Complex Analysis/Cauchy-Riemann equations

Introduction

In the following learning unit, an identification of the complex numbers ${\textstyle \mathbb {C} }$ with the two-dimensional ${\textstyle \mathbb {R} }$ vector space ${\textstyle \mathbb {R} ^{2}}$ is first performed and the classical real partial derivatives and the Jacobi matrix are considered and a relationship between complex differentiation and partial derivation of 4320 The Cauchy-Riemann differential equations are then proved with the preliminary considerations.

Identification of the complex numbers IR ²

Be ${\textstyle R:\mathbb {C} \rightarrow \mathbb {R} ^{2},\ x+iy\mapsto R(x+iy)={\begin{pmatrix}x\\y\end{pmatrix}}}$. Since the image ${\textstyle R}$ is bijective, you can use the reverse image Once again, vectors from ${\textstyle \mathbb {R} ^{2}}$ assign a complex number.

Real part and imaginary part function

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 ${\textstyle u:U_{R}\rightarrow \mathbb {R} }$,

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

Task

Specify the images ${\textstyle f:\mathbb {C} \rightarrow \mathbb {C} ,\ z\mapsto f(z)=z^{3}}$ for complex function ${\textstyle u,v}$ with ${\textstyle f\left(x+iy\right)=u\left(x,y\right)+i\,v\left(x,y\right)}$.

Evaluation of the Jacobimatrix in one point

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

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

Cauchy-Riemann differential equations

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 ${\textstyle u:U_{R}\rightarrow \mathbb {R} }$,

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

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

are fulfilled.

Relationship between the partial discharges

In the following explanations, the definition of the differentiation in ${\textstyle \mathbb {C} }$ is attributed to properties of the partial derivatives in the Jacobi matrix.

Part 1

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

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

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

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

is fulfilled.

Part 2

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

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

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

Part 3: Limitation process Real part

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

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

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

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

Part 4: Limit for Imaginary part

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

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

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

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

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

Part 5: Real part and imaginary part comparison

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

Part 6: Partial derivation towards real part

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

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

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

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

Part 7: Partial derivation towards the imaginary part

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 ${\textstyle {\mathfrak {Re}}(f):\mathbb {C} \rightarrow \mathbb {R} }$, ${\textstyle {\mathfrak {Im}}(f):\mathbb {C} \rightarrow \mathbb {R} }$

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