The Cauchy-Riemann equations


In order for a complex function of a  single complex variable to be differentiable it must be differentiable both parallell to the imaginary axis δy →0 and parallell to the real axis δx →0.

This condition leads to the CAuchy –Riemann equations-

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

(1a)     \dfrac{ \partial u }{ \partial x } = \dfrac{ \partial v }{ \partial y } \,

and

(1b)    \dfrac{ \partial u }{ \partial y } = -\dfrac{ \partial v }{ \partial x } \,
Proof:
Suppose that

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

is a function of a complex number z. Then the complex derivative of ƒ 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{h\in \mathbb{R}}{h\to 0}} \frac{f(z_0+ih)-f(z_0)}{ih} =\frac{1}{i}\frac{\partial f}{\partial y}(z_0).

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

i\frac{\partial f}{\partial x}(z_0)=\frac{\partial f}{\partial y}(z_0),
Holomorphy is the property of a complex function of being differentiable at every point of an open and connected subset of \mathbb{C} (this is called a domain in \mathbb{C}). 
 function that is complex-differentiable in a whole domain (holomorphic) is the same as an analytic function. This is not true for real differentiable functions.
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Om mattelararen

Licentiate of Philosophy in atomic Physics Master of Science in Physics
Det här inlägget postades i Advanced, Calculus, Imaginary numbers. Bokmärk permalänken.

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