GAMMA AND BETTA

FUNCTION

GROUP - 1

ENROLLMENT NO: 1-12

Gamma function :

Gamma function is define by the improper interval

𝑒−𝑥

∞

0

𝑥𝑛−1 𝑑𝑥 , 𝑛 > 0

And it is denoted by Γ n.

Alternate form of gamma function ,

Γ n = 𝑒−𝑥2∞

0 𝑥2𝑛−1 𝑑𝑥

Proof by definition

Γ n = 𝑒−𝑥∞

0 . 𝑥𝑛−1 𝑑𝑥

Let x=𝑡2 and 𝑑𝑥 = 2tdt

Γ n= 𝑒−𝑡2∞

0. 𝑡2𝑛−2 𝑑𝑡

=2 𝑒−𝑡2∞

0 . 𝑡2𝑡−1 𝑑𝑡

Now changing the variable t to x,

Γ n=2 𝑒−𝑥2∞

0 𝑥2𝑛−1 𝑑𝑥

Proprieties of GAMMA function :

Γ n+1 = n Γ n

Γ12 = Π

Proof of the property :

1] Γ n+1 = 𝑒−𝑥∞

0 . 𝑥𝑛 𝑑𝑥,

Inter changing by parts ,

Γ n+1 = | 𝑒−𝑥 . 𝑥𝑛|0∞ - 𝑒−𝑥

∞

0 . 𝑥𝑛−1 𝑑𝑥

= 𝑛 𝑒−𝑥∞

0 . 𝑥𝑛−1 𝑑𝑥

=n Γ n

Hence Γn+1 = n Γ n proved .

This is known as recurrence reduction formula for gamma function.

NOTE :

Γ n+1 = n! if n is a positive integer

Γn+1 = n Γn if n is a real number .

Γn = Γ𝑛+1

𝑛 if n is negative fraction.

Γn Γ1-n = Π

sin 𝑛 Π

(2) Γ1

2 = Π

PROOF :

By alternate form of gamma function ,

Γ1

2 = 2 𝑒−𝑥

2∞

0. 𝑥2(

1

2)−1 𝑑𝑥

= 𝑒−𝑥2. 𝑑𝑥

∞

0

Γ1

2Γ1

2 = 𝑒−𝑥

2𝑑𝑥

∞

0 . 𝑒−𝑦

2. 𝑑𝑦

∞

0

changing to polar coordinates x= 𝑟 cos 𝜃 , 𝑦 = 𝑟 sin 𝜃

∴ 𝑑𝑥𝑑𝑦 = 𝑟𝑑𝑎𝑑𝜃

Limits of x x=0 to x → ∞

Limits of y y=0 to y → ∞

This shows that the region of integration is the first quadrant.

Draw the elementary radius vector in the region which starts from

the pole extend up to ∞.

Limits of r r=0 to r → ∞

Limits of 𝜃 𝜃 =0 to 𝜃 → 𝜋

2 .

Γ1

2Γ1

2 = 4 𝑒−𝑟

2∞

0

𝜋

20

. 𝑟𝑑𝑟𝑑𝜃

= 4 𝑑𝜃 (−1

2)𝑒−𝑟

2∞

0

𝜋

20

. (−2𝑟) 𝑑𝑟

=4

−2 |𝜃|0

𝜋

2 |𝑒−𝑥2|0∞ [∵ 𝑒𝑓 𝑟 𝑓′ 𝑟 𝜃𝑟 = 𝑒𝑓 𝑟 ]

= -2.𝜋

2(0 − 1)

=𝜋

Γ1

2 = 𝜋

Examples : Find the value of Γ𝟓

𝟐

Γn = 𝛤𝑛+1

𝑛

Γ−5

2 =

𝛤−5

2+1

5

2

= −2

5 Γ−3

2

= −2

5

Γ−3

2+1

−3

2

=4

15 Γ−1

2

= 4

15

Γ−1

2+1 −1

2

=−8

15 Γ1

2

= -−8 𝜋

15 ∵ 𝑓𝑟𝑜𝑚 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦

THANK YOU