8.2 • TRIGONOMETRIC INTEGRALS 301- Find J (3z^4 + 10z^2 + 3)-
1
dz when
c
(a) c =ct (iv'3).
(b) C= Ct(~)·8. Find J (z^4 - z^3 - 2z^2 ) -
1
dz when
c(a) C =er (O).
(b) c =ct (O).
- Use residues to find the partial fraction representations of
1
(a) z 2 +3z+2·
(b) 3z - 3
z2 - z - 2 ·
z^2 -7z +4
(c) z^2 (z + 4)(d) lOz
(z^2 + 4) (z2 + 9)"
2z^2 - 3z - 1
(e) (z - 1)3.(
f) z3 + 3z^2 - z + 1
z(z+ 1)^2 (z^2 + l)°- Let f be analytic in a simply connected domain D , and let C be a simple
closed positively oriented contour in D. If z 0 is the only zero off in D and
z 0 lies interior to C , then show that 2 !; J LJttf dz = k, where k is the order
c
of t he zero at zo.- Let f be analytic at the points 0,±1,±2,. ... If g(z) = 7rf (z)cot7rz, then
show that Res[g, n ) = f (n) for n = 0, ±1, ±2, ....
8.2 Trigonometric Integrals
As indicated at the beginning of this chapter, we can evaluate certain definite
real integrals with the aid of the residue theorem. One way to do this is by
interpreting the definite integral as the parametric form of an integral of an
analytic function along a simple closed contour.