Winding number version of Cauchy integral formula - Revision history

Vipul: 5 revisions

2008-05-18T19:19:54Z

5 revisions

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Vipul at 19:03, 26 April 2008

2008-04-26T19:03:14Z

← Older revision		Revision as of 19:03, 26 April 2008
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			{{basic fact}}
	==Statement==		==Statement==

Vipul at 02:11, 20 April 2008

2008-04-20T02:11:09Z

← Older revision		Revision as of 02:11, 20 April 2008
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	==Proof==		==Proof==

			===Proof outline===

			* We define two functions: one on the open set <math>U</math>, and the other on the open set <math>V</math> of those points around which <math>c</math> has winding number zero.

			===Proof details===

	The idea here is to define analytic functions on two open subsets, show that they agree on the overlap, and hence obtain an analytic function on the whole of <math>\mathbb{C}</math>.		The idea here is to define analytic functions on two open subsets, show that they agree on the overlap, and hence obtain an analytic function on the whole of <math>\mathbb{C}</math>.

Vipul: /* Proof */

2008-04-19T14:31:46Z

Proof

← Older revision		Revision as of 14:31, 19 April 2008
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	Then <math>U \cup V = \mathbb{C}</math>, and <math>V</math> is an open subset. Consider the function <math>g_1, g_2:U, V \to \mathbb{C}</math>:		Then <math>U \cup V = \mathbb{C}</math>, and <math>V</math> is an open subset. Consider the function <math>g_1, g_2:U, V \to \mathbb{C}</math>:

	<math>g_1(z) := \oint_c \varphi(z,w) \, dz</math>		<math>g_1(z) := \oint_c \varphi(z,w) \, dw</math>

	and:		and:
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	which is zero for points in the intersection, as they are points in <math>U</math> about which the winding number is zero.		which is zero for points in the intersection, as they are points in <math>U</math> about which the winding number is zero.

			Thus, we can paste them together to get a single holomorphic function <math>g: \mathbb{C} \to \mathbb{C}</math>. It is clear from the expression that <math>g(z) \to 0</math> as <math>\|z\| \to \infty</math>; hence, using the fact that [[bounded and entire implies constant\|any bounded entire function is constant]], we obtain that <math>g \equiv 0</math>. This shows that:

			<math>\oint_c \frac{f(z)}{z - w} \, dw = \oint_c \frac{f(w)}{z - w} \, dw</math>

			The left side evaluates to <math>n(c;z)f(z)</math>: precisely what we want.

Vipul: /* Proof */

2008-04-19T14:25:37Z

Proof

← Older revision		Revision as of 14:25, 19 April 2008
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	Now define an open subset:		Now define an open subset:

	<math>V := \{ z \in \C \setminus c \mid n(c;z) = 0 \}</math>		<math>V := \{ z \in \mathbb{C} \setminus c \mid n(c;z) = 0 \}</math>

	Then <math>U \cup V = \mathbb{C}</math>, and <math>V</math> is an open subset. Consider the function <math>g_1, g_2:U, V \to \mathbb{C}</math>:		Then <math>U \cup V = \mathbb{C}</math>, and <math>V</math> is an open subset. Consider the function <math>g_1, g_2:U, V \to \mathbb{C}</math>:

Vipul: New page: ==Statement== Suppose $U$ is an open subset of $\mathbb{C}$ and $f:U \to \mathbb{C}$ is a holomorphic function. Suppose $c$ is a sum of or...

2008-04-19T14:25:08Z

New page: ==Statement== 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 or...

New page

==Statement==

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>\oint_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} \oint_c \frac{f(z)}{z - z_0} \, dz</math>

==Proof==

The idea here is to define analytic functions on two open subsets, show that they agree on the overlap, and hence obtain an analytic function on the whole of <math>\mathbb{C}</math>.

Define first:

<math>\varphi(z,w) = \frac{f(z) - f(w)}{z - w}</math> if <math>z \ne w</math>, <math>f'(z)</math> if <math>z = w</math>

Observe that <math>\varphi</math> is analytic in each variable. To see this, note that for fixed <math>z</math>, <math>\varphi</math> is analytic in <math>w</math> for <math>w</math> away from <math>z</math>, and the power series shows that it is analytic in a neighborhood of <math>z</math>.

Now define an open subset:

<math>V := \{ z \in \C \setminus c \mid n(c;z) = 0 \}</math>

Then <math>U \cup V = \mathbb{C}</math>, and <math>V</math> is an open subset. Consider the function <math>g_1, g_2:U, V \to \mathbb{C}</math>:

<math>g_1(z) := \oint_c \varphi(z,w) \, dz</math>

and:

<math>g_2(z) := \oint_c \frac{f(w)}{w - z} \, dw</math>

To see that <math>g_1</math> and <math>g_2</math> agree on the intersection, observe that on the intersection <math>U \cap V</math>, the difference is given by:

<math>\oint_c \frac{f(z)}{z - w} \, dz</math>

which is zero for points in the intersection, as they are points in <math>U</math> about which the winding number is zero.

Winding number version of Cauchy integral formula - Revision history

Vipul: 5 revisions

Vipul at 19:03, 26 April 2008

Vipul at 02:11, 20 April 2008

Vipul: /* Proof */

Vipul: /* Proof */

Vipul: New page: ==Statement== Suppose U is an open subset of \mathbb{C} and f:U \to \mathbb{C} is a holomorphic function. Suppose c is a sum of or...

Vipul: New page: ==Statement== Suppose $U$ is an open subset of $\mathbb{C}$ and $f:U \to \mathbb{C}$ is a holomorphic function. Suppose $c$ is a sum of or...