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2008-05-18T19:12:47Z

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Vipul: New page: ==Definition== ===For an open subset of the complex numbers=== Suppose $U$ is an open subset of $\mathbb{C}$ , and $f:U \to \mathbb{C}$ is a function. Let...

2008-04-18T20:14:53Z

New page: ==Definition== ===For an open subset of the complex numbers=== 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...

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==Definition==

===For an open subset of the complex numbers===

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 U</math> be a point. We say that <math>f</math> is '''real-differentiable''' at <math>z_0</math> if there exists a matrix <math>(Df) \in M_2(\R)</math> such that:

<math>\lim_{h \to 0} \frac{f(z + h) - f(z) - (Df)(h)}{\| h \|} = 0</math>

Here <math>(Df)(h)</math> denotes <math>Df</math> multiplied with <math>h</math> as a column vector.

Such a <math>Df</math> is termed a real differential of <math>f</math>.

Function real-differentiable at a point - Revision history

Vipul: 1 revision

Vipul: New page: ==Definition== ===For an open subset of the complex numbers=== Suppose U is an open subset of \mathbb{C}, and f:U \to \mathbb{C} is a function. Let...

Vipul: New page: ==Definition== ===For an open subset of the complex numbers=== Suppose $U$ is an open subset of $\mathbb{C}$ , and $f:U \to \mathbb{C}$ is a function. Let...