1549901369-Elements_of_Real_Analysis__Denlinger_

8.8 *Elementary Transcendental Functions (Project) 537

functions and derive their familiar properties as if we had never encountered

them before. We will use power series as our foundation.

EXPONENTIAL AND LOGARITHM FUNCTIONS

00 k

Definition 8.8.1 We define the function E: JR ---t JR by E(x) = L ~!.

k=O

(This series converges absolutely 't/x E R)

Theorem 8.8.2 The function E : JR ---t JR has the following properties:

(a) E(O) = 1.

(b) 't/x E JR, E(x) > 0.

(c) 't/x E JR, E is differentiable at x, and E'(x) = E(x).

(d) E is strictly increasing on R

Definition 8.8.3 e = E(l). Note that 2 < e < 3.

Remark 8.8.4 The series fore converges rapidly; in fact, le - f= k\ I < +.
k=O. n.n

(

00 1 1 00 1 )

To prove this, show that L -k 1 < I L ( )k •

k=n+l · n. k=O n + 1

Corollary 8.8.5 e is irrational.

Proof. For contradiction, suppose e is rational; say e = m/n where m, n E
n 1
N. Let Sn= L 1. Then, applying Remark 8.8.4,
k=O k.
0 < n!(e - Sn) < ~ < 1.
On the other hand, n!(e-Sn) = n! (~ - 1 - fr - ~ -if -· · · -~),which
must be a positive integer. In that case we would have a positive integer between
0 and 1 , a contradiction. Therefore, e is irrational. •

Theorem 8.8.6 The function E : JR ---t JR has the following properties:

(a) 't/x, y E JR, E(x)E(y) = E(x + y). [Use the Cauchy product formula.]

(b) 't/x,y E JR, E(x)/E(y) = E(x -y) and E(-x) = l/E(x).

(c) 't/x E JR, 'tin EN, E(nx) = [E(x)]n.