Re: Another Bug in Mathematica 3.0.0 definite integration

• To: mathgroup at smc.vnet.net
• Subject: [mg9254] Re: Another Bug in Mathematica 3.0.0 definite integration
• From: carlos at mars.Colorado.EDU (Carlos A. Felippa)
• Date: Fri, 24 Oct 1997 01:01:07 -0400
• Organization: University of Colorado, Boulder
• Sender: owner-wri-mathgroup at wolfram.com

```In article <62hfha\$m06\$4 at dragonfly.wolfram.com> "Gregor Overney" <overney at worldnet.att.net> writes:
>Mathematica 3.0.1.1x would give you at least a warning, suggesting to
>carefully check the convergence.
>
>
>Integrate::gener: Unable to check convergence
>
>and N[a] gives the obviously wrong value of -3.0123622967174799.
>
>GTO
>
>
>luca ciotti wrote in message <624fv1\$les at smc.vnet.net>...
>>Dear Users,
>>
>>unfortunately I found another erroneous result in  a definite integral
>>in Mathematica 3.0.0
>>
>>Let
>>
>>        a=Integrate[1/Sqrt[Sin[x]+Cos[x]], {x,0,Pi/2}]
>>
>>(Note that the integrand is definite positive in the integration range)
>>
>>Mathematica3.0.0 returns
>>
>>     a= -2 2^(3/4) HypergeometricPFQ[{1/4,3/4},{5/4},-1]
>>
>>and
>>     N[a]=-3.01236...
>>
>>With the standard change of variable t=Tan[x/2] the integral can be
>>easily evaluated symbolically,  and then the numeric evaluation returns
>>
>>              1.3974.....
>>
>>
>>in perfect agreement with the result obtained performing directly
>>NIntegrate on the original integrand.
>>

This is a pattern I have observed in many integrals involving
trigonometric functions that lead to hypergeometric and/or elliptic
functions.  The time comparison of 2.2 vs 3.0 is interesting.

Results from Mathematica 2.2.1 on above test function:

(Pi^(1/2)*Gamma[1/4])/(2*2^(1/4)*Gamma[3/4]) -
2*Hypergeometric2F1[1/2, 1, 5/4, -1] + Hypergeometric2F1[3/4, 1, 3/2,
-1]

1.397395299268851

Time: 0.95 seconds on Mac 8500/120.

Results from Mathematica 3.0.1, same machine:

-2*2^(3/4)*HypergeometricPFQ[{1/4, 3/4}, {5/4}, -1]

-3.012362296717479

Time: 10.4 seconds, a factor of 10.

Both 3.0 and 3.0.1 give the warning about convergence.

```

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