Relation Between Beta and Gamma functions

In summary, the Beta and Gamma functions are closely related mathematical functions used in calculus and complex analysis. The Beta function, denoted as B(x, y), is defined in terms of the Gamma function, Γ(n), which generalizes the factorial function. Specifically, B(x, y) can be expressed as B(x, y) = Γ(x)Γ(y) / Γ(x + y). This relationship allows for the evaluation of integrals and the study of properties related to probability distributions and combinatorial identities. The connection between these functions facilitates the computation of various mathematical problems, particularly in areas involving integration and series.
  • #1
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Proof of Beta Gamma function relation
Screenshot 2024-01-02 173019.png

So, my teacher showed me this proof and unfortunately it is vacation now. I don't understand what just happened in the marked line. Can someone please explain?
 
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  • #2
The line asserts that [tex]
\int_0^\infty e^{-x(1+ y)}x^{m+n-1}\,dx = \frac{\Gamma(m + n)}{(1 + y)^{m + n}}.[/tex] How would you apply the given formulae to arrive at that?
 

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