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Beta function - Exercise set 1

This exercise set contains some solved exercises on the Beta function. The theory needed to solve these exercises is introduced in the lecture entitled Beta function.

Exercise 1.1

Compute the following product:[eq1]where [eq2] is the Gamma function and $Bleft( {}ight) $ is the Beta function.

nav_button Solution

We need to write the Beta function in terms of Gamma functions:[eq3]where we have used several elementary facts about the Gamma function, that are explained in the lecture entitled Gamma function.

Exercise 1.2

Compute the following ratio[eq4]where $Bleft( {}ight) $ is the Beta function.

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This is achieved by rewriting the numerator of the ratio in terms of Gamma functions and using the recursive formula for the Gamma function:[eq5]

Exercise 1.3

Compute the following integral:[eq6]

nav_button Solution

We need to use the integral representation of the Beta function:[eq7]Now, write the Beta function in terms of Gamma functions:[eq8]Substituting this number into the previous expression for the integral, we obtain:[eq9]If you wish, you can check the above result using the following MATLAB commands:

syms x

f=(x^(3/2))*((1+2*x)^-5)

int(f,0,Inf)

by
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