Integral hangs G2
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03-30-2020, 05:16 PM
(This post was last modified: 06-30-2022 07:31 PM by Albert Chan.)
Post: #5
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RE: Integral hangs G2
(03-29-2020 09:56 PM)Albert Chan Wrote: Googled "wallis product", I found this: Mathematical Analysis and the Mathematics of Computation, p.404, eqn. 7.87 Just realized the integral is simply the beta function (see equation 18) Let \((1-u)^p = x\quad →\quad -p(1-u)^{p-1}\;du=dx\) \(\large \int_0^1 (1-x^{1/p})^q\;dx = p \int_0^1 u^q (1-u)^{p-1}\;du = p\;B(q+1,p) = {\Gamma(p+1)\Gamma(q+1) \over \Gamma(p+q+1)} = 1 / \binom{p+q}{p} \) Update: We can prove u integral is beta function by induction (via integration by parts) Foundations of Combinatorics with Applications: Appendix A, Example A.4 |
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Messages In This Thread |
Integral hangs G2 - lrdheat - 03-29-2020, 06:38 PM
RE: Integral hangs G2 - Albert Chan - 03-29-2020, 09:56 PM
RE: Integral hangs G2 - toml_12953 - 03-30-2020, 12:16 AM
RE: Integral hangs G2 - Albert Chan - 03-30-2020 05:16 PM
RE: Integral hangs G2 - roadrunner - 03-30-2020, 12:27 AM
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