Complex Analysis/Cauchy-Riemann equations

The total function ${\textstyle f:U\rightarrow \mathbb {C} }$ with ${\textstyle f\left(x+iy\right)=u\left(x,y\right)+i\,v\left(x,y\right)}$ in its real and imaginary parts with real functions ${\textstyle u:U_{R}\rightarrow \mathbb {R} }$,

$${\displaystyle {\begin{pmatrix}{\frac {\partial u}{\partial x}}&{\frac {\partial u}{\partial y}}\\{\frac {\partial v}{\partial x}}&{\frac {\partial v}{\partial y}}\end{pmatrix}}.}$$

The evaluation of the Jacobi matrix in one point ${\textstyle (x_{o},y_{o})\in \mathbb {R} ^{2}}$ provides total derivation in the point ${\textstyle x_{o}+iy_{o}\in \mathbb {C} }$

$${\displaystyle {\begin{pmatrix}{\frac {\partial u}{\partial x}}(x_{o},y_{o})&{\frac {\partial u}{\partial y}}(x_{o},y_{o})\\{\frac {\partial v}{\partial x}}(x_{o},y_{o})&{\frac {\partial v}{\partial y}}(x_{o},y_{o})\end{pmatrix}}}$$

A function ${\textstyle f}$ is complexly differentiable in ${\textstyle z_{o}:=x_{o}+iy_{o}}$ when it can be differentiated relatively and for ${\textstyle u,v}$ with ${\textstyle u:U_{R}\rightarrow \mathbb {R} }$,

$${\displaystyle {\frac {\partial u}{\partial x}}(x_{o},y_{o})={\frac {\partial v}{\partial y}}(x_{o},y_{o})}$$

$${\displaystyle \displaystyle {\frac {\partial u}{\partial y}}(x_{o},y_{o})=-{\frac {\partial v}{\partial x}}(x_{o},y_{o})}$$

are fulfilled.

If the following Limes exists for ${\textstyle f:G\rightarrow \mathbb {C} }$ for ${\textstyle z_{o}\in G}$ with ${\textstyle G\subset \mathbb {C} }$

$${\displaystyle f'(z_{o})=\lim _{z\rightarrow z_{o}}{\frac {f(z)-f(z_{o})}{z-z_{o}}}}$$,

means ${\textstyle \lim _{z\rightarrow z_{o}}...}$ that for any consequences ${\textstyle (z_{n})_{n\in \mathbb {N} }}$ definition range ${\textstyle G\subset \mathbb {C} }$ with ${\textstyle \lim _{n\rightarrow \infty }z_{n}=z_{o}}$ also

$${\displaystyle f'(z_{o})=\lim _{n\rightarrow \infty }{\frac {f(z_{n})-f(z_{o})}{z_{n}-z_{o}}}}$$

is fulfilled.

From these arbitrary consequences, one considers only the consequences for the two following limit processes with ${\textstyle h\in \mathbb {R} }$:

$${\displaystyle f'(z_{o})=\lim _{h\rightarrow 0}{\frac {f(z_{o}+h)-f(z_{o})}{(z_{o}+h)-z_{o}}}=\lim _{h\rightarrow 0}{\frac {f(z_{o}+h)-f(z_{o})}{h}}}$$,

$${\displaystyle f'(z_{o})=\lim _{ih\rightarrow 0}{\frac {f(z_{o}+ih)-f(z_{o})}{(z_{o}+ih)-z_{o}}}=\lim _{ih\rightarrow 0}{\frac {f(z_{o}+ih)-f(z_{o})}{ih}}}$$;

By inserting the component functions for the real part and imaginary part ${\textstyle u,v}$, the result is ${\textstyle h\in \mathbb {R} }$

$${\displaystyle f'(z_{o})=\lim _{h\rightarrow 0}{\frac {f(z_{o}+h)-f(z_{o})}{h}}=}$$

$${\displaystyle =\lim _{h\rightarrow 0}{\frac {u(x_{o}+h,y_{o})-u(x_{o},y_{o})}{h}}+i\lim _{h\rightarrow 0}{\frac {v(x_{o}+h,y_{o})-v(x_{o},y_{o})}{h}}}$$

$${\displaystyle ={\frac {\partial u}{\partial x}}(x_{o},y_{o})+i{\frac {\partial v}{\partial x}}(x_{o},y_{o})}$$

When applied to the second equation, ${\textstyle h\in \mathbb {R} }$

$${\displaystyle f'(z_{o})=\lim _{ih\rightarrow 0}{\frac {f(z_{o}+ih)-f(z_{o})}{ih}}}$$

$${\displaystyle =\lim _{h\rightarrow 0}{\frac {u(x_{o},y_{o}+h)-u(x_{o},y_{o})}{ih}}+i\lim _{h\rightarrow 0}{\frac {v(x_{o},y_{o}+h)-v(x_{o},y_{o})}{ih}}}$$

$${\displaystyle =-i\lim _{h\rightarrow 0}{\frac {u(x_{o},y_{o}+h)-u(x_{o},y_{o})}{h}}+\lim _{h\rightarrow 0}{\frac {v(x_{o},y_{o}+h)-v(x_{o},y_{o})}{h}}}$$,

$${\displaystyle =-i{\frac {\partial u}{\partial y}}(x_{o},y_{o})+{\frac {\partial v}{\partial y}}(x_{o},y_{o})}$$

The Cauchy Riemann differential equations are obtained by equation of the terms of (3) and (4) and comparison of the real part and the imaginary part.

• Real part: ${\textstyle {\frac {\partial u}{\partial x}}(x_{o},y_{o})={\frac {\partial v}{\partial y}}(x_{o},y_{o})}$
• Imaginary part: ${\textstyle \displaystyle {\frac {\partial u}{\partial y}}(x_{o},y_{o})=-{\frac {\partial v}{\partial x}}(x_{o},y_{o})}$

The partial derivations in ${\textstyle \mathbb {R} ^{2}}$ of the Cauchy-Riemann differential equations can also be shown in ${\textstyle \mathbb {C} }$ with ${\textstyle f:={\mathfrak {Re}}(f)+i{\mathfrak {Im}}(f)}$, ${\textstyle {\mathfrak {Re}}(f):\mathbb {C} \rightarrow \mathbb {R} }$,

$${\displaystyle {\frac {\partial f}{\partial x}}(z_{o})=\lim _{h\rightarrow 0}{\frac {f(z_{o}+h)-f(z_{o})}{h}}\in \mathbb {C} }$$,

$${\displaystyle {\frac {\partial {\mathfrak {Re}}(f)}{\partial x}}(z_{o})=\lim _{h\rightarrow 0}{\frac {{\mathfrak {Re}}(f)(z_{o}+h)-{\mathfrak {Re}}(f)(z_{o})}{h}}\in \mathbb {R} }$$,

$${\displaystyle {\frac {\partial {\mathfrak {Im}}(f)}{\partial x}}(z_{o})=\lim _{h\rightarrow 0}{\frac {{\mathfrak {Im}}(f)(z_{o}+h)-{\mathfrak {Im}}(f)(z_{o})}{h}}\in \mathbb {R} }$$,

The partial discharges in ${\textstyle \mathbb {R} ^{2}}$ of the Cauchy-Riemann differential equations can also be shown in ${\textstyle \mathbb {C} }$ with ${\textstyle f:={\mathfrak {Re}}(f)+i{\mathfrak {Im}}(f)}$ and ${\textstyle {\mathfrak {Re}}(f):\mathbb {C} \rightarrow \mathbb {R} }$, ${\textstyle {\mathfrak {Im}}(f):\mathbb {C} \rightarrow \mathbb {R} }$

$${\displaystyle {\frac {\partial f}{\partial y}}(z_{o})=\lim _{h\rightarrow 0}{\frac {f(z_{o}+ih)-f(z_{o})}{h}}\in \mathbb {C} }$$,