Beta in terms of power of t over power of (1+t)

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Theorem

The following formula holds: $$B(x,y)=\displaystyle\int_0^{\infty} \dfrac{t^{x-1}}{(t+1)^{x+y}} \mathrm{d}t,$$ where $B$ denotes the beta function.

Proof

From the definition, $$B(x,y)=\displaystyle\int_0^1 u^{x-1} (1-u)^{y-1} \mathrm{d}u.$$ We will proceed using substitution. Let $u=\dfrac{t}{t+1}$ so that $\mathrm{d}u=\dfrac{1}{(t+1)^2} \mathrm{d}t$. Since $u=0$ means $t=0$ and $u=1$ means $t=\infty$, we get $$\begin{array}{ll} B(x,y) &= \displaystyle\int_0^1 u^{x-1} (1-u)^{y-1} \mathrm{d}u \\ &= \displaystyle\int_0^{\infty} \left( \dfrac{t}{t+1} \right)^{x-1} \left( 1 -\dfrac{t}{t+1} \right)^{y-1} \dfrac{1}{(t+1)^2} \mathrm{d}t \\ &= \displaystyle\int_0^{\infty} \left( \dfrac{t}{t+1} \right)^{x-1} \left( \dfrac{1}{t+1} \right)^{y-1} \dfrac{1}{(t+1)^2} \mathrm{d}t \\ &= \displaystyle\int_0^{\infty} \dfrac{t^{x-1}}{(t+1)^{x+y}}, \end{array}$$ as was to be shown.

References

  • 1964: {{ #if: |{{{2}}}|Milton Abramowitz}}{{#if: Irene A. Stegun|{{#if: |, {{ #if: |{{{2}}}|Irene A. Stegun}}{{#if: |, [[Mathematician:{{{author3}}}|{{ #if: |{{{2}}}|{{{author3}}}}}]]{{#if: |, [[Mathematician:{{{author4}}}|{{ #if: |{{{2}}}|{{{author4}}}}}]]{{#if: |, [[Mathematician:{{{author5}}}|{{ #if: |{{{2}}}|{{{author5}}}}}]] and [[Mathematician:{{{author6}}}|{{ #if: |{{{2}}}|{{{author6}}}}}]]| and [[Mathematician:{{{author5}}}|{{ #if: |{{{2}}}|{{{author5}}}}}]]}}| and [[Mathematician:{{{author4}}}|{{ #if: |{{{2}}}|{{{author4}}}}}]]}}| and [[Mathematician:{{{author3}}}|{{ #if: |{{{2}}}|{{{author3}}}}}]]}}| and {{ #if: |{{{2}}}|Irene A. Stegun}}}}|}}: [[Book:Milton Abramowitz/Handbook of mathematical functions{{#if: |/Volume {{{volume}}}|}}{{#if: |/{{{edpage}}}}}|Handbook of mathematical functions{{#if: |: Volume {{{volume}}}|}}{{#if: |: {{{eddisplay}}}|{{#if: | ({{{ed}}} ed.)}}}}]]{{#if: | (translated by [[Mathematician:{{{translated}}}|{{ #if: |{{{2}}}|{{{translated}}}}}]])}}{{#if: |, {{{publisher}}}|}}{{#if: |, ISBN {{{isbn}}}|}}{{#if: Beta | ... (previous)|}}{{#if: Beta in terms of sine and cosine | ... (next)|}}{{#if: |: Entry: {{#if: |[[{{{entryref}}}|{{{entry}}}]]|{{{entry}}}}}|}}: $6.2.1$
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