Cauchy-Riemann-Differential equation

Introduction

In the following lesson, we first make an identification of the complex numbers $ℂ$ with the two-dimensional $ℝ$-vector space ${ℝ}^2$, 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 ${ℝ}^2$ to ${ℝ}^2$. After that, the Cauchy-Riemann differential equations are proven based on these preliminary considerations.

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

Let $R:ℂ\to {ℝ}^2,x+iy\mapsto R(x+iy)=\left(\begin{array}{c}x\\ y\end{array}\right)$. Since the mapping $R$ is bijective, the inverse mapping : $R^{-1}:{ℝ}^2\to ℂ,\left(\begin{array}{c}x\\ y\end{array}\right)\mapsto R^{-1}\left(\begin{array}{c}x\\ y\end{array}\right)=x+iy$ maps vectors from ${ℝ}^2$ one-to-one back to a complex number. File:Cauchy Riemann DGL audio0.ogg

Real and Imaginary Part Functions

Now, if we decompose a function $f:U\to ℂ$ with $f(x+iy)=u(x,y)+i,v(x,y)$ into its real and imaginary parts with real functions $u:U_R\to ℝ$, $v:U_R\to ℝ$ where $U_R\subset {ℝ}^2$ and $U=x+iy\in ℂ|(x,y)\in U_R$, then the total derivative of the function $f_R:U_R\to {ℝ}^2,(x,y)\mapsto \left(\begin{array}{c}u(x,y)v(x,y)\end{array}\right)$ has the following Jacobian matrix as its representation: $\left(\begin{array}{cc}\frac{\partial u}{\partial x}& \frac{\partial u}{\partial y}\\ \frac{\partial v}{\partial x}& \frac{\partial v}{\partial y}\end{array}\right).$ File:Cauchy Riemann DGL audio1.ogg

Task

For the complex-valued function $f:ℂ\to ℂ,z\mapsto f(z)=z^3$, give the mappings $u,v$ with $f(x+iy)=u(x,y)+iv(x,y)$ explicitly.

Evaluation of the Jacobian Matrix at a Point

The evaluation of the Jacobian matrix at a point $(x_o,y_o)\in {ℝ}^2$ gives the total derivative at the point $x_o+i{y}_o\in ℂ$ :$\left(\begin{array}{ccc}\frac{\partial u}{\partial x}(x_o,y_o)& \frac{\partial u}{\partial y}(x_o,y_o)\frac{\partial v}{\partial x}(x_o,y_o)& \frac{\partial v}{\partial y}(x_o,y_o)\end{array}\right)$

Cauchy-Riemann Differential Equations

A function $f$ is complex differentiable at $z_o:=x_o+i{y}_o$ if and only if it is real differentiable and the Cauchy-Riemann differential equations hold for $u,v$ with $u:U_R\to ℝ$, $v:U_R\to ℝ$ where $U_R\subset {ℝ}^2$: :$\frac{\partial u}{\partial x}(x_o,y_o)=\frac{\partial v}{\partial y}(x_o,y_o)$ :$\frac{\partial u}{\partial y}(x_o,y_o)=-\frac{\partial v}{\partial x}(x_o,y_o)$ are satisfied. File:Cauchy Riemann DGL audio4.ogg

Relationship Between the Partial Derivatives

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

Part 1

If the following limit exists for $f:G\to ℂ$ at $z_o\in G$ with $G\subset ℂ$ open: :$f^{\prime }(z_o)=\underset{z\to z_o}{\lim }\frac{f(z)-f(z_o)}{z-z_o}$, then for any sequences $(z_n)n\in \mathbb{N}$ in the domain $G\subset ℂ$ with $\lim n\to \mathrm{\infty }{z}_n=z_o$, we also have: :$f^{\prime }(z_o)=\underset{n\to \mathrm{\infty }}{\lim }\frac{f(z_n)-f(z_o)}{z_n-z_o}$ File:Cauchy Riemann DGL audio5.ogg

Part 2

Now consider only the sequences for the two following limit processes with $h\in ℝ$: :$f^{\prime }(z_o)=\underset{h\to 0}{\lim }\frac{f(z_o+h)-f(z_o)}{(z_o+h)-z_o}=\underset{h\to 0}{\lim }\frac{f(z_o+h)-f(z_o)}{h}$, :$f^{\prime }(z_o)=\underset{ih\to 0}{\lim }\frac{f(z_o+ih)-f(z_o)}{(z_o+ih)-z_o}=\underset{ih\to 0}{\lim }\frac{f(z_o+ih)-f(z_o)}{ih}$, File:Cauchy Riemann DGL audio6.ogg

Part 3: Limit Process for Real Part

By inserting the component functions for the real and imaginary parts $u,v$, we get with $h\in ℝ$: :$f^{\prime }(z_o)=\underset{h\to 0}{\lim }\frac{f(z_o+h)-f(z_o)}{h}=$ ::$=\underset{h\to 0}{\lim }\frac{u(x_o+h,y_o)-u(x_o,y_o)}{h}+i\underset{h\to 0}{\lim }\frac{v(x_o+h,y_o)-v(x_o,y_o)}{h}$ ::$=\frac{\partial u}{\partial x}(x_o,y_o)+i\frac{\partial v}{\partial x}(x_o,y_o)$ File:Cauchy Riemann DGL audio7.ogg

Part 4: Limit Process for Imaginary Part

Applying this to the second equation, we get with $h\in ℝ$: :$f^{\prime }(z_o)=\underset{ih\to 0}{\lim }\frac{f(z_o+ih)-f(z_o)}{ih}$ ::$=\underset{h\to 0}{\lim }\frac{u(x_o,y_o+h)-u(x_o,y_o)}{ih}+i\underset{h\to 0}{\lim }\frac{v(x_o,y_o+h)-v(x_o,y_o)}{ih}$ ::$=-i\underset{h\to 0}{\lim }\frac{u(x_o,y_o+h)-u(x_o,y_o)}{h}+\underset{h\to 0}{\lim }\frac{v(x_o,y_o+h)-v(x_o,y_o)}{h}$,

$=-i\frac{\partial u}{\partial y}(x_o,y_o)+\frac{\partial v}{\partial y}(x_o,y_o)$

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: $\frac{\partial u}{\partial y}(x_o,y_o)=-\frac{\partial v}{\partial x}(x_o,y_o)$

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

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

$\frac{\partial f}{\partial x}(z_o)=\underset{h\to 0}{\lim }\frac{f(z_o+h)-f(z_o)}{h}\in ℂ$,

$\frac{\partial \Re 𝔢(f)}{\partial x}(z_o)=\underset{h\to 0}{\lim }\frac{\Re 𝔢(f)(z_o+h)-\Re 𝔢(f)(z_o)}{h}\in ℝ$,

$\frac{\partial \Im 𝔪(f)}{\partial x}(z_o)=\underset{h\to 0}{\lim }\frac{\Im 𝔪(f)(z_o+h)-\Im 𝔪(f)(z_o)}{h}\in ℝ$.

File:Cauchy Riemann DGL audio10.ogg

Part 7: Partial Derivative in the Direction of the Imaginary Part

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

$\frac{\partial f}{\partial y}(z_o)=\underset{h\to 0}{\lim }\frac{f(z_o+ih)-f(z_o)}{h}\in ℂ$,

$\frac{\partial \Re 𝔢(f)}{\partial y}(z_o)=\underset{h\to 0}{\lim }\frac{\Re 𝔢(f)(z_o+ih)-\Re 𝔢(f)(z_o)}{h}\in ℝ$,

$\frac{\partial \Im 𝔪(f)}{\partial y}(z_o)=\underset{h\to 0}{\lim }\frac{\Im 𝔪(f)(z_o+ih)-\Im 𝔪(f)(z_o)}{h}\in ℝ$.

File:Cauchy Riemann DGL audio10.ogg

Part 8: Cauchy-Riemann DGL with Functions in $ℂ$

The partial derivatives of the Cauchy-Riemann differential equations can also be expressed in $ℂ$ with $f:=f_x+i{f}_y$, $f_x:=\Re 𝔢(f)$, $f_y:=\Im 𝔪(f)$: Real part: $\frac{\partial \Re 𝔢(f)}{\partial x}(z_o)=\frac{\partial \Im 𝔪(f)}{\partial y}(z_o)$ Imaginary part: $\frac{\partial \Re 𝔢(f)}{\partial y}(z_o)=-\frac{\partial \Im 𝔪(f)}{\partial x}(z_o)$

Theorem - Cauchy-Riemann DGL

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

$$\frac{\partial u}{\partial y}(x,y)=-\frac{\partial v}{\partial x}(x,y)$$

Remark on CR-DGL

In this case, the derivative of $f$ at the point $z\in ℂ$ can be represented in two ways using the component functions $u$ and $v$: $$f^{\prime }\left(z\right)=\frac{\partial u}{\partial x}(x,y)-i\frac{\partial u}{\partial y}(x,y)=\frac{\partial v}{\partial y}(x,y)+i\frac{\partial v}{\partial x}(x,y)$$ 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 $h\in ℂ$ converge to 0 in the direction of the real part. To achieve this, choose $h:=h_1+i\cdot 0$ with $h_1\in ℝ$. The decomposition of the function $f=u+iv$ into its real part $u$ and imaginary part $v$ then yields (DG1).

Step 2 - Calculation of the Derivative - Real Part

$$\begin{array}{ccc}{f}^{\prime }\left(z\right)& =& \lim {\mathrm{\backslash limits}}_{h\to 0}\frac{f(z+h)-f\left(z\right)}{h}\\ & =& \lim {\mathrm{\backslash limits}}_{h_1\to 0}\frac{u(x+h_1,y)+iv(x+h_1,y)-u(x,y)-iv(x,y)}{h_1}\\ & =& \lim {\mathrm{\backslash limits}}_{h_1\to 0}\frac{u(x+h_1,y)-u(x,y)}{h_1}+i\frac{v(x+h_1,y)-v(x,y)}{h_1}\\ & =& \frac{\partial u}{\partial x}(x,y)+i\frac{\partial v}{\partial x}(x,y)\end{array}$$

Step 3 - Derivative in the Direction of the Imaginary Part

Similarly, the partial derivative for the imaginary part can be considered with $h:=0+i\cdot h_2$ and $h_2\in ℝ$. This yields equation (DG2).

Step 4 - Calculation of the Derivative - Imaginary Part

$\begin{array}{ccc}{f}^{\prime }\left(z\right)& =& \lim {\mathrm{\backslash limits}}_{h\to 0}\frac{f(z+h)-f\left(z\right)}{h}\\ & =& \lim {\mathrm{\backslash limits}}_{h_2\to 0}\frac{u(x,y+h_2)+iv(x,y+h_2)-u(x,y)-iv(x,y)}{i\cdot h_2}\\ & =& \lim {\mathrm{\backslash limits}}_{l\to 0}\frac{1}{i}\frac{u(x,y+h_2)-u(x,y)}{h_2}+\frac{v(x,y+h_2)-v(x,y)}{h_2}\\ & =& \frac{\partial v}{\partial y}(x,y)-i\frac{\partial u}{\partial y}(x,y)\end{array}$

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): $$f^{\prime }\left(z\right)=\frac{\partial u}{\partial x}(x,y)+i\frac{\partial v}{\partial x}(x,y)=\frac{\partial v}{\partial y}(x,y)-i\frac{\partial u}{\partial y}(x,y)$$

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.

See also

• Complex Analysis
• Wirtinger derivatives or Wirtinger calculus
• Maximum Principle
• Real and imaginary part functions

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