
Chapter 2 5
Elementar y Functions
and Their Properties
⋆
Throughout Chapter 25 it is assumed that
n
is a positive integer unless otherwise spec-
ified.
25.1 Power, Exponential, and Logarithmic Functions
25.1.1 Properties of the Power Function
Basic properties of the power function:
x
α
x
β
= x
α+β
, (x
1
x
2
)
α
= x
α
1
x
α
2
, (x
α
)
β
= x
αβ
,
for any α and β, where x > 0, x
1
> 0, x
2
> 0.
Differentiation and integration formulas:
(x
α
)
′
= αx
α−1
,
Z
x
α
dx =
x
α+1
α + 1
+ C if α 6= −1,
ln |x| + C if α = −1.
The Taylor series expansion in a neighborhood of an arbitrary point:
x
α
=
∞
X
n=0
C
n
α
x
α−n
0
(x −x
0
)
n
for |x −x
0
| < |x
0
|,
where C
n
α
=
α(α − 1) . . . (α −n + 1)
n!
are binomial coefficients.
25.1.2 Properties ...