log(sin(πz))の微分

$\log (\sin \left(\pi z\right))$の微分

$$\begin{array}{rcl}\frac{\mathrm{d}}{\mathrm{d}z}\log (\sin \left(\pi z\right))& =& \frac{\mathrm{d}}{\mathrm{d}f}\log (\sin \left(f\right))\frac{\mathrm{d}f}{\mathrm{d}z}\dots f=\pi z,f\in \mathbb{C}\\ & =& \frac{\mathrm{d}}{\mathrm{d}g}\log \left(g\right)\frac{\mathrm{d}g}{\mathrm{d}f}\frac{\mathrm{d}f}{\mathrm{d}z}\dots g=\sin \left(f\right),g\in \mathbb{C}\\ & =& \frac{1}{g}\cos \left(f\right)\pi \\ & & \dots \frac{\mathrm{d}}{\mathrm{d}g}\log \left(g\right)=\frac{1}{g}\\ & & \dots \frac{\mathrm{d}}{\mathrm{d}f}\sin \left(f\right)=\cos \left(f\right)\\ & & \dots \frac{\mathrm{d}}{\mathrm{d}z}\pi z=\pi \\ & =& \frac{1}{\sin \left(f\right)}\cos \left(f\right)\pi \\ & =& \frac{1}{\sin \left(\pi z\right)}\cos \left(\pi z\right)\pi \\ & =& \pi \frac{\cos \left(\pi z\right)}{\sin \left(\pi z\right)}\\ & =& \pi \frac{1}{\tan \left(\pi z\right)}\dots \tan \left(\pi z\right)=\frac{\sin \left(\pi z\right)}{\cos \left(\pi z\right)}\\ & =& \pi \cot \left(\pi z\right)\end{array}$$