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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.

10.1   Definition (Derivative.) Let be a complex valued function with , let be a point such that , and is a limit point of . We say is differentiable at if the limit

exists. In this case, we denote this limit by and call the derivative of at .

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, .

10.4   Remark. It follows immediately from (10.3) that if is differentiable at , then is continuous at .

10.5   Example. Let be given by

and let . Then for all ,

If we define by

then is continuous at , so is differentiable at and

We could also write this calculation as

Hence is differentiable at and for all .

10.6   Example. Let for and let . Then for all

Let be defined by

Then is continuous at , so is differentiable at , and

for all .

10.7   Warning. The function should not be confused with . In the example above

Also it is not good form to say
 (10.8)

without specifying the condition  for ," since someone reading (10.8) would assume is undefined at .

10.9   Example. Let for all , and let . Let

I claim that does not have a limit at , and hence is a nowhere differentiable function.

Let

Then and are sequences in both of which converge to . For all ,

so and , and hence does not have a limit at .

10.10   Exercise. A Investigate the following functions for differentiability at an arbitrary point . Calculate the derivatives of any differentiable functions.
a)
are given complex numbers.
b)
.
c)
, i.e. .

10.11   Theorem (Sum theorem for differentiable functions.)Let
be complex functions, and suppose and are differentiable at . Suppose is a limit point of . Then is differentiable at and .

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

10.12   Theorem. Let be a complex function and let . If is differentiable at , then is differentiable at and .

Proof: The proof is left to you.

10.13   Theorem (Chain Rule.) Let be complex functions, and let . Suppose is differentiable at , and is differentiable at , and that is a limit point of . Then the composition is differentiable at , and

Proof: From our hypotheses, there exist functions

such that is continuous at , is continuous at and
 (10.14) (10.15)

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

10.16   Theorem (Reciprocal rule.) Let be a complex function, and let . If is differentiable at and , then 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

10.17   Exercise (Product rule.) A Let be complex functions. Suppose and are both differentiable at , and that is a limit point of . Show that is differentiable at , and that

10.18   Exercise (Power rule.) Let be a complex function, and suppose that is differentiable at . Show that is differentiable at for all and

(Use induction.)

10.19   Exercise (Power rule.) A Let be a complex function. suppose that is differentiable at , and . Show that is differentiable at for all , and that

for all

10.20   Exercise (Quotient rule.) A Let be complex functions and let
. Suppose and are differentiable at and , and is a limit point of . Show that is differentiable at and

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