Calculus: Analytic Geometry and Calculus, with Vectors

686 Iterated and multiple integrals

Changing the variable of integration by setting u = t/(n - 1) then gives

1 z+1 n-1 t n 1

(6) F-(2;) = (1 Jof (1-n - 1/ dt

when n > 1. Therefore

(7)

where

(g)

Z! = lim dt

I

-1

Gn(t) = tz (1 -n I/ (0<t5n-1)

and 0 when t > n. It can be shown that

(9) lim G (t) = tze-1, JG(t)J S I tze t 1.

While full exploration of the matter lies beyond the scope of elementary calculus,

(1) is a consequence of (7), (9), and the Lebesgue criterion of dominated con-

vergence for taking limits under integral signs. When m and s are numbers for

which s > m, we can put t = (s - nz)x in (1) to obtain the formula

(10) e-"zxze'"s dx.

z.

(s - m).'1= 10,

Particularly when it is recognized that (10) is valid even when m is complex, this

single formula (10) is the equivalent of a huge table of Laplace transforms and is

therefore very important.

19 We examine the formulas by which ideas of this chapter are used to start

with the Euler gamma integral formula

(1) at = fo0 tie e dt

of the preceding problem and derive the beta integral formula

(2) fo tn(1- t)Q dt

p !q!

• (p + q + 1)!

It is supposed that z, p, and q are complex numbers having real parts exceeding

-1. Use of (1) gives

(3) p!q! = f , xpe= dx 0 f0`° ygeydy

where the right side is the product of two integrals. Writing this as an iterated

integral and putting y = u - x give

(4) p!q! = fog dx fog xpyse (rty) dy=f00 dx

fxo0

xp(u - x)9e " du

The Fubini theorem justifies change of order of integration to obtain the first

equality in

U m 1

(5) p!q! = ooe-°' du fo xp(u - x)4 dx = fo up+4+1e du fo tp(1 - t)- dt