alternate integral representation of beta function
AlternateIntegralRepresentationOfBetaFunction
2007-05-28 16:49:25
2007-05-28 16:49:25
Result
beta function
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By making the change of variable $x^p = y$, we see that
\[
\int_0^1 x^{p-1} (1-x)^{q-1} \, dx = {1 \over p}
\int_0^1 (1 - y^{1 \over p})^{q-1} \, dy .
\]
Hence, we have
\[
\int_0^1 (1 - y^{1 \over p})^{q-1} \, dy =
p {\Gamma (p) \Gamma (q) \over \Gamma (p + q)} .
\]