# What is the unique special function in this integral problem?

• A
• Fred Wright
In summary, the conversation discusses a special function found while studying a specific integral problem online. The function is the Euler's Beta function with specific arguments, which can be simplified by replacing the arguments and then reverting at the end. References for this function are provided.
Fred Wright
While studying the solution to a integral problem I found online I ran across a special function I am unfamiliar with. The integral is
$$\int_0^{\infty}\frac{t^{\frac{m+1}{n}-1}}{1+t}dt=\mathcal{B}(\frac{m+1}{n},1-\frac{m+1}{n})$$
This certainly isn't the normal beta function. What is it? Can anyone direct me to a reference on this function?

benorin and Delta2
It's just the "normal Euler's Beta function" in the RHS but with "intimidating arguments" . One trick is to simply replace ##\frac{m+1}{n} \equiv \alpha## in your exponent under the integral sign, do the integral and revert to ##m,n## at the end.

benorin
$$\int_0^1 t^{p-1}(1-t)^{q-1}\,dt = \int_0^\infty \frac{z^{p-1}}{(z + 1)^{p+q}}\,dz$$ where $z = t/(1-t)$.

benorin and Delta2

## 1. What is a special function?

A special function is a mathematical function that has a specific purpose and is used to solve a particular problem. These functions are often more complex and specialized than basic mathematical functions, and are commonly used in fields such as physics, engineering, and statistics.

## 2. How do I know if I need help with a special function?

If you are working on a problem that requires a specific mathematical function to solve, or if you are struggling to find a solution using basic functions, you may need help with a special function. It is always a good idea to consult with a knowledgeable colleague or seek assistance from a professional in the field.

## 3. Where can I find resources for help with special functions?

There are many resources available for help with special functions, including textbooks, online tutorials, and forums where experts and other users can offer guidance and assistance. You can also consult with a math or science tutor for personalized help.

## 4. Are there any common mistakes when using special functions?

Yes, there are common mistakes that can occur when using special functions. Some of these include incorrect input values, using the wrong function for the problem, and not understanding the limitations of the function. It is important to carefully read the documentation and understand the function before using it.

## 5. Can special functions be used in programming?

Yes, special functions can be used in programming. Many programming languages have built-in functions for common special functions, and there are also libraries and packages available for more complex functions. It is important to understand the syntax and usage of the function within the specific programming language you are using.

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