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Physical Mathematics by Kevin Cahill

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5

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Complex-variable theory

5.1 Analytic functions

A complex-valued function f(z) of a complex variable z is differentiable at z with derivative f′(z) if the limit

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(5.1)

exists as z′ approaches z from any direction in the complex plane. The limit must exist no matter how or from what direction z′ approaches z.

If the function f(z) is differentiable in a small disk around a point z0, then f(z) is said to be analytic at z0 (and at all points inside the disk).

Example 5.1 (Polynomials) If f(z) = zn for some integer n, then for tiny dz and ...

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