1550251515-Classical_Complex_Analysis__Gonzalez_

634 Chapters

12. Show that

{

0

(^1) xn dx Vi I'((n + 1)/2)
lo Vl=X2 = 2 r((n/2) + 1)
_ n!! 2
{

(n-1)!! '.'.:.

if n > 0 is even

• (n -1)!!
  n!!

if n > 0 is odd

13. Show that

7r/2 7r/2 " 2

{

(n-l)!!rr

f sinn () d() = f cosn () d() = ( : .. )II

lo lo n 1 ..

n!!

14. Show that

ifn>Oiseven

if n > 0 is odd

lb (x - a)P-^1 (b -x)q-l dx = (b - a)P+q-l B(p, q)

where a< bare real and Rep> 0, Req > O. Hint: Let x-a = (b-a)t.

15. Show that

sinP () cosq d() = -^2 2

1

7r/^2 1 r( tl! )r( tl.!)

o 2 r(.e.p: +1)

(Rep> -l,Req > -1)

16. Prove that

1

1

(1 -xP)l/q dx = fo\1 -xq)l/p dx

where p > O, q > 0.

17. Prove the Dirichlet formula

J

r lj i-1 m-1 n-1 d d d albmcn r ( ~) r ( ~) r ( ~)

l. x y z x y z = ---pg;:-r ( 1 + .!!!. + 11 + l)

V p q r

where V is the volume bounded in the first octant by the coordinate

planes and the surface

(~Y +(tY+(~r =l

and the parameters a, b, c, l, m, n, p, q, r are real and positive. Use the

Dirichlet formula to find the volume, centroid, and moments of inertia

with respect to the coordinate planes of an octant of the ellipsoid

x2 y2 z2

a2 + b2 + c2 = 1