Cauchy-Riemann equations

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In mathematics, the Cauchy-Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two partial differential equations which provide a necessary but not necessarily sufficient condition for a function to be holomorphic (it is a sufficient condition if, for example, the functions u and v have continuous partial derivatives). This system of equations was first published in 1814 by Cauchy, in his paper Sur les intégrales définies.

Contents 1 Formulation 2 Example 3 Derivation 4 Alternative formulation 5 Polar representation 6 Several variables

Formulation

Let f(x + iy) = u + iv be a function from an open subset of the complex numbers C to C, where x, y, u, and v are real, and regard u and v as real-valued functions defined on an open subset of R². Then f is holomorphic if and only if u and v are continuously differentiable and their partial derivatives satisfy the Cauchy-Riemann equations, which are:

<math>{ \partial u \over \partial x } = { \partial v \over \partial y}</math>

and

<math>{ \partial u \over \partial y } = -{ \partial v \over \partial x}.</math>

The equations correspond structurally to the condition that the Jacobian matrix is of the form

<math>

\begin{pmatrix}

 a &   -b  \\
 b & \;\; a  

\end{pmatrix}, </math>

the matrix representation of a complex number. Geometrically, this expresses the conformal nature by a combination of rotation and enlargement, for any analytic function at a point where its derivative isn't zero. It does so by a first-order picture (small discs are rotated and enlarged to other small approximate discs).

It follows from the equations, if they can be differentiated twice, that u and v must be harmonic functions since they satisfy Laplace's equation. The equations can therefore be seen as the conditions on a given pair of harmonic functions to come as real and imaginary parts of a complex-analytic function. For a given harmonic function u a corresponding harmonic function v is called a harmonic conjugate. If it exists it is unique up to a constant term.

Example

The equations give a direct insight into antiholomorphic functions.

Suppose a complex function f analytic on an open set D. Then f satisfies Cauchy-Riemann equations; that is, if <math>f(x + iy) = u(x, y) + iv(x, y)</math>, then:

<math>{\partial u \over \partial x} = {\partial v \over \partial y}</math> and <math>{\partial v \over \partial x} = - {\partial u \over \partial y}</math>.

Now suppose <math>\bar f</math> is also analytic on D. Then since <math>f(x + iy) = u(x, y) - iv(x, y)</math>,

<math>{\partial u \over \partial x} = -{\partial v \over \partial y}</math> and <math>{\partial v \over \partial x} = {\partial u \over \partial y}</math>.

Combining this with the early equations, we get:

<math>{\partial u \over \partial x} = {\partial u \over \partial y} = {\partial v \over \partial x} ={\partial v \over \partial y} = 0</math>.

This shows that f is locally constant on D, and constant if D is connected.

Derivation

Consider a function f(z) = u(x, y) + i v(x, y) over C, and we wish to calculate its derivative at some point, z₀. We can essentially approach z₀ along the real axis towards 0, or down the imaginary axis towards 0.

If we take the first path:

<math>f'(z)\,</math>	<math>=\lim_{h\rightarrow 0} {f(z+h)-f(z) \over h}</math>
	<math>=\lim_{h\rightarrow 0}{u(x+h,y)+iv(x+h,y)-[u(x,y)+iv(x,y)]\over h}</math>
	<math>=\lim_{h\rightarrow 0}{[u(x+h,y)-u(x,y)]+i[v(x+h,y)-v(x,y)]\over h}</math>
	<math>=\lim_{h\rightarrow 0}{\left[\frac{u(x+h,y)-u(x,y)}{h}+i\frac{v(x+h,y)-v(x,y)}{h}\right]}.</math>

This is now in the form of two difference quotients, so now

<math>f'(z)={\partial u \over \partial x} + i {\partial v \over \partial x}.</math>

Taking the second path:

<math>f'(z)\,</math>	<math>=\lim_{h\rightarrow 0} {f(z+ih)-f(z) \over ih}</math>
	<math>=\lim_{h\rightarrow 0}{u(x,y+h)+iv(x,y+h)-[u(x,y)+iv(x,y)]\over ih}</math>
	<math>=\lim_{h\rightarrow 0}{\left[\frac{u(x,y+h)-u(x,y)}{ih} +i\frac{v(x,y+h)-v(x,y)}{ih}\right]}</math>
	<math>=\lim_{h\rightarrow 0}{\left[-i\frac{u(x,y+h)-u(x,y)}{h}+\frac{v(x,y+h)-v(x,y)}{h}\right]}</math>
	<math>=\lim_{h\rightarrow 0}{\left[\frac{v(x,y+h)-v(x,y)}{h}-i\frac{u(x,y+h)-u(x,y)}{h}\right]}.</math>

Again, this is now in the form of two difference quotients, so

<math>f'(z)={\partial v \over \partial y} - i {\partial u \over \partial y}.</math>

Equating these two we get

<math>{\partial u \over \partial x} + i {\partial v \over \partial x} = {\partial v \over \partial y} - i {\partial u \over \partial y}.</math>

Equating real and imaginary parts, then

<math>{\partial u \over \partial x} = {\partial v \over \partial y}</math>

<math>{\partial u \over \partial y} = - {\partial v \over \partial x}. \quad\square</math>

Alternative formulation

Suppose <math>z = x + iy</math> for real variables x and y. Then we can write <math>x = (z + \bar z)/2</math> and <math>y = (z - \bar z)/(2i)</math>. Now <math>\mathit{x}</math> and <math>\mathit{y}</math> can be thought of as real functions of complex independent variables <math>\mathit{z}</math> and <math>\bar z</math>. Differentiating <math>\mathit{x}</math> and <math>\mathit{y}</math> gives:

<math>{\partial x \over \partial z} = {1 \over 2}\ \mathrm{and}\ {\partial y \over \partial z} = {1 \over 2i}</math>

as well as

<math>{\partial x \over \partial \bar z} = {1 \over 2}\ \mathrm{and}\ {\partial y \over \partial \bar z} = -{1 \over 2i}.</math>

Differentiate a function <math>f (x, y) = u(x, y)+iv(x, y)</math>:

<math>{\partial f \over \partial z} = {\partial f \over \partial x}{\partial x \over \partial z} + {\partial f \over \partial y}{\partial y \over \partial z}\ \mathrm{and}\ {\partial f \over \partial \bar z} = {\partial f \over \partial x}{\partial x \over \partial \bar z} + {\partial f \over \partial y}{\partial y \over \partial \bar z}.</math>

Finally, substitution yields:

<math>{\partial f \over \partial z} = {1 \over 2}\left({\partial f \over \partial x} + {1 \over i}{\partial f \over \partial y}\right)\ \mathrm{and}\ {\partial f \over \partial \bar z} = {1 \over 2}\left({\partial f \over \partial x} - {1 \over i}{\partial f \over \partial y}\right).</math>

If we let <math>{\partial f \over \partial \bar z} = 0</math>, then, since <math>{\partial f \over \partial x} = {\partial u \over \partial x} + i{\partial v \over \partial x}\ \mathrm{and}\ {\partial f \over \partial y} = {\partial u \over \partial y} + i{\partial v \over \partial y}</math>,

<math>{\partial u \over \partial x} + i{\partial v \over \partial x} = {1 \over i}\left({\partial u \over \partial y} + i{\partial v \over \partial y}\right).</math>

This represents Cauchy-Riemann equations.

The relation has this interpretation: <math>\mathit{x}</math> and <math>\mathit{y}</math> must be constant with respect to <math>\bar z</math>. This expresses the concept that an analytic function is "truly" a function of a single complex variable, rather than of a real vector.

Polar representation

Considering the polar representation <math>z=re^{i\theta}</math>, the equations take the form

<math>{ \partial u \over \partial r } = {1 \over r}{ \partial v \over \partial \theta},</math>

<math>{ \partial v \over \partial r } = -{1 \over r}{ \partial u \over \partial \theta}.</math>

Several variables

There are Cauchy-Riemann equations, appropriately generalized, in the theory of several complex variables. They form a significant system of overdetermined PDEs. As often formulated, the d-bar operator

<math>\bar{\partial}</math>

annihilates holomorphic functions. This generalizes most directly the formulation

<math>{\partial f \over \partial \bar z} = 0</math>,

where

<math>{\partial f \over \partial \bar z} = {1 \over 2}\left({\partial f \over \partial x} - {1 \over i}{\partial f \over \partial y}\right).</math>ca:Equacions de Cauchy-Riemann

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