Re: Troubles with Integrate

• To: mathgroup at smc.vnet.net
• Subject: [mg70236] Re: Troubles with Integrate
• From: dimmechan at yahoo.com
• Date: Sun, 8 Oct 2006 02:05:02 -0400 (EDT)
• References: <eg84hk\$nir\$1@smc.vnet.net>

```Hello Roman.

\$VersionNumber
5.2

psi = -(((Sin[(1/3)*Pi*(t - 5/4)]*Cos[Pi*(t - 5/4)])/(t - 5/4) +
(Sin[Pi*(t + 1/4)]*Cos[Pi*(t + 1/4)])/(t + 1/4))/Pi) +
((Sin[(2/3)*Pi*(t - 7/8)]*Sin[2*Pi*(t - 7/8)])/(t - 7/8) -
(Sin[(2/3)*Pi*(t - 1/8)]*Sin[2*Pi*(t - 1/8)])/(t - 1/8))/Pi;

First of all as regards the integral of psi over
{t,-infinitty,infinity} I think Mathematica 5.2
is right.

Integrate[psi, t]
(1/Pi)*((1/2)*CosIntegral[(1/6)*Pi*(-7 + 8*t)] -
(1/2)*CosIntegral[(1/3)*Pi*(-7 + 8*t)] -
(1/2)*CosIntegral[(1/6)*(Pi - 8*Pi*t)] + (1/2)*CosIntegral[(1/3)*(Pi
- 8*Pi*t)] - (1/2)*SinIntegral[(1/6)*Pi*(5 - 4*t)] +
(1/2)*SinIntegral[(1/3)*Pi*(5 - 4*t)] - (1/2)*SinIntegral[Pi/2 +
2*Pi*t])

Timing[Integrate[psi, {t, -Infinity, Infinity}]]
{36.437999999999995*Second, -(1/2)}

Timing[NIntegrate[psi3, {t, -Infinity, Infinity}, Method ->
DoubleExponential,
MaxRecursion -> 16]]
NIntegrate::slwcon :(...)
NIntegrate::ncvi :(...)
{33.827999999999975*Second, -0.5089996793160453}

Second I din't get the error message you mention to your post (no
matter how I tried!).

I hope someone will help you to evaluate the integral of psi^2 over {t,
-Infinity, Infinity}.
I failed to evalute it even I try two or three approaches.

I believe you cannot obtain a closed form result, but I wish I do
mistake.

Here is a numerical approxiamation:

Timing[NIntegrate[psi^2, {t, -Infinity, Infinity}, MaxRecursion -> 20]]
(messages are not displayed)
{17.046999999999997*Second, 0.9331917391109911}

```

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