Page 394 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 15] GAMMA AND BETA FUNCTIONS 385

To this point the analysis has been rigorous. The following formal steps can be made rigorous by
incorporating appropriate limiting procedures; however, because of the difficulty of the proofs, they shall be
omitted.
In (2)introduce the logarithmic expansion
y y y y
2 3
x x 2x 3x
ln 1 þ ¼ 2 þ 3 þ ð3Þ
and also let
x v; x dv
p ffiffiffi p ffiffiffi
y ¼ dy ¼
Then
ð
1 2 3 p ffiffi
x x
ffiffiffi
ðx þ 1Þ¼ x e p x e v =2þðv =3Þ x dv ð4Þ
x
For large values of x
ð
1 2 p ffiffiffiffiffiffiffiffi
x x
x x
x
ðx þ 1Þ x e p ffiffiffi e v =2 dv ¼ x e 2 x
x
When x is replaced by integer values n,then the Stirling relation
ffiffiffiffiffiffiffiffi
p x x
2 x x e
n! ¼ ðx þ 1Þ ð5Þ
is obtained.
It is of interest that from (4)wecan also obtain the result (12)on Page 378. See Problem 15.72.
DIRICHLET INTEGRALS
ðð ð
z
x 1 y 1
1 dx dy dz where V is
15.21. Evaluate I ¼
V
the region in the first octant bounded by the sphere
2 2 2
x þ y þ z ¼ 1 and the coordinate planes.
2
2
2
Let x ¼ u; y ¼ v; z ¼ w. Then
ððð dv dw
v
u ð 1Þ=2 ð 1Þ=2 w ð
1Þ=2 du
I ¼ p ffiffiffi p ffiffiffi p ffiffiffiffi
2 u 2 v 2 w
r
1 ðð ð ð =2Þ 1 ð =2Þ 1 ð
=2Þ 1
u v w du dv dw
¼ ð1Þ
8
r
where r is the region in the uvw space bounded by the plane
u þ v þ w ¼ 1 and the uv; vw, and uw planes as in Fig. 15-2.
Thus,
1 ð 1 ð 1 u ð 1 u v ð =2Þ 1 ð =2Þ 1 ð
=2Þ 1
u v w du dv dw Fig. 15-2
I ¼ ð2Þ
8 u¼0 v¼0 w¼0
1 ð 1 ð 1 u ð =2Þ 1 ð =2Þ 1
=2
4
u¼0 v¼0
¼ u v ð1 u vÞ du dv
1 ð 1 ð =2Þ 1 ð 1 u ð =2Þ 1
=2
4
u¼0 v¼0
¼ u v ð1 u vÞ dv du
Letting v ¼ð1 uÞt,we have
ð 1 u ð 1
v ð =2Þ 1 ð1 u vÞ
=2 dv ¼ð1 uÞ ð þ
Þ=2 t ð =2Þ 1 ð1 tÞ
=2 dt
v¼0 t¼0
ð þ
Þ=2 ð =2Þ ð
=2 þ 1Þ
¼ð1 uÞ
½ð þ
Þ=2 þ 1Š