Begin2.DVI

Example 12-4.

Show that Γ(

1

2 ) =

√

π.

Solution

Substitute x= 1/ 2 into equation (12.79) and show

Γ(^1

2

) =

∫∞

0

ξ−^1 /^2 e−ξdξ =

∫∞

0

e√−ξ

ξ

dξ (12 .86)

In equation (12.86) make the substitution ξ=x^2 with dξ = 2 x dx to obtain

Γ(^1

2

) =

∫∞

0

e−x^2

x

2 x dx = 2

∫∞

0

e−x^2 dx (12 .87)

Let Idenote the integrals

I=

∫∞

0

e−x

2

dx and I=

∫∞

0

e−y

2

dy

and form the double integral

I^2 =

∫∞

0

∫∞

0

e−(x

(^2) +y (^2) )
dxdy = limT→∞
∫T
0
∫T
0
e−(x
(^2) +y (^2) )
dxdy

and observe that as T increases without bound the area of

integration fills up the first quadrant.

Change the double integral for I^2 from rectangular to polar

coordinates where

x=rcos θ, y =rsin θ, dxdy =rdrdθ

and write

I^2 = limR→∞

∫R

r=0

∫π/ 2

0

e−r

2

rdrdθ

and observe that as Rincreases without bound the area of

integration is still over the first quadrant. Now integrate with

respect to θand then integrate with respect to rto show

I^2 =π

4

or I=

∫∞

0

e−x

2

dx =

√

π

2

Substitute this result into the equation (12.87) to show that Γ(^1

2

) =

√

π