Difference between revisions of "Holomorphic function"
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| − | {{ | + | A '''holomorphic''' function <math>f: \mathbb{C} \to \mathbb{C}</math> is a differentiable [[complex number|complex]] [[function]]. That is, just as in the [[real number|real]] case, <math>f</math> is holomorphic at <math>z</math> if <math>\lim_{h\to 0} \frac{f(z+h)-f(z)}{h}</math> exists. This is a much stronger than in the real case since we must allow <math>h</math> to approach zero from any direction in the [[complex plane]]. |
| − | A | + | == Cauchy-Riemann Equations == |
| + | |||
| + | Let us break <math>f</math> into its real and imaginary components by writing <math>f(z)=u(x,y)+iv(x,y)</math>, where <math>u</math> and <math>v</math> are real functions. Then it turns out that <math>f</math> is holomorphic at <math>z</math> [[iff]] <math>u</math> and <math>v</math> have continuous partial derivatives and the following equations hold: | ||
| + | * <math>\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}</math> | ||
| + | * <math>\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}</math> | ||
| + | |||
| + | These equations are known as the [[Cauchy-Riemann Equations]]. | ||
| + | |||
| + | == Analytic Functions == | ||
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| + | A related notion to that of homolorphicity is that of analyticity. A function <math>f:\mathbb{C}\to\mathbb{C}</math> is said to be '''analytic''' at <math>z</math> if <math>f</math> has a convergent [[power series]] expansion on some [[neighborhood]] of <math>z</math>. Amazingly, it turns out that a function is holomorphic at <math>z</math> if and only if it is analytic at <math>z</math>. | ||
Revision as of 12:23, 12 July 2006
A holomorphic function 
is a differentiable complex function. That is, just as in the real case, 
is holomorphic at 
if 
exists. This is a much stronger than in the real case since we must allow 
to approach zero from any direction in the complex plane.
Cauchy-Riemann Equations
Let us break into its real and imaginary components by writing 
, where 
and 
are real functions. Then it turns out that is holomorphic at iff and have continuous partial derivatives and the following equations hold:
These equations are known as the Cauchy-Riemann Equations.
Analytic Functions
A related notion to that of homolorphicity is that of analyticity. A function 
is said to be analytic at if has a convergent power series expansion on some neighborhood of . Amazingly, it turns out that a function is holomorphic at if and only if it is analytic at .
