316 CHAPTER 8 • RESIDUE THEORY(^1) J
·oo cos 2x dx
1..
-oo x^2 + 2x + 2
(^2) J
oo x^3 sin 2x dx
- oo x (^4) + 4.
1 3. Why do you need to use the exponential function when evaluating improper
integrals involving the sine and cosine functions?
- oo x (^4) + 4.
8.5 Indented Contour Integrals
If f is continuous on the interval b < x :::; c, but discontinuous at b, then the
improper integral of f over [b, c] is defined byt J (x) dx = lim Jc J (x) d.x,
} b r-b+ r
provided the limit exists. Similarly, if f is continuous on the interval a :::; x < b,
but discontinuous at b, then the improper integral of f over (a, b) is defined by
b R{ f(x)dx= lim f J(x)dx,
la R.~b-} 0
provided the limit exists. For example,f
9
dx = Jim 19
d.x = Jim ( v'X 1 ;~~) = 3 - Jim Jr= 3.
Jo 2..;X r-+O+ r 2..;X r-+O+ r-o+If we let f be continuous for all values of x in the interval [a, cJ. except at
the value x = b, where a< b < c, then the Cauchy principal value off over [a,c)
is defined byP.V. (" f (x) dx = Jim+ [ {b-r f (x) dx + (" f (x) dx] ,
la r-O la l b+r
provided the limit exists.- EXAMPLE 8. 19
1
P.V.^8 - dx = lim [1-r -dx +^18 - dx 1 ] •
-I x! r-+O+ -I x! r X'
Evaluating the integrals and computing limits givehm. [3 -r?i^2 --^3 +6--^3 rs 2] =^9 -.
r-+O+ 2 2 2 2