Complex differential equals de Rham derivative - Revision history

Vipul: 5 revisions

2008-05-18T19:11:03Z

5 revisions

← Older revision	Revision as of 19:11, 18 May 2008
(No difference)

Vipul at 19:09, 26 April 2008

2008-04-26T19:09:24Z

← Older revision		Revision as of 19:09, 26 April 2008
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			{{companal-derham fact}}
	==Statement==		==Statement==

Vipul: /* Proof */

2008-04-26T19:08:02Z

Proof

← Older revision		Revision as of 19:08, 26 April 2008
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	<math>f'(z)dz = f'(z) (dx + idy) = f'(z)dx + if'(z)dy</math>		<math>f'(z)dz = f'(z) (dx + idy) = f'(z)dx + if'(z)dy</math>

	We now expand <math>f'(z) dx</math> using the first description of <math>f'(z)</math>, and <math>f'(z) dy</math? using the second description, and observe that we get the precise expression for <math>df</math>.		We now expand <math>f'(z) dx</math> using the first description of <math>f'(z)</math>, and <math>f'(z) dy</math> using the second description, and observe that we get the precise expression for <math>df</math>.

Vipul: /* Definitions used */

2008-04-26T19:07:15Z

Definitions used

← Older revision		Revision as of 19:07, 26 April 2008
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	Then, we define:		Then, we define:

	{{quotation\|<math>df := \frac{\partial u}{\partial x}dx + i \frac{\partial v}{\partial x} dx + \frac{\partial u}{\partial y} dy + i \frac{\partial v}{\partial y} dy</math>		{{quotation\|<math>df := \frac{\partial u}{\partial x}dx + i \frac{\partial v}{\partial x} dx + \frac{\partial u}{\partial y} dy + i \frac{\partial v}{\partial y} dy</math>}}

	And we define:		And we define:

Vipul: /* Definitions used */

2008-04-26T19:07:03Z

Definitions used

← Older revision		Revision as of 19:07, 26 April 2008
Line 17:		Line 17:
	Then, we define:		Then, we define:

	<math>df := \frac{\partial u}{\partial x}dx + i \partial{v}{\partial x} dx + \frac{\partial u}{\partial y} dy + i \frac{\partial v}{\partial y} dy</math>		{{quotation\|<math>df := \frac{\partial u}{\partial x}dx + i \frac{\partial v}{\partial x} dx + \frac{\partial u}{\partial y} dy + i \frac{\partial v}{\partial y} dy</math>

	And we define:		And we define:

	<math>f'(z)dz = f'(z)(dx + i dy)</math>		{{quotation\|<math>f'(z)dz = f'(z)(dx + i dy)</math>}}

	==Facts used==		==Facts used==

Vipul: New page: ==Statement== Suppose $U \subset \mathbb{C}$ is an open subset, and $f: U \to \mathbb{C}$ is a holomorphic function. Let $f'$ denote the [[complex dif...

2008-04-18T21:40:26Z

New page: ==Statement== Suppose <math>U \subset \mathbb{C}</math> is an open subset, and <math>f: U \to \mathbb{C}</math> is a holomorphic function. Let <math>f'</math> denote the [[complex dif...

New page

==Statement==

Suppose <math>U \subset \mathbb{C}</math> is an open subset, and <math>f: U \to \mathbb{C}</math> is a [[holomorphic function]]. Let <math>f'</math> denote the [[complex differential]] of <math>f</math>. Then, we have:

<math>df = f'(z) dz</math>

Here, <math>df</math> denotes the de Rham derivative of <math>f</math>.

==Definitions used==

Let us write:

<math>f(z) = u(z) + iv(z)</math>

where <math>u,v</math> are respectively the real and imaginary parts of <math>f</math>.

Then, we define:

<math>df := \frac{\partial u}{\partial x}dx + i \partial{v}{\partial x} dx + \frac{\partial u}{\partial y} dy + i \frac{\partial v}{\partial y} dy</math>

And we define:

<math>f'(z)dz = f'(z)(dx + i dy)</math>

==Facts used==

We use the fact that since <math>f</math> is holomorphic, then:

<math>f'(z) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} = \frac{\partial v}{\partial y} - i \frac{\partial u}{\partial y}</math>

==Proof==

We observe that:

<math>f'(z)dz = f'(z) (dx + idy) = f'(z)dx + if'(z)dy</math>

We now expand <math>f'(z) dx</math> using the first description of <math>f'(z)</math>, and <math>f'(z) dy</math? using the second description, and observe that we get the precise expression for <math>df</math>.

Complex differential equals de Rham derivative - Revision history

Vipul: 5 revisions

Vipul at 19:09, 26 April 2008

Vipul: /* Proof */

Vipul: /* Definitions used */

Vipul: /* Definitions used */

Vipul: New page: ==Statement== Suppose U \subset \mathbb{C} is an open subset, and f: U \to \mathbb{C} is a holomorphic function. Let f' denote the [[complex dif...

Vipul: New page: ==Statement== Suppose $U \subset \mathbb{C}$ is an open subset, and $f: U \to \mathbb{C}$ is a holomorphic function. Let $f'$ denote the [[complex dif...