Solved examples on beta and gamma functions

SOLVED EXAMPLES ON BETA AND GAMMA FUNCTIONS

MFMCF-FUTA

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Recall: ⌈ = ∫ . Evaluate the following integrals as a function of gamma

1. ∫ (Let n – 1 = 2/3)

Thus, n = 2/3 + 1 = 5/3. ∫ ⌈( )



2. ∫ (Let n – 1 = -1/2)

Thus, n = -1/2 + 1 = 1/2. ∫ ⌈( )



3. ∫ (Let u = , solve the integration by substitution)

;

.

When When
(now we will substitute all the factors of u for x)

∫ ∫ =∫

∫ ∫ (Let y=3u, u=y/3)

,

When u=0, y=0 When u=∞, y=∞

∫ ∫ ∫

∫ (Now the expression can written as gamma, if n – 1=1)

Thus, n = 1+1 = 2. ∫ ∫ ⌈


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, 4. ∫ √ (√ )

⁄
∫ √ ∫ (Let n – 1= 1/2)

Thus, n = ½ + 1 = 3/2. ∫ √ ⌈



5. ∫ (Let u=x2 and x=√u: when x=0, u=0, when x=∞, u=∞)


√

(Substitute for x)

∫ ∫ ( ) ∫ ( ) ∫ (√ )
√ √



∫ √ ∫


Let n – 1 = 1/2. Thus, n = ½ + 1 = 3/2

∫ ∫ ⌈ (MFMCF-FUTA)



Evaluate the following and Recall: ⌈ ⌈ , ⌈( ) √

⌈
6. ⌈( ) Recall; ⌈

⌈( ) ⌈( ). ⌈( ) ⌈( )

⌈( ) ⌈( ). ⌈( ) ⌈( )

⌈( ) ⌈( ). ⌈( ) √



7. ∫ Let n – 1 = 3, therefore n = 4


∫ ⌈

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