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 and sufficient condition for a differentiable function to be holomorphic in an open set. This system of equations first appeared in the work of Jean le Rond d'Alembert (d'Alembert 1752). Later, Leonhard Euler connected this system to the analytic functions (Euler 1777). Cauchy (1814) then used these equations to construct his theory of functions. Riemann's dissertation (Riemann 1851) on the theory of functions appeared in 1851.

The Cauchy-Riemann equations on a pair of real-valued functions u(x,y) and v(x,y) are the two equations:

(1a) ${ \partial u \over \partial x } = { \partial v \over \partial y }$

and

(1b) ${ \partial u \over \partial y } = -{ \partial v \over \partial x } .$

Typically the pair u and v are taken to be the real and imaginary parts of a complex-valued function f(x + iy) = u(x,y) + iv(x,y). Suppose that u and v are continuously differentiable on an open subset of C. Then f=u+iv is holomorphic if and only if the partial derivatives of u and v satisfy the Cauchy-Riemann equations (1a) and (1b).

1 Interpretation and reformulation
2 Inhomogeneous equations
3 Generalizations
- 3.1 Goursat's theorem
- 3.2 Several variables
4 References
5 External links

[edit] Interpretation and reformulation

[edit] Conformal mappings

The Cauchy-Riemann equations are often reformulated in a variety of ways. Firstly, they may be written in complex form

(2) ${ i { \partial f \over \partial x } } = { \partial f \over \partial y } .$

In this form, the equations correspond structurally to the condition that the Jacobian matrix is of the form

$\begin{pmatrix} a & -b \\ b & \;\; a \end{pmatrix},$

where $\scriptstyle a=\partial u/\partial x=\partial v/\partial y$ and $\scriptstyle b=\partial v/\partial x=-\partial u/\partial y$ . A matrix of this form is the matrix representation of a complex number. Geometrically, such a matrix is always the composition of a rotation with a scaling, and in particular preserves angles. Consequently, a function satisfying the Cauchy-Riemann equations, with a nonzero derivative, preserves the angle between curves in the plane. That is, the Cauchy-Riemann equations are the conditions for a function to be conformal.

[edit] Independence of the complex conjugate

The equations are sometimes written as a single equation

(3) $\frac{\partial f}{\partial\bar{z}} = 0$

where the differential operator $\frac{\partial}{\partial\bar{z}}$ is defined by

$\frac{\partial}{\partial\bar{z}} = \frac{1}{2}\left(\frac{\partial}{\partial x} + i\frac{\partial}{\partial y}\right).$

In this form, the Cauchy-Riemann equations can be interpreted as the statement that f is independent of the variable $\bar{z}$ .

[edit] Complex differentiability

The Cauchy-Riemann equations are necessary and sufficient conditions for the complex differentiability (or holomorphicity) of a function (Ahlfors 1953, §1.2). Specifically, suppose that

f (z) = u (z) + i v (z)

if a function of a complex number z∈C. Then the complex derivative of f at a point z₀ is defined by

$\lim_{\underset{h\in\mathbb{C}}{h\to 0}} \frac{f(z_0+h)-f(z_0)}{h} = f'(z_0)$

provided this limit exists.

If this limit exists, then it may be computed by taking the limit as h→0 along the real axis or imaginary axis; in either case it should give the same result. Approaching along the real axis, one finds

$\lim_{\underset{h\in\mathbb{R}}{h\to 0}} \frac{f(z_0+h)-f(z_0)}{h} = \frac{\partial f}{\partial x}(z_0).$

On the other hand, approaching along the imaginary axis,

$\lim_{\underset{ih\in i\mathbb{R}}{h\to 0}} \frac{f(z_0+ih)-f(z_0)}{ih} = \lim_{\underset{ih\in i\mathbb{R}}{h\to 0}} -i\frac{f(z_0+ih)-f(z_0)}{h} =-i\frac{\partial f}{\partial y}(z_0).$

The equality of the derivative of f taken along the two axes is

$\frac{\partial f}{\partial x}(z_0)=-i\frac{\partial f}{\partial y}(z_0),$

which are the Cauchy-Riemann equations (2) at the point z₀.

Conversely, if f:C → C is a function which is differentiable when regarded as a function into R², then f is complex differentiable if and only if the Cauchy-Riemann equations hold.

[edit] Physical interpretation

One interpretation of the Cauchy-Riemann equations (Pólya & Szegö 1978) does not involve complex variables directly. Suppose that u and v satisfy the Cauchy-Riemann equations in an open subset of R², and consider the vector field

$\bar{f} = \begin{bmatrix}u\\ -v\end{bmatrix}$

regarded as a (real) two-component vector. Then the first Cauchy-Riemann equation (1a) asserts that $\bar{f}$ is irrotational:

$\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} = 0.$

The second Cauchy-Riemann equation (1b) asserts that the vector field is solenoidal (or divergence-free):

$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}=0.$

Owing respectively to Green's theorem and the divergence theorem, such a field is necessarily conserved and free from sources or sinks, having net flux equal to zero through any open domain. These two observations combine as real and imaginary parts in Cauchy's integral theorem.

[edit] Other representations

Other representations of the Cauchy-Riemann equations occasionally arise in other coordinate systems. If (1a) and (1b) hold for a continuously differentiable pair of functions u and v, then so do

$\frac{\partial u}{\partial s} = \frac{\partial u}{\partial n},\quad \frac{\partial u}{\partial n} = -\frac{\partial u}{\partial s}$

for any coordinates (n(x,y), s(x,y)) such that the pair $\scriptstyle (\nabla n, \nabla s)$ is orthonormal and positively oriented. As a consequence, in particular, in the system of coordinates given by the polar representation z=re^iθ the equations then take the form

${ \partial u \over \partial r } = {1 \over r}{ \partial v \over \partial \theta},\quad{ \partial v \over \partial r } = -{1 \over r}{ \partial u \over \partial \theta}.$

Combining these into one equation for f gives

${\partial f \over \partial r} = {1 \over i r}{\partial f \over \partial \theta}.$

[edit] Inhomogeneous equations

The inhomogeneous Cauchy-Riemann equations consist of the two equations for a pair of unknown functions u(x,y) and v(x,y) of two real variables

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

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

for some given functions α(x,y) and β(x,y) defined in an open subset of R². These equations are usually combined into a single equation

$\frac{\partial f}{\partial\bar{z}} = \phi(z,\bar{z})$

where f=u+iv and φ=(α+iβ)/2.

If φ is C^k, then the inhomogeneous equation is explicitly solvable in any bounded domain D, provided φ is continuous on the closure of D. Indeed, by the Cauchy integral formula,

$f(\zeta,\bar{\zeta}) = \frac{1}{2\pi i}\iint_D \phi(z,\bar{z})\frac{dz\wedge d\bar{z}}{z-\zeta}$

for all ζ∈D.

[edit] Generalizations

[edit] Goursat's theorem

See also: Cauchy-Goursat theorem

Suppose that f = u+iv is a complex-valued function which is differentiable as a function f : R² → R². Then Goursat's theorem asserts that f is analytic if and only if it satisfies the Cauchy-Riemann equation on an open disk surrounding the point of interest (Rudin 1966, Theorem 11.2). In particular, continuous differentiability of f need not be assumed (Dieudonné 1969, §9.10, Ex. 1).

The hypotheses of Goursat's theorem can be weakened significantly. If f=u+iv is continuous in an open set Ω and the partial derivatives of f with respect to x and y exist, then f is holomorphic (and thus analytic). This result is the Looman–Menchoff theorem. It is important to emphasize that hypothesis of the open set (or an open disk) in this theorem is essential. It is possible to find functions where the Cauchy-Riemann equations are satisfied at a point, but not on a disk surrounding it, and these functions are not analytic functions.

[edit] Several variables

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

$\bar{\partial}$

annihilates holomorphic functions. This generalizes most directly the formulation

${\partial f \over \partial \bar z} = 0$ ,

where

${\partial f \over \partial \bar z} = {1 \over 2}\left({\partial f \over \partial x} - {1 \over i}{\partial f \over \partial y}\right).$

[edit] References

Ahlfors, Lars (1953), Complex analysis (3rd ed.), McGraw Hill (published 1979), ISBN 0-07-000657-1 .
d'Alembert, J. (1752), Essai d'une nouvelle théorie de la résistance des fluides, Paris, <http://gallica2.bnf.fr/ark:/12148/bpt6k206036b.modeAffichageimage.f1.langFR.vignettesnaviguer> .
Cauchy, A.L. (1814), Mémoire sur les intégrales définies,, vol. 1, Oeuvres complètes Ser. 1, Paris (published 1882), pp. 319–506
Dieudonné, Jean Alexander (1969), Foundations of modern analysis, Academic Press .
Euler, L. (1797), Nova Acta Acad. Sci. Petrop. 10: 3–19
Pólya, George & Szegö, Gabor (1978), Problems and theorems in analysis I, Springer, ISBN 3-540-63640-4
Riemann, B. (1851), “Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen komplexen Grösse”, in H. Weber, Riemann's gesammelte math. Werke, Dover, 1953, pp. 3–48
Rudin, Walter (1966), Real and complex analysis (3rd ed.), McGraw Hill (published 1987), ISBN 0-07-054234-1 .
Solomentsev, E.D. (2001), “Cauchy–Riemann conditions”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104

[edit] External links