The given mathematical expression is:

$B(m, n) = \int_{0}^{\infty} \frac{y^{n-1}}{(1 + y)^{m+n}} \, dy$

This represents the Beta function in an alternative form. The Beta function $B(m, n)$ is generally expressed as:

$B(m, n) = \int_{0}^{1} t^{m-1}(1-t)^{n-1} \, dt$

However, the expression provided in your image relates to the Beta function by a substitution. To see the relationship, you can use the substitution $y = \frac{t}{1-t}$, transforming the limits of integration and the form of the integrand.

This form is commonly used in the context of probability, combinatorics, or mathematical physics, where $m$ and $n$ are typically real or complex numbers.

Would you like me to:

1. Derive the relationship between this form and the standard Beta function?
2. Solve the integral explicitly for specific values of $m$ and $n$?
3. Explain where such integrals are applied in mathematics or physics?
4. Work through the substitution to reduce this to the Beta function standard form?
5. Provide properties or examples of the Beta function?

Tip: The Beta function and Gamma function are closely related: $B(m, n) = \frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}$.