偏角の微分

$$\begin{array}{rcl}z& =& x+iy\dots z\in \mathbb{Z},x,y\in \mathbb{R}\\ & =& r{e}^{i\theta }\dots r,\theta \in \mathbb{R},r\ge 0,-\pi <\theta \le \pi \\ r& =& |z|=\sqrt{x^2+y^2}\\ \arg \left(z\right)& =& \theta +2n\pi (n\in \mathbb{Z})\\ & =& \mathrm{Arg}\left(z\right)+2n\pi (n\in \mathbb{Z})\end{array}$$ $$\begin{array}{rcl}\mathrm{Arg}\left(z\right)& =& \mathrm{Arg}(x+iy)\\ & =& \{\begin{array}{ll}{\tan }^{-1}\left(\frac{y}{x}\right)& (x>0)\\ {\tan }^{-1}\left(\frac{y}{x}\right)+\pi & (x<0\mathrm{か}\mathrm{つ}y\ge 0)\\ {\tan }^{-1}\left(\frac{y}{x}\right)-\pi & (x<0\mathrm{か}\mathrm{つ}y<0)\\ \frac{\pi }{2}& (x=0\mathrm{か}\mathrm{つ}y>0)\\ -\frac{\pi }{2}& (x=0\mathrm{か}\mathrm{つ}y<0)\\ \mathrm{不}\mathrm{定}& (x=0\mathrm{か}\mathrm{つ}y=0)\end{array}\end{array}$$

xでの偏微分

$$\begin{array}{rcl}\frac{\partial }{\partial x}\arg \left(z\right)& =& \frac{\partial }{\partial x}\{\theta +2n\pi \}\\ & =& \frac{\partial }{\partial x}\{\mathrm{Arg}\left(z\right)+2n\pi \}\\ & =& \frac{\partial }{\partial x}\mathrm{Arg}\left(z\right)+\frac{\partial }{\partial x}2n\pi \\ & =& \frac{\partial }{\partial x}\mathrm{Arg}\left(z\right)\dots \frac{\partial }{\partial x}C=0(C\mathrm{は}\mathrm{定}\mathrm{数}．x\mathrm{の}\mathrm{凾}\mathrm{数}\mathrm{で}\mathrm{な}\mathrm{い})\end{array}$$ $$\begin{array}{rcl}\frac{\partial }{\partial x}\mathrm{Arg}\left(z\right)& =& \frac{\partial }{\partial x}\mathrm{Arg}(x+iy)\\ & =& \{\begin{array}{llll}\frac{\partial }{\partial x}{\tan }^{-1}\left(\frac{y}{x}\right)& =& \frac{-y}{x^2+y^2}& (x>0)\\ \frac{\partial }{\partial x}{\tan }^{-1}\left(\frac{y}{x}\right)+\pi & =& \frac{-y}{x^2+y^2}& (x<0\mathrm{か}\mathrm{つ}y\ge 0)\\ \frac{\partial }{\partial x}{\tan }^{-1}\left(\frac{y}{x}\right)-\pi & =& \frac{-y}{x^2+y^2}& (x<0\mathrm{か}\mathrm{つ}y<0)\\ \frac{\partial }{\partial x}\frac{\pi }{2}& =& 0& (x=0\mathrm{か}\mathrm{つ}y>0)\\ \frac{\partial }{\partial x}-\frac{\pi }{2}& =& 0& (x=0\mathrm{か}\mathrm{つ}y<0)\\ \mathrm{不}\mathrm{定}& & & (x=0\mathrm{か}\mathrm{つ}y=0)\end{array}\dots \frac{\partial }{\partial x}{\tan }^{-1}\left(\frac{y}{x}\right)=\frac{-y}{x^2+y^2}\end{array}$$

yでの偏微分

$$\begin{array}{rcl}\frac{\partial }{\partial y}\arg \left(z\right)& =& \frac{\partial }{\partial y}\{\theta +2n\pi \}\\ & =& \frac{\partial }{\partial y}\{\mathrm{Arg}\left(z\right)+2n\pi \}\\ & =& \frac{\partial }{\partial y}\mathrm{Arg}\left(z\right)+\frac{\partial }{\partial y}2n\pi \\ & =& \frac{\partial }{\partial y}\mathrm{Arg}\left(z\right)\dots \frac{\partial }{\partial y}C=0(C\mathrm{は}\mathrm{定}\mathrm{数}．x\mathrm{の}\mathrm{凾}\mathrm{数}\mathrm{で}\mathrm{な}\mathrm{い})\end{array}$$ $$\begin{array}{rcl}\frac{\partial }{\partial y}\mathrm{Arg}\left(z\right)& =& \frac{\partial }{\partial y}\mathrm{Arg}(x+iy)\\ & =& \{\begin{array}{llll}\frac{\partial }{\partial y}{\tan }^{-1}\left(\frac{y}{x}\right)& =& \frac{x}{x^2+y^2}& (x>0)\\ \frac{\partial }{\partial y}{\tan }^{-1}\left(\frac{y}{x}\right)+\pi & =& \frac{x}{x^2+y^2}& (x<0\mathrm{か}\mathrm{つ}y\ge 0)\\ \frac{\partial }{\partial y}{\tan }^{-1}\left(\frac{y}{x}\right)-\pi & =& \frac{x}{x^2+y^2}& (x<0\mathrm{か}\mathrm{つ}y<0)\\ \frac{\partial }{\partial y}\frac{\pi }{2}& =& 0& (x=0\mathrm{か}\mathrm{つ}y>0)\\ \frac{\partial }{\partial y}-\frac{\pi }{2}& =& 0& (x=0\mathrm{か}\mathrm{つ}y<0)\\ \mathrm{不}\mathrm{定}& & & (x=0\mathrm{か}\mathrm{つ}y=0)\end{array}\dots \frac{\partial }{\partial y}{\tan }^{-1}\left(\frac{y}{x}\right)=\frac{x}{x^2+y^2}\end{array}$$