Cauchy-Riemann Equations

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Theorem

Let $D \subseteq \C$ be an open subset of the set of complex numbers $\C$.

Let $f: D \to \C$ be a complex function on $D$.

Let $u, v: \set {\tuple {x, y} \in \R^2: x + i y = z \in D} \to \R$ be the two real-valued functions defined as:

where:

$\map \Re {\map f z}$ denotes the real part of $\map f z$

$\map \Im {\map f z}$ denotes the imaginary part of $\map f z$.

Then $f$ is complex-differentiable in $D$ if and only if:

$u$ and $v$ are differentiable in their entire domain

and:

The following two equations, known as the Cauchy-Riemann equations, hold for the continuous partial derivatives of $u$ and $v$:

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If the conditions are true, then for all $z \in D$:

$\map {f'} z = \map {\dfrac {\partial f} {\partial x} } z = -i \map {\dfrac {\partial f} {\partial y} } z$

Polar Form

The Cauchy-Riemann equations can be expressed in polar form as:

where $z$ is expressed in exponential form as:

$z = r e^{i \theta}$

Proof

Necessary Condition

Let $z = x + i y \in D$.

The Epsilon-Function Complex Differentiability Condition shows that there exists $r \in \R_{>0}$ such that for all $t \in \map {B_r} 0 \setminus \set 0$:

$(\text a): \quad \map f {z + t} = \map f z + t \paren {\map {f'} z + \map \epsilon t}$

where:

$\map {B_r} 0$ denotes an open ball of $0$

$\epsilon: \map {B_r} 0 \setminus \set 0 \to \CC$ is a function with $\ds \lim_{t \mathop \to 0} \map \epsilon t = 0$.

Define $a, b, h, k \in \R$ by $a + i b = \map {f'} z$, and $h + i k = t$.

By taking the real parts of both sides of equation $(\text a)$, it follows that:

To find the partial derivative $\dfrac {\partial u} {\partial x}$, assume that $y$ is fixed, and let $t$ be wholly real.

Then $t = h$, and $k = 0$, so:

$\map u {x + h, y} = \map u {x, y} + h \paren {a + \map \Re {\map \epsilon h} }$

From Limits of Real and Imaginary Parts, it follows that:

$\ds \lim_{h \mathop \to 0} \map \Re {\map \epsilon h} = \map \Re {\lim_{h \mathop \to 0} \map \epsilon h} = \map \Re 0 = 0$

Then the Epsilon-Function Differentiability Condition proves that:

$\map {\dfrac {\partial u} {\partial x} } {x, y} = a$

To find the partial derivative $\dfrac {\partial u} {\partial y}$, assume that $x$ is fixed, and let $t$ be wholly imaginary.

Then $t = i k$, and $h = 0$, so:

$\map u {x, y + k} = \map u {x, y} + k \paren {-b + \map \Re {i \map \epsilon {i k} } }$

From Limits of Real and Imaginary Parts, it follows that:

$\ds \lim_{k \mathop \to 0} \map \Re {i \map \epsilon k} = \map \Re {i \lim_{k \mathop \to 0} \map \epsilon k} = 0$

Then the Epsilon-Function Differentiability Condition proves that:

$\map {\dfrac {\partial u} {\partial y} } {x, y} = -b$

$\Box$

By taking the imaginary parts of both sides of equation $(\text a)$, it follows that:

To find the partial derivative $\dfrac {\partial v} {\partial x}$, assume that $y$ is fixed, and let $t$ be wholly real.

Then $t = h$, and $k = 0$, so:

$\map v {x + h, y} = \map v {x, y} + h \paren {b + \map \Im {\map \epsilon h} }$

A similar argument to the ones above shows that the Epsilon-Function Differentiability Condition can be applied to prove that:

$\map {\dfrac {\partial v} {\partial x} } {x, y} = b = -\map {\dfrac {\partial u} {\partial y} } {x, y}$

This proves the second Cauchy-Riemann equation.

To find the partial derivative $\dfrac {\partial v} {\partial y}$, assume that $x$ is fixed, and let $t$ be wholly imaginary.

Then $t = ik$, and $h = 0$, so:

$\map v {x, y + k} = \map v {x, y} + k \paren {a + \map \Im {i \map \epsilon {i k} } }$

Here, the Epsilon-Function Differentiability Condition shows that:

$\map {\dfrac {\partial v} {\partial y} } {x, y} = a = \map {\dfrac {\partial u} {\partial x} } {x, y}$

This proves the first Cauchy-Riemann equation.

$\Box$

From Holomorphic Function is Continuously Differentiable, it follows that $f'$ is continuous.

From Composite of Continuous Mappings is Continuous and Real and Imaginary Part Projections are Continuous, it follows that $\map \Re {f'}$ and $\map \Im {f'}$ are continuous.

Then all four partial derivatives are continuous, as:

and:

By definition of differentiable real-valued function, it follows that $u$ and $v$ are differentiable.

$\Box$

Sufficient Condition

Suppose that the Cauchy-Riemann equations hold for $u$ and $v$ in their entire domain.

Let $h, k \in \R \setminus \set 0$, and put $t = h + i k \in \C$.

Let $\tuple {x, y} \in \R^2$ be a point in the domain of $u$ and $v$.

Put:

$a = \map {\dfrac {\partial u} {\partial x} } {x, y} = \map {\dfrac {\partial v} {\partial y} } {x, y}$

and:

$b = -\map {\dfrac {\partial u} {\partial y} } {x, y} = \map {\dfrac {\partial v} {\partial x} } {x, y}$

From the Epsilon-Function Differentiability Condition, it follows that:

$\map u {x + h, y} = \map u {x, y} + h \paren {a + \map {\epsilon_{u x} } h}$

$\map u {x, y + k} = \map u {x, y} + k \paren {-b + \map {\epsilon_{u y} } k}$

$\map v {x + h, y} = \map v {x, y} + h \paren {b + \map {\epsilon_{v x} } k}$

$\map v {x, y + k} = \map v {x, y} + k \paren {a + \map {\epsilon_{v y} } h}$

where $\epsilon_{u x}, \epsilon_{u y}, \epsilon_{v x}, \epsilon_{v y}$ are real functions that converge to $0$ as $h$ and $k$ tend to $0$.

We have by hypothesis that the partial derivatives are continuous.

Then Epsilon-Function Differentiability Condition shows that $\epsilon_{u x}, \epsilon_{u y}, \epsilon_{v x}, \epsilon_{v y}$ are continuous real functions.

With $z = x + i y$, it follows that:

Now, define $\map \epsilon t$ as:

By Modulus Larger than Real Part and Imaginary Part:

$\cmod k \le \cmod t$

and:

$\cmod h \le \cmod t$

Hence dividing by $\cmod t$ we have:

$\cmod {\dfrac h t} \le 1$

and:

$\cmod {\dfrac k t} \le 1$

From Composite of Continuous Mappings is Continuous, it follows that $\map \epsilon t$ is continuous.

Thus:

This shows that:

$\ds \lim_{t \mathop \to 0} \map \epsilon t = 0$

Then the Epsilon-Function Complex Differentiability Condition shows that:

$\map {f'} x = a + i b$

$\blacksquare$

Expression of Derivative

Let $z = x + i y$.

Then:

Similarly:

$\blacksquare$

Source of Name

This entry was named for Augustin Louis Cauchy and Georg Friedrich Bernhard Riemann.

Sources

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.7$ Complex Numbers and Functions: Complex Functions, Cauchy-Riemann Equations: $3.7.30$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Cauchy-Riemann equations
• 2001: Christian Berg: Kompleks funktionsteori: $\S 1.3$