unit 4

Question 1

integration tips and what is improper integral

Answer 1

limits in integration show be low on bottom and high limits on top, else limits should be this way resulting in a negative sign inside the integral

properties
=> symmetric property=> B(m,n) = B(n,m)
=> we know from the beta function B(m,n)=

Question 2

gamma formulas

Answer 2

[n+1 =n[n
[n =(n-1)[n-1
n[n=n!
[n =n[n OR [n =[n+1/n
[n=(n-1)!

NOTE: gamma of n is not defined for when n=0 and n is an negative integer, ex=n=-3

EX if its [2.5
then use [n= (n-1)[n-1
thus = 3/2[3/2 (split it until it reaches closest positive number above 0)
=> 3/2*1/2[1/2 (cant be simplified anymore)

[1/2 =sqrt(pi)

Question 3

Legendre’s duplication formula

how to find the n for gamma formulas

Answer 3

For Gamma function:
𝛤(2𝑝)√𝜋 =
2^(2𝑝−1)𝛤(𝑝)𝛤(𝑝 +1/2)

For Beta function:
𝛽 (𝑝,1/2) = 2^(2𝑝−1)𝛽(𝑝, 𝑝)

in all the equations relating from beta=>gamma
ex t^-1/2 and eqn part is x^n-1

solve it in terms of
n-1=-1/2, solve and find n hence use it in gamma part of eqn