Cauchy integral formula - Revision history

Vipul at 18:43, 12 September 2008

2008-09-12T18:43:03Z

← Older revision		Revision as of 18:43, 12 September 2008
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	In other words, we can determine the value of <math>f</math> at ''any point in the interior'', by knowing its value at all points on the boundary.		In other words, we can determine the value of <math>f</math> at ''any point in the interior'', by knowing its value at all points on the boundary.

	===For any ~~simple closed curve~~ and any point in the ~~interior~~===		===For any union of piecewise smooth curves and any point===

			{{further\|[[Winding number version of Cauchy integral formula]]}}

			==Related facts==

			* [[Cauchy integral formula for derivatives]]
			* [[Winding number version of Cauchy integral formula]]
			* [[Homotopy-invariance formulation of Cauchy's theorem]]

			==Facts used==

			# [[Cauchy integral formula for constant functions]]: In other words, the fact that for a constant function, the Cauchy integral formula holds.
			# A version of Fubini's theorem, that allows the exchange of a real and a complex integral when the total quantities are absolutely integrable.
			# [[uses::fundamental theorem of complex calculus]].
			==Proof==

			'''Given''': <math>U</math> is a [[domain]] in <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a [[holomorphic function]]. <math>z_0 \in U</math> is such that the disk <math>D</math> of radius <math>r</math> about <math>z_0</math> lies completely inside <math>U</math>, and <math>z</math> is such that <math>\|z - z_0\| < r</math>.

			'''To prove''':

			<math>f(z) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(\xi)}{\xi - z} \, d\xi</math>

			'''Proof''': The proof proceeds in two steps. We first use the Cauchy integral formula for ''constant'' functions (fact (1)) to show that for the constant function <math>w \mapsto f(z)</math>, we have:

			<math>f(z) = \oint_{\partial D} \frac{f(z)}{\xi - z} \, d\xi</math>

			We then show that:

			<math> \oint_{\partial D} \frac{f(\xi)}{\xi - z} \, d\xi = \oint_{\partial D} \frac{f(z)}{\xi - z} \, d\xi</math>.

			Let's do the second step now. We do this by a ''homotopy'' method. Namely, consider the function <math>[0,1] \times \overline{D} \setminus \{ z \} \to \mathbb{C}</math>:

			<math>F(t,w) = \frac{f((1-t)z + tw)}{w - z}</math>.

			Clearly, <math>F</math> is holomorphic in <math>w</math>, and we have:

			<math>\frac{\partial F}{\partial t} = f'((1 - t)z + tw)</math>.

	This ~~is the same integral formula~~, ~~this time applied to ''~~any~~'' simple closed curve and any point~~ in ~~the interior.~~		This yields, by fact (3), that for any <math>t \in [0,1]</math>:

	~~Suppose <math>U</math> is a [[domain]] in~~ <math>\~~mathbb~~{C}~~</math> and <math>f:U~~ \to \~~mathbb~~{C}~~</math> is a [[holomorphic function]]. The '''Cauchy integral formula''' states that if <math>~~\~~gamma</math> is a smooth simple closed curve lying completely inside <math>U</math>~~, ~~and <math>z_0</math> is in the component of the complement of <math>~~\~~gamma</math> that lies completely inside <math>U~~</math>~~, then we have:~~		<math>\oint_{\partial D} \frac{\partial F(t,\xi)}{\partial t} \, d\xi = 0</math>

	<math>~~f(z_0) =~~ \~~frac{~~1~~}{2\pi i} \oint_{\partial D} \frac{f(z)}{z - z_0} \, dz~~</math>		Integrating over <math>t \in [0,1]</math> yields:

	~~In other words~~, ~~the curve doesn'~~t ~~need to be a circle and the point can be anywhere inside.~~		<math>\int_0^1 \oint_{\partial D} \frac{\partial F(t,\xi)}{\partial t} \, d\xi \, dt = 0</math>

	~~===For any union of piecewise smooth curves and any point===~~		We now exchange the two integrals by fact (2). We get:

	{{~~further\|[[Winding number version of Cauchy integral formula]]~~}}		<math>\oint_{\partial D} \int_0^1 \frac{\partial F(t,\xi)}{\partial t} \, dt \, d\xi = 0</math>.

	~~Suppose <math>U</math> is an open subset~~ of ~~<math>\mathbb{C}</math>~~ and ~~<math>f~~:~~U \to \mathbb{C}</math> is a [[holomorphic function]].~~		Next, we apply the fundamental theorem of ''real'' calculus to evaluate the inner integral, and obtain:

	~~Suppose~~ <math>~~c</math> is a sum of oriented piecewise smooth loops <math>c_1, c_2~~, \~~ldots, c_r</math>, each completely inside <math>U</math>. In other words~~, ~~when we use the symbol <math>~~\~~int_c g(z~~) ~~dz</math> we actually mean <math>~~\~~sum_1^r~~ \~~int_{c_r} g(z) dz~~</math>.		<math>\oint_{\partial D} F(1,\xi) - F(0,\xi) \, d\xi = 0</math>.

	~~Denote by <math>n(c;z_0)</math> the sum of the winding numbers of the <math>c_k</math> about <math>z_0</math>. Suppose <math>c</math> is zero-homologous, i.e. <math>n(c;z) = 0</math> for <math>z \~~in ~~\mathbb{C}\setminus U</math>. Then we have, for any <math>z_0 \in U</math>~~:		Plugging in values yields:

	<math>~~n(c;z_0)f(z_0) =~~ \~~frac{1}~~{2 \~~pi i~~} ~~\int_c~~ \frac{f(z)}{z - ~~z_0~~} ~~\, dz~~</math>		<math>\oint_{\partial D} \frac{f(z) - f(\xi)}{z - \xi} = 0</math>

	~~==Related facts==~~		which, on rearrangement, gives the desired equality:

	* [[Cauchy integral formula for derivatives]]		<math> \oint_{\partial D} \frac{f(\xi)}{\xi - z} \, d\xi = \oint_{\partial D} \frac{f(z)}{\xi - z} \, d\xi</math>.
	* [[Winding number version of Cauchy integral formula]]
	* [[Homotopy-~~invariance formulation of Cauchy's theorem]]~~

Vipul at 16:27, 12 September 2008

2008-09-12T16:27:30Z

← Older revision		Revision as of 16:27, 12 September 2008
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	===For a circle with respect to any point in the interior===		===For a circle with respect to any point in the interior===

	Suppose <math>U</math> is a [[domain]] in <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a [[holomorphic function]]. The '''Cauchy integral formula''' states that if <math>z_0 \in U</math> is such that the ~~disc~~ of radius <math>r</math> about <math>z_0</math> lies completely inside <math>U</math>, and if <math>z</math> is such that <math>\|z - z_0\| < r</math>, we have:		Suppose <math>U</math> is a [[domain]] in <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a [[holomorphic function]]. The '''Cauchy integral formula''' states that if <math>z_0 \in U</math> is such that the disk <math>D</math> of radius <math>r</math> about <math>z_0</math> lies completely inside <math>U</math>, and if <math>z</math> is such that <math>\|z - z_0\| < r</math>, we have:

	<math>f(z) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(\xi)}{\xi - z} \, d\xi</math>		<math>f(z) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(\xi)}{\xi - z} \, d\xi</math>

	In other words, we can determine the value of <math>f</math> at the ~~center~~, by knowing its value on the boundary.		In other words, we can determine the value of <math>f</math> at ''any point in the interior'', by knowing its value at all points on the boundary.

	===For any simple closed curve and any point in the interior===		===For any simple closed curve and any point in the interior===

Vipul: 11 revisions

2008-05-18T19:10:34Z

11 revisions

← Older revision	Revision as of 19:10, 18 May 2008
(No difference)

Vipul: /* Related facts */

2008-04-26T18:46:47Z

Related facts

← Older revision		Revision as of 18:46, 26 April 2008
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	* [[Cauchy integral formula for derivatives]]		* [[Cauchy integral formula for derivatives]]
			* [[Winding number version of Cauchy integral formula]]
			* [[Homotopy-invariance formulation of Cauchy's theorem]]

Vipul: /* For any union of piecewise smooth curves and any point */

2008-04-26T18:42:02Z

For any union of piecewise smooth curves and any point

← Older revision		Revision as of 18:42, 26 April 2008
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	Suppose <math>U</math> is an open subset of <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a [[holomorphic function]].		Suppose <math>U</math> is an open subset of <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a [[holomorphic function]].

	Suppose <math>c</math> is a sum of oriented piecewise smooth loops <math>c_1, c_2, \ldots, c_r</math>, each completely inside <math>U</math>. In other words, when we use the symbol <math>\int_c g(z) dz</math>, we actually mean <math>\sum_1^r \int_{c_r} g(z) dz</math>.		Suppose <math>c</math> is a sum of oriented piecewise smooth loops <math>c_1, c_2, \ldots, c_r</math>, each completely inside <math>U</math>. In other words, when we use the symbol <math>\int_c g(z) dz</math> we actually mean <math>\sum_1^r \int_{c_r} g(z) dz</math>.

	Denote by <math>n(c;z_0)</math> the sum of the winding numbers of the <math>c_k</math> about <math>z_0</math>. Suppose <math>c</math> is zero-homologous, i.e. <math>n(c;z) = 0</math> for <math>z \in \mathbb{C}\setminus U</math>. Then we have, for any <math>z_0 \in U</math>:		Denote by <math>n(c;z_0)</math> the sum of the winding numbers of the <math>c_k</math> about <math>z_0</math>. Suppose <math>c</math> is zero-homologous, i.e. <math>n(c;z) = 0</math> for <math>z \in \mathbb{C}\setminus U</math>. Then we have, for any <math>z_0 \in U</math>:

Vipul at 18:41, 26 April 2008

2008-04-26T18:41:26Z

← Older revision		Revision as of 18:41, 26 April 2008
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			{{basic fact}}
	==Statement==		==Statement==

Vipul: /* For a circle with respect to any point in the interior */

2008-04-26T18:41:10Z

For a circle with respect to any point in the interior

← Older revision		Revision as of 18:41, 26 April 2008
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	===For a circle with respect to any point in the interior===		===For a circle with respect to any point in the interior===

	Suppose <math>U</math> is a [[domain]] in <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a [[holomorphic function]]. The '''Cauchy integral formula''' states that if <math>z_0 \in U</math> is such that the disc of radius <math>r</math> about <math>z_0</math> lies completely inside <math>U</math>, and if <math>z</math> in such that <math>\|z - z_0\| < r</math>, we have:		Suppose <math>U</math> is a [[domain]] in <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a [[holomorphic function]]. The '''Cauchy integral formula''' states that if <math>z_0 \in U</math> is such that the disc of radius <math>r</math> about <math>z_0</math> lies completely inside <math>U</math>, and if <math>z</math> is such that <math>\|z - z_0\| < r</math>, we have:

	<math>f(z) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(\xi)}{\xi - z} \, d\xi</math>		<math>f(z) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(\xi)}{\xi - z} \, d\xi</math>

Vipul: /* For any union of piecewise smooth curves and any point */

2008-04-19T14:15:42Z

For any union of piecewise smooth curves and any point

← Older revision		Revision as of 14:15, 19 April 2008
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	===For any union of piecewise smooth curves and any point===		===For any union of piecewise smooth curves and any point===

			{{further\|[[Winding number version of Cauchy integral formula]]}}

	Suppose <math>U</math> is an open subset of <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a [[holomorphic function]].		Suppose <math>U</math> is an open subset of <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a [[holomorphic function]].

Vipul: /* For any union of piecewise smooth curves and any point */

2008-04-19T14:14:39Z

For any union of piecewise smooth curves and any point

← Older revision		Revision as of 14:14, 19 April 2008
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	===For any union of piecewise smooth curves and any point===		===For any union of piecewise smooth curves and any point===

	{{~~fillin~~}} ~~This is in terms~~ of the winding numbers		Suppose <math>U</math> is an open subset of <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a [[holomorphic function]].

			Suppose <math>c</math> is a sum of oriented piecewise smooth loops <math>c_1, c_2, \ldots, c_r</math>, each completely inside <math>U</math>. In other words, when we use the symbol <math>\int_c g(z) dz</math>, we actually mean <math>\sum_1^r \int_{c_r} g(z) dz</math>.

			Denote by <math>n(c;z_0)</math> the sum of the winding numbers of the <math>c_k</math> about <math>z_0</math>. Suppose <math>c</math> is zero-homologous, i.e. <math>n(c;z) = 0</math> for <math>z \in \mathbb{C}\setminus U</math>. Then we have, for any <math>z_0 \in U</math>:

			<math>n(c;z_0)f(z_0) = \frac{1}{2 \pi i} \int_c \frac{f(z)}{z - z_0} \, dz</math>

	==Related facts==		==Related facts==

	* [[Cauchy integral formula for derivatives]]		* [[Cauchy integral formula for derivatives]]

Vipul at 13:49, 19 April 2008

2008-04-19T13:49:34Z

← Older revision		Revision as of 13:49, 19 April 2008
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	{{fillin}} This is in terms of the winding numbers		{{fillin}} This is in terms of the winding numbers

			==Related facts==

			* [[Cauchy integral formula for derivatives]]