Cauchy-Riemann-Differential equation

Introduction

In the following lesson, we first make an identification of the complex numbers $\mathbb {C}$ with the two-dimensional $\mathbb {R}$ -vector space $\mathbb {R} ^{2}$ , then we consider the classical real partial derivatives and the Jacobian matrix, and investigate the relationship between complex differentiability and partial derivatives of component functions of a map from $\mathbb {R} ^{2}$ to $\mathbb {R} ^{2}$ . After that, the Cauchy-Riemann differential equations are proven based on these preliminary considerations.

Identification of Complex Numbers with $\mathbb {R} ^{2}$

Let $R:\mathbb {C} \rightarrow \mathbb {R} ^{2},\ x+iy\mapsto R(x+iy)={\begin{pmatrix}x\\y\end{pmatrix}}$ . Since the mapping $R$ is bijective, the inverse mapping : $R^{-1}:\mathbb {R} ^{2}\rightarrow \mathbb {C} ,\ {\begin{pmatrix}x\\y\end{pmatrix}}\mapsto R^{-1}{\begin{pmatrix}x\\y\end{pmatrix}}=x+iy$ maps vectors from $\mathbb {R} ^{2}$ one-to-one back to a complex number.

Real and Imaginary Part Functions

Now, if we decompose a function $f:U\rightarrow \mathbb {C}$ with $f\left(x+iy\right)=u\left(x,y\right)+i,v\left(x,y\right)$ into its real and imaginary parts with real functions $u:U_{R}\rightarrow \mathbb {R}$ , $v:U_{R}\rightarrow \mathbb {R}$ where $U_{R}\subset \mathbb {R} ^{2}$ and $U={x+iy\in \mathbb {C} \ |\ (x,y)\in U_{R}}$ , then the total derivative of the function $f_{R}:U_{R}\rightarrow \mathbb {R} ^{2},(x,y)\mapsto {\begin{pmatrix}u\left(x,y\right)\ v\left(x,y\right)\end{pmatrix}}$ has the following Jacobian matrix as its representation: ${\begin{pmatrix}{\frac {\partial u}{\partial x}}&{\frac {\partial u}{\partial y}}\\{\frac {\partial v}{\partial x}}&{\frac {\partial v}{\partial y}}\end{pmatrix}}.$

Task

For the complex-valued function $f:\mathbb {C} \rightarrow \mathbb {C} ,\ z\mapsto f(z)=z^{3}$ , give the mappings $u,v$ with $f\left(x+iy\right)=u\left(x,y\right)+i\,v\left(x,y\right)$ explicitly.

Evaluation of the Jacobian Matrix at a Point

The evaluation of the Jacobian matrix at a point $(x_{o},y_{o})\in \mathbb {R} ^{2}$ gives the total derivative at the point $x_{o}+iy_{o}\in \mathbb {C}$ : ${\begin{pmatrix}{\frac {\partial u}{\partial x}}(x_{o},y_{o})&{\frac {\partial u}{\partial y}}(x_{o},y_{o})\ {\frac {\partial v}{\partial x}}(x_{o},y_{o})&{\frac {\partial v}{\partial y}}(x_{o},y_{o})\end{pmatrix}}$

Cauchy-Riemann Differential Equations

A function $f$ is complex differentiable at $z_{o}:=x_{o}+iy_{o}$ if and only if it is real differentiable and the Cauchy-Riemann differential equations hold for $u,v$ with $u:U_{R}\rightarrow \mathbb {R}$ , $v:U_{R}\rightarrow \mathbb {R}$ where $U_{R}\subset \mathbb {R} ^{2}$ : : ${\frac {\partial u}{\partial x}}(x_{o},y_{o})={\frac {\partial v}{\partial y}}(x_{o},y_{o})$ : $\displaystyle {\frac {\partial u}{\partial y}}(x_{o},y_{o})=-{\frac {\partial v}{\partial x}}(x_{o},y_{o})$ are satisfied.

Relationship Between the Partial Derivatives

In the following explanations, the definition of differentiability in $\mathbb {C}$ to properties of the partial derivatives in the Jacobian matrix.

Part 1

If the following limit exists for $f:G\rightarrow \mathbb {C}$ at $z_{o}\in G$ with $G\subset \mathbb {C}$ open: : $f'(z_{o})=\lim _{z\rightarrow z_{o}}{\frac {f(z)-f(z_{o})}{z-z_{o}}}$ , then for any sequences $(z_{n}){n\in \mathbb {N} }$ in the domain $G\subset \mathbb {C}$ with $\lim {n\rightarrow \infty }z_{n}=z_{o}$ , we also have: : $f'(z_{o})=\lim _{n\rightarrow \infty }{\frac {f(z_{n})-f(z_{o})}{z_{n}-z_{o}}}$

Part 2

Now consider only the sequences for the two following limit processes with $h\in \mathbb {R}$ : : $f'(z_{o})=\lim _{h\rightarrow 0}{\frac {f(z_{o}+h)-f(z_{o})}{(z_{o}+h)-z_{o}}}=\lim _{h\rightarrow 0}{\frac {f(z_{o}+h)-f(z_{o})}{h}}$ , : $f'(z_{o})=\lim _{ih\rightarrow 0}{\frac {f(z_{o}+ih)-f(z_{o})}{(z_{o}+ih)-z_{o}}}=\lim _{ih\rightarrow 0}{\frac {f(z_{o}+ih)-f(z_{o})}{ih}}$ ,

Part 3: Limit Process for Real Part

By inserting the component functions for the real and imaginary parts $u,v$ , we get with $h\in \mathbb {R}$ : : $f'(z_{o})=\lim _{h\rightarrow 0}{\frac {f(z_{o}+h)-f(z_{o})}{h}}=$ :: $=\lim _{h\rightarrow 0}{\frac {u(x_{o}+h,y_{o})-u(x_{o},y_{o})}{h}}+i\lim _{h\rightarrow 0}{\frac {v(x_{o}+h,y_{o})-v(x_{o},y_{o})}{h}}$ :: $={\frac {\partial u}{\partial x}}(x_{o},y_{o})+i{\frac {\partial v}{\partial x}}(x_{o},y_{o})$

Part 4: Limit Process for Imaginary Part

Applying this to the second equation, we get with $h\in \mathbb {R}$ : : $f'(z_{o})=\lim _{ih\rightarrow 0}{\frac {f(z_{o}+ih)-f(z_{o})}{ih}}$ :: $=\lim _{h\rightarrow 0}{\frac {u(x_{o},y_{o}+h)-u(x_{o},y_{o})}{ih}}+i\lim _{h\rightarrow 0}{\frac {v(x_{o},y_{o}+h)-v(x_{o},y_{o})}{ih}}$ :: $=-i\lim _{h\rightarrow 0}{\frac {u(x_{o},y_{o}+h)-u(x_{o},y_{o})}{h}}+\lim _{h\rightarrow 0}{\frac {v(x_{o},y_{o}+h)-v(x_{o},y_{o})}{h}}$ ,

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

Remark on Part 4

In the first summand, the fraction is extended by $i$ , and in the second summand $i$ , the is canceled so that the denominator becomes real-valued and $h$ corresponds.

Part 5: Comparison of Real and Imaginary Parts

By equating the terms from (3) and (4) and comparing the real and imaginary parts, we obtain the Cauchy-Riemann differential equations.

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

Part 6: Partial Derivative in the Direction of the Real Part

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

{\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 Derivative in the Direction of the Imaginary Part

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

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

,

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

,

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

.

Part 8: Cauchy-Riemann DGL with Functions in $\mathbb {C}$

The partial derivatives of the Cauchy-Riemann differential equations can also be expressed in $\mathbb {C}$ with $f:=f_{x}+if_{y}$ , $f_{x}:={\mathfrak {Re}}(f)$ , $f_{y}:={\mathfrak {Im}}(f)$ : Real part: ${\frac {\partial {\mathfrak {Re}}(f)}{\partial x}}(z_{o})={\frac {\partial {\mathfrak {Im}}(f)}{\partial y}}(z_{o})$ Imaginary part: $\displaystyle {\frac {\partial {\mathfrak {Re}}(f)}{\partial y}}(z_{o})=-{\frac {\partial {\mathfrak {Im}}(f)}{\partial x}}(z_{o})$

Theorem - Cauchy-Riemann DGL

Let ${\textstyle G\subseteq \mathbb {C} }$ be an open subset. The function ${\textstyle f=u+iv}$ is complex differentiable at a point ${\textstyle z=x+iy\in G}$ . Then, the partial derivatives of ${\textstyle u}$ and ${\textstyle v}$ exist at ${\textstyle \left(x,y\right)\in \mathbb {R} ^{2}}$ , and the following Cauchy-Riemann differential equations hold: ${\dfrac {\partial u}{\partial x}}\left(x,y\right)={\dfrac {\partial v}{\partial y}}\left(x,y\right)$

${\dfrac {\partial u}{\partial y}}\left(x,y\right)=-{\dfrac {\partial v}{\partial x}}\left(x,y\right)$

Remark on CR-DGL

In this case, the derivative of ${\textstyle f}$ at the point ${\textstyle z\in \mathbb {C} }$ can be represented in two ways using the component functions ${\textstyle u}$ and ${\textstyle v}$ : $f'\left(z\right)={\dfrac {\partial u}{\partial x}}\left(x,y\right)-i{\dfrac {\partial u}{\partial y}}\left(x,y\right)={\dfrac {\partial v}{\partial y}}\left(x,y\right)+i{\dfrac {\partial v}{\partial x}}\left(x,y\right)$ The proof of the Cauchy-Riemann differential equations uses a comparison of the real and imaginary parts to derive the above equations.

Proof

The proof considers two directional derivatives:

(DG1) the derivative in the direction of the real part and

(DG2) the derivative in the direction of the imaginary part.

Since these coincide for complex differentiability, the Cauchy-Riemann differential equations are obtained by setting them equal and comparing the real and imaginary parts.

Step 1 - Derivative in the Direction of the Real Part

In the first step, let $h\in \mathbb {C}$ converge to 0 in the direction of the real part. To achieve this, choose ${\textstyle h:=h_{1}+i\cdot 0}$ with ${\textstyle h_{1}\in \mathbb {R} }$ . The decomposition of the function ${\textstyle f=u+iv}$ into its real part $u$ and imaginary part $v$ then yields (DG1).

Step 2 - Calculation of the Derivative - Real Part

${\begin{array}{rcl}f'\left(z\right)&=&\lim \limits _{h\to 0}{\dfrac {f\left(z+h\right)-f\left(z\right)}{h}}\\&=&\lim \limits _{h_{1}\to 0}{\dfrac {u\left(x+h_{1},y\right)+iv\left(x+h_{1},y\right)-u\left(x,y\right)-iv\left(x,y\right)}{h_{1}}}\\&=&\lim \limits _{h_{1}\to 0}{\dfrac {u\left(x+h_{1},y\right)-u\left(x,y\right)}{h_{1}}}+i{\dfrac {v\left(x+h_{1},y\right)-v\left(x,y\right)}{h_{1}}}\\&=&{\dfrac {\partial u}{\partial x}}\left(x,y\right)+i{\dfrac {\partial v}{\partial x}}\left(x,y\right)\end{array}}$

Step 3 - Derivative in the Direction of the Imaginary Part

Similarly, the partial derivative for the imaginary part can be considered with ${\textstyle h:=0+i\cdot h_{2}}$ and ${\textstyle h_{2}\in \mathbb {R} }$ . This yields equation (DG2).

Step 4 - Calculation of the Derivative - Imaginary Part

${\textstyle {\begin{array}{rcl}f'\left(z\right)&=&\lim \limits _{h\to 0}{\dfrac {f\left(z+h\right)-f\left(z\right)}{h}}\\&=&\lim \limits _{h_{2}\to 0}{\dfrac {u\left(x,y+h_{2}\right)+iv\left(x,y+h_{2}\right)-u\left(x,y\right)-iv\left(x,y\right)}{i\cdot h_{2}}}\\&=&\lim \limits _{l\to 0}{\dfrac {1}{i}}{\dfrac {u\left(x,y+h_{2}\right)-u\left(x,y\right)}{h_{2}}}+{\dfrac {v\left(x,y+h_{2}\right)-v\left(x,y\right)}{h_{2}}}\\&=&{\dfrac {\partial v}{\partial y}}\left(x,y\right)-i{\dfrac {\partial u}{\partial y}}\left(x,y\right)\end{array}}}$

Step 5 - Equating the Derivatives

By equating the two derivatives, one can compare the real and imaginary parts of the two derivatives (DG1) and (DG2): $f'\left(z\right)={\dfrac {\partial u}{\partial x}}\left(x,y\right)+i{\dfrac {\partial v}{\partial x}}\left(x,y\right)={\dfrac {\partial v}{\partial y}}\left(x,y\right)-i{\dfrac {\partial u}{\partial y}}\left(x,y\right)$

Step 6 - Comparison of Real and Imaginary Parts

Two complex numbers are equal if and only if their real and imaginary parts are equal. This results in the Cauchy-Riemann differential equations. The two representation formulas follow from the above equation and the Cauchy-Riemann equations.

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Date: 12/26/2024

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