Holomorphic function: Difference between revisions

Latest revision as of 19:13, 18 May 2008

Definition

Definition for one-variable function

Let $U$ be an open subset (not necessarily connected, though we can for practical purposes restrict ourselves to domains -- connected open subsets) of $\mathbb {C}$ . A function $f:U\to \mathbb {C}$ is termed a holomorphic function if it satisfies the following equivalent conditions:

$f$ is complex-differentiable at every point of $U$
$f$ is complex-differentiable at every point of $U$ , and the function $f':U\to \mathbb {C}$ we obtain as the derivative, is a continuous function.
$f$ is infinitely often complex-differentiable. In other words, we can take the $n^{th}$ derivative of $f$ for any $n$
For any point $z_{0}\in U$ , and any disc centered at $z_{0}$ of radius $r$ , that lies completely inside $U$ , $f$ can be expressed using a power series in $(z-z_{0})$
$f$ is a continuous function and satisfies the Cauchy-Riemann differential equations at every point in $U$
$f$ is a continuous function and its integral along the boundary of any triangle contained completely inside $U$ is zero.

Equivalence of definitions

For equivalence of (2), (3), (4), refer complex-analytic implies holomorphic and holomorphic implies complex-analytic
Goursat's lemma and Morera's theorem establish the equivalence of (1), (2) and (6)
Looman-Menchoff theorem establishes the equivalence with (5).

Definition on Riemann surfaces

Let $M$ be a Riemann surface. A holomorphic function on $M$ is a map from $M$ to $\mathbb {C}$ whose restriction to any coordinate chart is a holomorphic function in the usual sense.

@@ Line 9: / Line 9: @@
 # <math>f</math> is infinitely often complex-differentiable. In other words, we can take the <math>n^{th}</math> derivative of <math>f</math> for any <math>n</math>
 # For any point <math>z_0 \in U</math>, and any disc centered at <math>z_0</math> of radius <math>r</math>, that lies completely inside <math>U</math>, <math>f</math> can be expressed using a power series in <math>(z - z_0)</math>
+# <math>f</math> is a continuous function and [[function satisfying Cauchy-Riemann differential equations at a point|satisfies the Cauchy-Riemann differential equations]] at ''every'' point in <math>U</math>
+# <math>f</math> is a continuous function and [[integral of complex-valued function along a curve|its integral]] along the boundary of any triangle contained completely inside <math>U</math> is zero.
-===Definition for functions in several variables===
+===Equivalence of definitions===
-Let <math>U</math> be an open subset (not necessarily connected, though we may restrict attention to connected subsets). A function <math>f:U \to \mathbb{C}</math> is termed a '''holomorphic function''' if it satisfies the following equivalent conditions:
+* For equivalence of (2), (3), (4), refer [[complex-analytic implies holomorphic]] and [[holomorphic implies complex-analytic]]
+* [[Goursat's lemma]] and [[Morera's theorem]] establish the equivalence of (1), (2) and (6)
+* [[Looman-Menchoff theorem]] establishes the equivalence with (5).
-# <math>f</math> is [[complex-differentiable function|complex-differentiable]] at every point of <math>U</math>
+===Definition on Riemann surfaces===
-# <math>f</math> is complex-differentiable, and its partial derivatives in all directions are continuous functions
-# <math>f</math> is infinitely often complex-differentiable. In other words, we can take any sequence of mixed partial derivatives of <math>f</math>
-# For any point <math>z_0 \in U</math>, and any disc centered at <math>z_0</math> of radius <math>r</math>, that lies completely inside <math>U</math>, <math>f</math> can be expressed using a power series in <math>z - z_0</math>
-===Equivalence of definitions===
-Definitions (1) and (2) are equivalent as a consequence of [[Morera's theorem]].
+Let <math>M</math> be a [[Riemann surface]]. A '''holomorphic function''' on <math>M</math> is a map from <math>M</math> to <math>\mathbb{C}</math> whose restriction to any coordinate chart is a [[holomorphic function]] in the usual sense.