Exploring the Generalized Beta Function: Integrals and Partial Derivatives

In summary, we discussed the integrals of the form \beta(a,b,c;s) and its relation to the beta function with s=1. We assumed \Re(a+b)>0 and \Re(1-c)>0 and found that \beta(a,b,c;\,1) can be expressed as \frac{\Gamma(1-c)\Gamma(1-(a+b))}{\Gamma(2-(a+b+c))}. We also suggested exploring the reflection substitution x \to 1-x and considering partial derivatives with respect to the parameters. This topic is very interesting and worth further exploration.
  • #1
alyafey22
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In this thread we consider the integrals of the form

\(\displaystyle \beta(a,b,c;s) = \int^1_0 \frac{1}{(1-x)^a(s-x)^b x^c}\, dx \,\,\,\,\,\,-1<a,b,c<1 \,\,\,s\geq 0 \)

This is NOT a tutorial , all suggestions are encouraged.
 
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  • #2
We can relate it to the beta function \(\displaystyle s=1\)
Assume \(\displaystyle \Re(a+b)>0 , \Re(1-c)>0\)

\(\displaystyle \beta(a,b,c;\,1) = \int^1_0 (1-x)^{-(a+b)} x^{-c}dx= \beta(1-c,1-(a+b))=\frac{\Gamma(1-c)\Gamma(1-(a+b))}{\Gamma(2-(a+b+c))}\)

\(\displaystyle \tag{1} \beta(a,b,c;\,1) = \frac{\Gamma(1-c)\Gamma(1-(a+b))}{\Gamma(2-(a+b+c))} \,\,\,\, \Re(a+b)>0 , \Re(1-c)>0\)​
 
  • #3
ZaidAlyafey said:
In this thread we consider the integrals of the form

\(\displaystyle \beta(a,b,c;s) = \int^1_0 \frac{1}{(1-x)^a(s-x)^b x^c}\, dx \,\,\,\,\,\,-1<a,b,c<1 \,\,\,s\geq 0 \)

This is NOT a tutorial , all suggestions are encouraged.
You're on tip-top form, Zaid! I like it! (Heidy)

A few things jump out at me, which might be worth exploring...

Firstly, the reflection substitution \(\displaystyle x \to 1-x\) might be worth a look... Also, being the logarithmic fiend that I am, I think it might be worth considering partial derivatives wrt any/all of the parameters.

I'll definitely come back to this topic when it's not so close to bed time. Very interesting! (heart)
 

Related to Exploring the Generalized Beta Function: Integrals and Partial Derivatives

1. What is the generalized beta function?

The generalized beta function, also known as the Euler integral of the first kind, is a mathematical function that generalizes the beta function. It is defined as B(a,b) = ∫01 xa-1(1-x)b-1 dx, where a and b are positive real numbers.

2. What is the purpose of the generalized beta function?

The generalized beta function is used in many areas of mathematics and science, including probability theory, statistics, and physics. It is particularly useful in evaluating integrals and in solving problems related to probability distributions.

3. How is the generalized beta function related to the beta function?

The beta function, denoted by B(a,b), is a special case of the generalized beta function when both a and b are positive integers. In other words, B(a,b) = (a-1)!(b-1)! / (a+b-1)!. This relationship allows for the extension of the properties and applications of the beta function to the generalized beta function.

4. Can the generalized beta function be expressed in terms of other mathematical functions?

Yes, the generalized beta function can be expressed in terms of other mathematical functions, such as the gamma function and the hypergeometric function. In fact, the beta function can be written as a special case of the hypergeometric function, which further demonstrates the close relationship between these functions.

5. Are there any real-life applications of the generalized beta function?

Yes, the generalized beta function has many practical applications in fields such as economics, engineering, and biology. It is commonly used in statistical analyses and in modeling various phenomena, such as the distribution of income, the growth of populations, and the behavior of fluids.

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