User:Silly rabbit/sandbox2

From Wikipedia, the free encyclopedia

[1] $\int x\,dx = \frac{x^2}{2}+C.$

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.

[2] Testing.

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 Goursat's theorem asserts that 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

[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 typically 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 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

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