Complex differential of a complex-valued function: Difference between revisions
(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...) |
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Revision as of 20:49, 13 April 2008
Definition
Definition as a general limit
Suppose is an open subset of and is a function. Let . Then, the complex differential of at is given by:
The ratio and limit are evaluated as complex numbers.
Definition as limits from the real and imaginary directions
If is complex-differentiable at , then we can compute its derivative by using a linear direction of approach. For instance, we can look at , where , and take the limit as . Thus, if we write:
where are real-valued functions, then we get:
Similarly, we can consider approach along the imaginary direction, namely, , where , and let . We then get:
It turns out that if is continuously differentiable in the real sense at , and the two notions of differential above coincide at , then is complex-differentiable at , and the complex differential equals either of the expressions. The equality of the two expressions is termed the Cauchy-Riemann differential equations: