Beta and Gamma Function Higher Engineering Mathematics full concept covered

Beta Function
Definition
The Beta function is denoted by 𝛽(𝑚, 𝑛) where 𝑚, 𝑛 > 0. And defined by a definite integral as:
1
𝛽(𝑚, 𝑛) = ∫ 𝑥 𝑚−1 (1 − 𝑥)𝑛−1 𝑑𝑥 ; (𝑚 > 0, 𝑛 > 0)
0

It is an area function. Basically, in engineering energy is calculated by using Beta function.


Properties of Beta function
1. 𝛽(𝑚, 𝑛) = 𝛽(𝑛, 𝑚) (Symmetry of Beta fn)
1
By definition, 𝛽(𝑚, 𝑛) = ∫0 𝑥 𝑚−1 (1 − 𝑥)𝑛−1 𝑑𝑥 ; (𝑚 > 0, 𝑛 > 0)
𝑎 𝑎
We know that, ∫0 𝑓(𝑥) 𝑑𝑥 = ∫0 𝑓(𝑎 − 𝑥) 𝑑𝑥
1 1
∴ 𝛽(𝑚, 𝑛) = ∫ (1 − 𝑥)𝑚−1 (1 − (1 − 𝑥))𝑛−1 𝑑𝑥 = ∫ (1 − 𝑥)𝑚−1 𝑥𝑛−1 𝑑𝑥 ; (𝑚 > 0, 𝑛 > 0)
0 0




∞
𝑥 𝑛−1 ∞
𝑥 𝑚−1
2. 𝛽(𝑚, 𝑛) = ∫ 𝑑𝑥 = ∫ 𝑑𝑥
0 (1 + 𝑥)𝑚+𝑛 0 (1 + 𝑥)
𝑚+𝑛


1
By definition, 𝛽(𝑚, 𝑛) = ∫0 𝑥 𝑚−1 (1 − 𝑥)𝑛−1 𝑑𝑥 ; (𝑚 > 0, 𝑛 > 0)

1 −1 1 ∞ ; 𝑥=0
put, 𝑥 = 1+𝑦 ⇒ 𝑑𝑥 = (1+𝑦)2 𝑑𝑦 And, 𝑦 = 𝑥 − 1 ⇒ 𝑦 = {
0 ; 𝑥=1
0
1 𝑚−1 1 𝑛−1 (−1)𝑑𝑦 ∞
1 𝑦 𝑛−1 𝑑𝑦
𝛽(𝑚, 𝑛) = ∫ ( ) (1 − ) 2
= ∫ 𝑚−1 𝑛−1
∞ 1+𝑦 1+𝑦 (1 + 𝑦) 0 (1 + 𝑦) (1 + 𝑦) (1 + 𝑦)2
∞
𝑦 𝑛−1 ∞
𝑥 𝑛−1 𝑏 𝑏
𝛽(𝑚, 𝑛) = ∫ 𝑑𝑦 = ∫ 𝑑𝑥 [∵ ∫ 𝑓(𝑥) 𝑑𝑥 = ∫ 𝑓(𝑦) 𝑑𝑦]
0 (1 + 𝑦)𝑚+𝑛 0 (1 + 𝑥)
𝑚+𝑛
𝑎 𝑎
∞ 𝑛−1 ∞ 𝑚−1
𝑥 𝑥
𝛽(𝑚, 𝑛) = ∫ 𝑑𝑥 = ∫ 𝑑𝑥 [∵ 𝛽(𝑚, 𝑛) = 𝛽(𝑛, 𝑚)]
0 (1 + 𝑥)𝑚+𝑛 0 (1 + 𝑥)
𝑚+𝑛




𝜋
2

3. 𝛽(𝑚, 𝑛) = 2 ∫(sin 𝜃)2𝑚−1 (cos 𝜃)2𝑛−1 𝑑𝜃
0

1
By definition, 𝛽(𝑚, 𝑛) = ∫0 𝑥 𝑚−1 (1 − 𝑥)𝑛−1 𝑑𝑥 ; (𝑚 > 0, 𝑛 > 0)

0 ; 𝑥=0
Put, 𝑥 = sin2 𝜃 ⇒ 𝑑𝑥 = 2 sin 𝜃 cos 𝜃 𝑑𝜃 And, 𝜃 = {𝜋 ; 𝑥 = 1
2

, 𝜋
2
∴ 𝛽(𝑚, 𝑛) = ∫ (sin2 𝜃)𝑚−1 (cos2 𝜃)𝑛−1 ⋅ 2 sin 𝜃 cos 𝜃 𝑑𝜃
0
𝜋 𝜋
2 2
2𝑚−2 2𝑛−2
= 2 ∫ (sin 𝜃) (cos 𝜃) ⋅ sin 𝜃 cos 𝜃 𝑑𝜃 = 2 ∫ (sin 𝜃)2𝑚−1 (cos 𝜃)2𝑛−1 𝑑𝜃
0 0



𝜋
2
𝑝+1 𝑞+1
4. 𝛽 ( , ) = 2 ∫(sin 𝜃)𝑝 (cos 𝜃)𝑞 𝑑𝜃
2 2
0

𝜋
2

We have got, 𝛽(𝑚, 𝑛) = 2 ∫(sin 𝜃)2𝑚−1 (cos 𝜃)2𝑛−1 𝑑𝜃
0

𝑝+1 𝑞+1
If we put 2𝑚 − 1 = 𝑝 ⇒ 𝑚 = and 2𝑛 − 1 = 𝑞 ⇒ 𝑛 = in previous relation. Then,
2 2
𝜋
2
𝑝+1 𝑞+1
⇒ 𝛽( , ) = 2 ∫(sin 𝜃)𝑝 (cos 𝜃)𝑞 𝑑𝜃
2 2
0




Gamma function
Definition
The Gamma fn is denoted by ′ Γ(𝑛) ′ (read as gamma ′𝑛′) where 𝑛 > 0, and denoted by definite
integral:
∞
Γ(𝑛) = ∫ 𝑒 −𝑥 𝑥 𝑛−1 𝑑𝑥 ; (𝑛 > 0)
0

Properties of Gamma function
1. Γ(1) = 1
∞
By definition, Γ(𝑛) = ∫0 𝑒 −𝑥 𝑥 𝑛−1 𝑑𝑥
∞
−𝑥 1−1
∞
−𝑥
𝑒 −𝑥 ∞
Γ(1) = ∫ 𝑒 𝑥 𝑑𝑥 = ∫ 𝑒 𝑑𝑥 = [ ] = −(0 − 1) [∵ lim 𝑒 −𝑥 = 0] = 1
0 0 −1 0 𝑥→∞



2. Γ(𝑛 + 1) = 𝑛 ⋅ Γ(𝑛) = 𝑛!
∞
By definition, Γ(𝑛) = ∫0 𝑒 −𝑥 𝑥 𝑛−1 𝑑𝑥
∞ ∞ ∞
−𝑥 𝑛−1+1 −𝑥 𝑛 −𝑥 𝑛−1
Γ(𝑛 + 1) = ∫ 𝑒 𝑥 𝑑𝑥 = ∫ 𝑒 𝑥 𝑑𝑥 = ∫ 𝑒⏟ 𝑥⏟ 𝑑𝑥
0 0 0 II I


𝑛
𝑒 −𝑥 ∞ ∞
𝑒 −𝑥
= [𝑥 ⋅ ] − ∫ 𝑛 ⋅ 𝑥 𝑛−1 ⋅ 𝑑𝑥
−1 0 0 −1

∞
= −(0 − 0) + 𝑛 ∫ 𝑥 𝑛−1 𝑒 −𝑥 𝑑𝑥 = 𝑛Γ(𝑛)
0