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 '''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 much stronger than in the real case since we must allow <math>h</math> to approach zero from any direction in the [[complex plane]]. |
== Cauchy-Riemann Equations == | == Cauchy-Riemann Equations == | ||
Revision as of 14:03, 18 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 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 .
