Re: NIntegrate HypergeometricU
- To: mathgroup at smc.vnet.net
- Subject: [mg84878] Re: NIntegrate HypergeometricU
- From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
- Date: Thu, 17 Jan 2008 07:02:02 -0500 (EST)
- References: <fmkf3a$9fg$1@smc.vnet.net> <fmmk6e$eog$1@smc.vnet.net>
sashap <pavlyk at gmail.com> wrote:
> On Jan 16, 2:26 am, "subo_... at yahoo.com" <subodh... at gmail.com> wrote:
> > I am trying to NIntegrate the function HypergeometricU[a,b,z] with
> > a>=1, b>=1,z=R/sin[x]^2 where integration is from 0 to Pi/2. For lower
> > values of R this gives answers and for Higher values of R this is
> > giving errors .
>
> You could avoid using NIntegrate altogether, by finding the integral
> exactly.
> Do the substitution Sin[t]^2 = x,
A slightly different substitution from the one I had used.
> and then it reduces to
>
> Integrate[
> HypergeometricU[a, b, R/x] (2 Sqrt[x] Sqrt[1 - x]), {x, 0, 1},
> Assumptions -> R > 0 && a > 1 && b > 1]
No. Divide by (2 Sqrt[x] Sqrt[1 - x]), rather than multiply by it. Doing
that, I think the result of your integration will then be the same as my
result.
Can that then be expressed in terms of MeijerG? If so, can we get
Mathematica to make that conversion for us?
> which gives a complicated expansion of hypergeometric functions that
> disguises
>
> res[a_,b_,R_] := Sqrt[Pi]/(Gamma[a] Gamma[a + 1 - b])
> MeijerG[{{1 - a}, {3}}, {{3/2, 0, 1 - b}, {}}, R]
>
> This result is correct under your assumptions (a>=1, b>=1 and R>0)
> provided a+1-b is not a non-negative integer.
I should take this opportunity to mention that, in my previous post, due to
an oversight, I failed to mention that proviso. Sorry.
David W. Cantrell
> The answer admits the limit at this point.
>
> Let b = a + n + 1. Then
>
> Table[HypergeometricU[a, a + n + 1, R/x] ==
> Sum[Binomial[n, n - k] Pochhammer[a, k] (x/R)^(k + a), {k, 0,
> n}], {n, 0, 5}] // Simplify[#, x > 0 && R > 0] &
>
> Using this definition
>
> Integrate[
> HypergeometricU[a, a+n+1, R/x] (2 Sqrt[x] Sqrt[1 - x]), {x, 0, 1},
> Assumptions -> R > 0 && a > 1 && n > 0] ==
> (Sqrt[Pi]*(1/R)^a*Gamma[3/2 + a]*
> HypergeometricPFQ[{a, 3/2 + a, -n}, {3 + a}, -(1/R)])/Gamma[3 + a]
>
> Indeed:
>
> In[141]:= Sum[
> Binomial[n, n - k]*Pochhammer[a, k]*
> Integrate[(x/R)^(a + k)*2*Sqrt[x]*Sqrt[1 - x], {x, 0, 1},
> Assumptions -> a >= 1 && k > 0], {k, 0, n}]
>
> Out[141]= (Sqrt[Pi]*(1/R)^a*Gamma[3/2 + a]*
> HypergeometricPFQ[{a, 3/2 + a, -n}, {3 + a}, -(1/R)])/Gamma[3 + a]
>
> Hope this helps,
> Oleksandr Pavlyk
> Special Functions Developer
> Wolfram Research Inc
>
> >
> > \!\(NIntegrate::"singd" \(\(:\)\(\ \)\) "NIntegrate's singularity
> > handling
> > has failed at point \
> > \!\({x}\)=\!\({8.145014137541`5.15492657977389*^-15}\) for the
> > specified \
> > precision goal. Try using larger values for any of $MaxExtraPrecision
> > or the \
> > options WorkingPrecision, or SingularityDepth and MaxRecursion"\)
> >
> > Even though I change the working precisio,SingularityDepth and
> > MaxRecursion it is not helping me much. Please advice me how to solve
> > this problem.
> >
> > Subodha