140 Chapter 5Integration
In this case the value of sin 1 boscillates between + 1 and − 1 as bincreases, and no
unique value can be assigned to the integral as defined here. On the other hand, it is
shown in Example 6.13 that, fora 1 > 10 ,
This result is valid for all positive values of the parameter a, however small, and it
follows that
We note that the limit must be taken afterintegration; taking the limit before
integration leads to a different, and divergent, integral.
Even and odd functions
When a function has the property
f(−x) 1 = 1 f(x) (5.20)
it is called an even functionof x; it has even parity(or parity + 1 ) and is symmetric
with respect to the axis x 1 = 10 (the y-axis). Examples of even functions are x
2, e
−|x|(figure 5.8) and cos 1 x(shown in Figure 5.10a); in each case the value of the function
is unchanged when xis replaced by −x. On the other hand, a function with the
property
f(−x) 1 = 1 −f(x) (5.21)
is said to be an odd function; it has odd parity(or parity − 1 ) and is antisymmetric
with respect to the axisx 1 = 10. Examples of odd functions arex
3,sin 1 x(shown in
Figure 5.10b), andx 1 cos 1 x; in each case the value of the function changes sign when x
is replaced by−x. The product of two even functions or of two odd functions is even;
the product of an even function and an odd function is odd. Thusx 1 cos 1 xis an odd
function, withxodd andcos 1 xeven.
lim cos lim
aa
exdx
a
a
ax→→
−=
=
00
021
Z 0
∞Z
021
∞exdx
a
a
−ax=
cos
- 1
− 1
−π
0 πx
−x
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(b) y=sinx;o ddfunct ion
− 1
+1
−π −x 0 πx
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(a ) y=cosx;evenfuncti on
Figure 5.10