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Let $\alpha,\beta,\gamma,\delta > -1$, and $A\le B\le C\le D$. Is there an analytical expression, or an efficient numerical technique, to estimate the following integral?

$$\int_{B}^{C}\left(x-A\right)^{\alpha}\left(x-B\right)^{\beta}\left(C-x\right)^{\gamma}\left(D-x\right)^{\delta}\mathrm{d}x$$

a06e
  • 6,665

1 Answers1

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$\int_B^C(x-A)^\alpha(x-B)^\beta(C-x)^\gamma(D-x)^\delta~dx$

$=\int_0^{C-B}(x+B-A)^\alpha x^\beta(C-B-x)^\gamma(D-B-x)^\delta~dx$

$=\int_0^1((C-B)x+B-A)^\alpha((C-B)x)^\beta(C-B-(C-B)x)^\gamma(D-B-(C-B)x)^\delta~d((C-B)x)$

$=(B-A)^\alpha(C-B)^{\beta+\gamma+1}(D-B)^\delta\int_0^1x^\beta(1-x)^\gamma\left(1+\dfrac{(C-B)x}{B-A}\right)^\alpha\left(1-\dfrac{(C-B)x}{D-B}\right)^\delta~dx$

$=\dfrac{(B-A)^\alpha(C-B)^{\beta+\gamma+1}(D-B)^\delta~\Gamma(\beta+1)\Gamma(\gamma+1)}{\Gamma(\beta+\gamma+2)}F_1\left(\beta+1,-\alpha,-\delta,\beta+\gamma+2;\dfrac{B-C}{B-A},\dfrac{C-B}{D-B}\right)$ (according to http://en.wikipedia.org/wiki/Appell_series#Integral_representations)

Harry Peter
  • 7,819
  • +1 That's great! Too bad I haven't found any numerical implementation of Appell's series available for C++/C. – a06e May 14 '14 at 17:38
  • Numerical computation of Appell series: http://scicomp.stackexchange.com/q/11627/988 – a06e May 15 '14 at 00:11