3 2 2 CHAPTER 8 • RESIDUE THEORYco x^4 dx6. P.V. f -cox - 6 - - l.
7. P.V. f~oo sinx dx.
xS. P.V. I"'° cosx dx.
J-oo x2 - xco sinx dx9.P.V.f -cox (11' (^2) - x 2 ).
co cosx dx
1 0. P.V. f 2 2.
- co 11' - 4x
co sinx dx- P.V. f - (^00) xx ( (^2) + 1 ).
- P.V. Joo xcosx dx.
- co x^2 + 3x + 2
co sinx dx- P.V. J_ 00 ( 2 ).
x 1 - x
J
oo COSX dx
14. P.V. _ 00 2 2.
a - x
- P.V. J~ 00 sin
2
x~ dx. Hint: Use trigonometric identity sin^2 x = ~ - ~cos 2x.8.6 Integrands with Branch Points
We now show how to evaluate certain improper real integrals involving the inte-
grand x"' ~~=~. The complex function z"' is multivalued, so we must first specify
the branch to b e used.
Let a be a real number with 0 < a < I. In this section we use the branch
of z"' corresponding to the branch of the logarithm log 0 (see Equation (5-20)) as
follows:z"' = e"'{log.,(z)] = e<>(lnlzl+iarg 0 z) = ea(lnr+i8) = r" (cosa8 + i sin a-8), {8-28)
where z = rei^8 f= 0 and 0 < 8 ~ 211'. Note that this is not the traditional
principal branch of z" and that, as defined, the function z" is analytic in the
domain {re'^9 : r· > 0, 0 < 8 < 211' }.