80 Chapter 3Transcendental functions
3.6 The exponential function
An exponential function has the form
f(x) 1 = 1 a
x(3.29)
in which the number ais the baseof the function and the variable xis the exponent.
The best known example is the function 10
xwhich, when xis an integer, is used for
the representation of numbers in the decimal system. The exponential function that
occurs in statistics and in the physical sciences, theexponential function, is
exp(x) 1 = 1 e
x(3.30)
in which the base eis the Euler number (see Section 1.4).
0 Exercise 32
The function is defined by the infinite series (see Chapter 7)
5(3.31)
The value of the exponential function can be computed from the series to any desired
accuracy, although the number of terms required increases rapidly as |x|increases
(see Example 1.10 forx 1 = 11 ).
EXAMPLE 3.19Calculateexp(− 2 x
2)to 8 significant figures forx 1 = 1 0.1.
The value of the exponential can be obtained directly by substituting (−0.02)for xin
(3.31). Alternatively,
Substituting forx 1 = 1 0.1, and holding 2 significant figures more than the required 8 for
intermediate values,
exp(−0.02) 1 ≈ 111 − 1 0.02 1 + 1 0.000 121 − 1 0.000 10011333131 + 1 0.000 1000100617
− 1 0.3 1 × 110
− 101 +1-
exp( )
()()()(
−=+
−
!
−
!
−
!
−
21
2
1
2
2
2
3
222223x
xx x 22
4
2
5
12 2
4
3
2
3
4
1
24 2524 6 8xx
xx x x
)( )
!
−
!
=−+− + −
55
10x +
e
x
n
xx x x
xnn=
!
=+
!
!
!
!
=∑
02341
12 3 4
∞5In his influential textbook Introductio in analysin infinitorumof 1748, Euler defined the exponential and
logarithmic functions in terms of infinite series, and showed them to be inverse functions. In this book Euler laid
the foundation for the branch of mathematics called analysis, the study of infinite processes, as in infinite series
and in the calculus.