What is a complex derivative


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10.1 Derivatives of Complex Functions

You are familiar with derivatives of functions from to , and with the motivation of the definition of derivative as the slope of the tangent to a curve. For complex functions, the geometrical motivation is missing, but the definition is formally the same as the definition for derivatives of real functions.

By the definition of limit, we can say that is differentiable at if , and is a limit point of and there exists a function such that is continuous at , and such that

(10.2)

and in this case is equal to .

It is sometimes useful to rephrase condition (10.2) as follows: is differentiable at if , is a limit point of , and there is a function such that is continuous at , and

(10.3)

In this case, .

Proof: Since are differentiable at , there are functions , such that , are continuous at , and




It follows that

and is continuous at .

We can let and we see is differentiable at and


Proof: The proof is left to you.

Proof: From our hypotheses, there exist functions


such that is continuous at , is continuous at and

If , then , so we can replace in (10.15) by to get

Using (10.14) to rewrite , we get

Hence we have

and is continuous at . Hence is differentiable at and



Proof: If , we saw above that is differentiable and . Let be a complex function, and let . Suppose is differentiable at , and . Then . By the chain rule is differentiable at , and




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