       Re: Indefinite Integrals?

• To: mathgroup at smc.vnet.net
• Subject: [mg26162] Re: [mg26127] Indefinite Integrals?
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Thu, 30 Nov 2000 01:04:06 -0500 (EST)
• References: <200011280656.BAA02421@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```drek wrote:
>
> Hi,
> I have defined some parameters as follows:
> k0:=1.52
> u0 := Sqrt[x^2 - k0^2]
> u :=Sqrt[x^2 - 2.56 * k0^2]
> DTE := u0 + u * Coth[u*2.0]
> J0[b_] := BesselJ[0, b]
>
> I then try to integrate a function as follows:
> GA=Integrate[J0[x * 2] * x / DTE, {x, 0, 100}]
>
> Errors occur as a singularity exists at DTE for x=0, resulting in indefinite
> integrals. I believe that it may be possible to solve the integration
> problem using the residue theorem. However, I would like to know if there
> are any functions available in Mathematica 4.0 which may be able to solve
> such indefinite integrals.
>
> Thanks to all who may be able to enlighten me on this question.
>
> Derek

Your integrand vanishes at x==0, hence is not singular there.

You will likely do better with NIntegrate since this is posed as a
numeric rather than symbolic problem, and moreover I do not see a way to
get a symbolic result for either definite or indefinite integral. You
will need to play with various NIntegrate options in order to get an
accurate result. Here is something that may be reasonable (Michael Trott
suggested some of the option settings to me).

k0 = 38/25;
u0 = Sqrt[x^2 - k0^2];
u = Sqrt[x^2 - 32/25*k0^2];
DTE = u0 + u*Coth[2*u];
J0[b_] = BesselJ[0, b];
integrand = J0[2*x]*x/DTE;

In:= GA = NIntegrate[Evaluate[integrand], {x,0,100},
(*WorkingPrecision->50, *)
PrecisionGoal->8, MinRecursion->8,
MaxRecursion->14, Method->DoubleExponential]

NIntegrate::ncvi:
NIntegrate failed to converge to prescribed accuracy after 15
iterated refinements in x in the interval {{x, 0., 100.}}.

Out= -0.389906 + 0.121605 I

Due to the highly oscillatory nature of the integrand it may be quite
difficult to get a trustworthy value. One possibility might be to break
into (many) subregions bounded by the zeroes of BesselJ[0,...]. Then try
to get very accurate (signed) integrals in each region to evade
cancellation error when you sum these.

Daniel Lichtblau
Wolfram Research

```

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