Next: 12.5 Logarithms Up: 12. Power Series Previous: 12.3 Differentiation of Power   Index

# 12.4 The Exponential Function

12.22   Example. Suppose we had a complex function such that is everywhere differentable and
 (12.23)

Let for all . By the chain and product rules,

on , so is constant. Since , we conclude
 (12.24)

In particular is never , and

Now let and define a function by

We have

for all , so is constant, and . Thus

and by (12.24),
 (12.25)

Next suppose you know some function such that for all and . (You do know such a function from your previous calculus course.) Let

Then by the product and chain rules,

so is constant on , and since , we have . By (12.24),

Now I will try to construct a function satisfying the differential equation (12.23) by hoping that is given by a power series. Suppose

Since , we must have , and

By the differentiation theorem,

and

By the differentiation theorem again,

so

Hence

Repeating the process, we get

so

I see a pattern here: .

12.26   Definition (Exponential function.) Let denote the power series . We will show in exercise 12.31 that has infinite radius of convergence. We write

12.27   Theorem. and .

Proof: It is clear that . The formal derivative of is

so the [still unproved] differentiation theorem says that . It follows from our discussion above that is never ,
 (12.28)

and
 (12.29)

It is clear that is real for all . In fact, we must have for all , since is continuous (differentiable functions are continuous) and if for some , the intermediate value theorem would say for some between and . Since on , is strictly increasing on .

12.30   Definition (.) We define to be the number ; i.e., .

12.31   Exercise. Show that has infinite radius of convergence.

12.32   Exercise. Use the definition of to show that .

12.33   Exercise. A
a) Show that for all , .
b) Show that for all , .

12.34   Exercise. A From the previous exercise, it follows that

Use this to prove that

i.e.,

(Note that for , this says

12.35   Notation (.) Another notation for is . This notation is motivated by the previous exercise. With this notation, we have

12.36   Theorem. Every number can be written as for a unique .

Proof: The uniqueness of follows from the fact that is strictly increasing on . Let . From the expansion , we see that . Since is continuous, we can apply the intermediate value theorem to on to conclude for some . If , then , so for some , and where . Since , the theorem has been proved in all cases.

Next: 12.5 Logarithms Up: 12. Power Series Previous: 12.3 Differentiation of Power   Index