\left(Re(v)\right)^{2}+\left(Im(v)\right)^{2}}+\frac{i\times 2\pi n_{1}Im(v)}{\left(Re(v)\right)^{2}+\left(Im(v)\right)^{2}}-\frac{2\pi n_{1}Re(v)}{\left(Re(v)\right)^{2}+\left(Im(v)\right)^{2}}}$$

$$n_{1}\in \mathrm{Z}$$

$$x\neq 0$$

Solve for v

$$\left\{\begin{matrix}v=-i\log_{f}\left(\frac{\cosh(x)}{x}\right)+\frac{2\pi n_{1}}{\ln(f)}\text{, }n_{1}\in \mathrm{Z}\text{, }\nexists n_{1}\in \mathrm{Z}\text{ : }x=2\pi n_{1}i+\frac{3\pi i}{2}\text{, }&\nexists n_{2}\in \mathrm{Z}\text{ : }x=2\pi n_{2}i+\frac{\pi i}{2}\text{ and }x\neq 0\text{ and }f\neq 1\text{ and }f\neq 0\\v\in \mathrm{C}\text{, }&\left(f=0\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }x=2\pi n_{1}i+\frac{3\pi i}{2}\right)\text{ or }\left(f=0\text{ and }\exists n_{2}\in \mathrm{Z}\text{ : }x=2\pi n_{2}i+\frac{\pi i}{2}\right)\text{ or }\left(f=1\text{ and }\frac{\cosh(x)}{x}=1\text{ and }x\neq 0\right)\end{matrix}\right.$$