Vipul: 2 revisions

2008-05-18T19:11:08Z

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Vipul at 17:58, 16 April 2008

2008-04-16T17:58:19Z

← Older revision		Revision as of 17:58, 16 April 2008
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	The ratio and limit are evaluated as complex numbers.		The ratio and limit are evaluated as complex numbers.

			If <math>f</math> has a complex differential at <math>z_0</math>, we say that <math>f</math> is [[function complex-differentiable at a point\|complex-differentiable]] at <math>z_0</math>.

	===Definition as limits from the real and imaginary directions===		===Definition as limits from the real and imaginary directions===

Vipul: New page: ==Definition== ===Definition as a general limit=== Suppose $U$ is an open subset of $\mathbb{C}$ and $f:U \to \mathbb{C}$ is a function. Let $z_0 \i...$

2008-04-13T20:49:38Z

New page: ==Definition== ===Definition as a general limit=== Suppose <math>U</math> is an open subset of <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a function. Let <math>z_0 \i...

New page

==Definition==

===Definition as a general limit===

Suppose <math>U</math> is an open subset of <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a function. Let <math>z_0 \in \mathbb{C}</math>. Then, the '''complex differential''' of <math>f</math> at <math>z_0</math> is given by:

<math>f'(z_0) := \lim_{z \to z_0} \frac{f(z) - f(z_0)}{z - z_0}</math>

The ratio and limit are evaluated as complex numbers.

===Definition as limits from the real and imaginary directions===

If <math>f</math> is complex-differentiable at <math>z_0 \in \mathbb{C}</math>, then we can compute its derivative by using a linear direction of approach. For instance, we can look at <math>z = z_0 + h</math>, where <math>h \in \R</math>, and take the limit as <math>h \to 0</math>. Thus, if we write:

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

where <math>u,v</math> are real-valued functions, then we get:

<math>f'(z_0) = \frac{\partial u}{\partial x}(z_0) + i \frac{\partial v}{\partial x}(z_0)</math>

Similarly, we can consider approach along the ''imaginary'' direction, namely, <math>z = z_0 + ih</math>, where <math>h \in \R</math>, and let <math>h \to 0</math>. We then get:

<math>f'(z_0) = \frac{\partial v}{\partial y}(z_0) - i \frac{\partial u}{\partial y}(z_0)</math>

It turns out that if <math>f</math> is continuously differentiable in the real sense at <math>z_0</math>, ''and'' the two notions of differential above coincide at <math>z_0</math>, then <math>f</math> is complex-differentiable at <math>z_0</math>, and the complex differential equals either of the expressions. The equality of the two expressions is termed the [[Cauchy-Riemann differential equations]]:

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

Complex differential of a complex-valued function - Revision history

Vipul: 2 revisions

Vipul at 17:58, 16 April 2008

Vipul: New page: ==Definition== ===Definition as a general limit=== Suppose U is an open subset of \mathbb{C} and f:U \to \mathbb{C} is a function. Let z_0 \i...

Vipul: New page: ==Definition== ===Definition as a general limit=== Suppose $U$ is an open subset of $\mathbb{C}$ and $f:U \to \mathbb{C}$ is a function. Let $z_0 \i...$