Cauchy-Riemann differential equations - Revision history

Vipul: 3 revisions

2008-05-18T19:10:12Z

3 revisions

← Older revision	Revision as of 19:10, 18 May 2008
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Vipul at 00:49, 18 April 2008

2008-04-18T00:49:17Z

← Older revision		Revision as of 00:49, 18 April 2008
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	{{quotation\|<math>u_x = v_y, \qquad u_y = -v_x</math>}}		{{quotation\|<math>u_x = v_y, \qquad u_y = -v_x</math>}}

	Note that if we are given that <math>u</math> ~~and~~ <math>v</math> ~~are both continuously differentiable~~ in the real sense, then satisfying the Cauchy-Riemann differential equations at <math>z_0</math> is equivalent to being complex-differentiable at <math>z_0</math>.		We can talk of a [[function satisfying Cauchy-Riemann differential equations at a point]].
			Note that if we are given that <math>f</math> is differentiable at <math>z_0</math> in the real sense, then satisfying the Cauchy-Riemann differential equations at <math>z_0</math> is equivalent to being complex-differentiable at <math>z_0</math>. {proofat\|[[Real-differentiable and Cauchy-Riemann equals complex-differentiable]]}}

Vipul: /* Definition */

2008-04-13T21:16:05Z

Definition

← Older revision		Revision as of 21:16, 13 April 2008
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	We say that <math>f</math> satisfies the '''Cauchy-Riemann differential equations''' at a point <math>z_0 \in U</math> if the partial derivatives of the <math>u,v</math> in both the <math>x</math> and <math>y</math> directions exist, and satisfy the following conditions:		We say that <math>f</math> satisfies the '''Cauchy-Riemann differential equations''' at a point <math>z_0 \in U</math> if the partial derivatives of the <math>u,v</math> in both the <math>x</math> and <math>y</math> directions exist, and satisfy the following conditions:

	<math>\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}</math>		{{quotation\|<math>\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}</math>}}

	Using the subscript notation for partial derivatives, we can write this more compactly as:		Using the subscript notation for partial derivatives, we can write this more compactly as:

	<math>u_x = v_y, u_y = -v_x</math>		{{quotation\|<math>u_x = v_y, \qquad u_y = -v_x</math>}}

	Note that if we are given that <math>u</math> and <math>v</math> are both continuously differentiable in the real sense, then satisfying the Cauchy-Riemann differential equations at <math>z_0</math> is equivalent to being complex-differentiable at <math>z_0</math>.		Note that if we are given that <math>u</math> and <math>v</math> are both continuously differentiable in the real sense, then satisfying the Cauchy-Riemann differential equations at <math>z_0</math> is equivalent to being complex-differentiable at <math>z_0</math>.

Vipul: New page: ==Definition== Suppose $U$ is an open subset of $\mathbb{C}$ and $f:U \to \mathbb{C}$ is a function. Write $f$ in terms of its real and imagina...

2008-04-13T21:05:40Z

New page: ==Definition== Suppose <math>U</math> is an open subset of <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a function. Write <math>f</math> in terms of its real and imagina...

New page

==Definition==

Suppose <math>U</math> is an open subset of <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a function. Write <math>f</math> in terms of its real and imaginary parts as follows:

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

We say that <math>f</math> satisfies the '''Cauchy-Riemann differential equations''' at a point <math>z_0 \in U</math> if the partial derivatives of the <math>u,v</math> in both the <math>x</math> and <math>y</math> directions exist, and satisfy the following conditions:

<math>\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}</math>

Using the subscript notation for partial derivatives, we can write this more compactly as:

<math>u_x = v_y, u_y = -v_x</math>

Note that if we are given that <math>u</math> and <math>v</math> are both continuously differentiable in the real sense, then satisfying the Cauchy-Riemann differential equations at <math>z_0</math> is equivalent to being complex-differentiable at <math>z_0</math>.

Cauchy-Riemann differential equations - Revision history

Vipul: 3 revisions

Vipul at 00:49, 18 April 2008

Vipul: /* Definition */

Vipul: New page: ==Definition== Suppose U is an open subset of \mathbb{C} and f:U \to \mathbb{C} is a function. Write f in terms of its real and imagina...

Vipul: New page: ==Definition== Suppose $U$ is an open subset of $\mathbb{C}$ and $f:U \to \mathbb{C}$ is a function. Write $f$ in terms of its real and imagina...