Holomorphic function - Revision history

Vipul: 5 revisions

2008-05-18T19:13:31Z

5 revisions

← Older revision	Revision as of 19:13, 18 May 2008
(No difference)

Vipul at 22:03, 3 May 2008

2008-05-03T22:03:16Z

← Older revision		Revision as of 22:03, 3 May 2008
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	* [[Goursat's lemma]] and [[Morera's theorem]] establish the equivalence of (1), (2) and (6)		* [[Goursat's lemma]] and [[Morera's theorem]] establish the equivalence of (1), (2) and (6)
	* [[Looman-Menchoff theorem]] establishes the equivalence with (5).		* [[Looman-Menchoff theorem]] establishes the equivalence with (5).

			===Definition on Riemann surfaces===

			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.

Vipul: /* Definition */

2008-04-18T21:07:07Z

Definition

← Older revision		Revision as of 21:07, 18 April 2008
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	# <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>		# <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>		# 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>
	~~===Definition for functions in several variables===~~		# <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.

	~~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:~~

	~~# <math>f</math> is~~ [[~~complex~~-~~differentiable function~~\|~~complex~~-~~differentiable~~]] at every point of <math>U</math>
	# <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===		===Equivalence of definitions===

	~~Definitions~~ (1) ~~and~~ (2) ~~are equivalent as a consequence of~~ [[Morera's theorem]].		* 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).

Vipul at 18:19, 16 April 2008

2008-04-16T18:19:50Z

← Older revision		Revision as of 18:19, 16 April 2008
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	==Definition==		==Definition==

	===Definition ~~with symbols~~===		===Definition for one-variable function===

	Let <math>~~\Omega~~</math> be an open subset of <math>\mathbb{C}</math>. A function <math>f:U \to \mathbb{C}</math> is termed a '''holomorphic function''' if it satisfies the following equivalent conditions:		Let <math>U</math> be an open subset (not necessarily connected, though we can for practical purposes restrict ourselves to [[domain]]s -- connected open subsets) of <math>\mathbb{C}</math>. A function <math>f:U \to \mathbb{C}</math> is termed a '''holomorphic function''' if it satisfies the following equivalent conditions:

	# <math>f</math> is [[complex-differentiable function\|complex-differentiable]] at every point of <math>U</math>		# <math>f</math> is [[complex-differentiable function\|complex-differentiable]] at every point of <math>U</math>
	# <math>f</math> is [[complex-differentiable function\|complex-differentiable]] at every point of <math>U</math>, and the function <math>f':U \to \mathbb{C}</math> we obtain as the derivative, is a continuous function.		# <math>f</math> is [[complex-differentiable function\|complex-differentiable]] at every point of <math>U</math>, and the function <math>f':U \to \mathbb{C}</math> we obtain as the derivative, is a continuous function.
	# <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>		# <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</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>		# 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>

			===Definition for functions in several variables===

			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:

			# <math>f</math> is [[complex-differentiable function\|complex-differentiable]] at every point of <math>U</math>
			# <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===		===Equivalence of definitions===

	Definitions (1) and (2) are equivalent as a consequence of [[Morera's theorem]].		Definitions (1) and (2) are equivalent as a consequence of [[Morera's theorem]].

Vipul: /* Definition with symbols */

2008-04-13T19:29:16Z

Definition with symbols

← Older revision		Revision as of 19:29, 13 April 2008
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	# <math>f</math> is [[complex-differentiable function\|complex-differentiable]] at every point of <math>U</math>		# <math>f</math> is [[complex-differentiable function\|complex-differentiable]] at every point of <math>U</math>
	# <math>f</math> is [[complex-differentiable function\|complex-differentiable]] at every point of <math>U</math>, and the function <math>f':U \to mathbb{C}</math> we obtain as the derivative, is a continuous function.		# <math>f</math> is [[complex-differentiable function\|complex-differentiable]] at every point of <math>U</math>, and the function <math>f':U \to \mathbb{C}</math> we obtain as the derivative, is a continuous function.
	# <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>		# <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</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>		# For any point <math>z_0 \in U</math>, and any disc centered at <math>z</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>

Vipul: New page: ==Definition== ===Definition with symbols=== Let $\Omega$ be an open subset of $\mathbb{C}$ . A function $f:U \to \mathbb{C}$ is termed a '''holomorphic f...

2008-04-13T19:28:02Z

New page: ==Definition== ===Definition with symbols=== Let <math>\Omega</math> be an open subset of <math>\mathbb{C}</math>. A function <math>f:U \to \mathbb{C}</math> is termed a '''holomorphic f...

New page

==Definition==

===Definition with symbols===

Let <math>\Omega</math> be an open subset of <math>\mathbb{C}</math>. A function <math>f:U \to \mathbb{C}</math> is termed a '''holomorphic function''' if it satisfies the following equivalent conditions:

# <math>f</math> is [[complex-differentiable function|complex-differentiable]] at every point of <math>U</math>
# <math>f</math> is [[complex-differentiable function|complex-differentiable]] at every point of <math>U</math>, and the function <math>f':U \to mathbb{C}</math> we obtain as the derivative, is a continuous function.
# <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</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]].