Re: Another Bug in Mathematica 3.0.0 definite integration

• To: mathgroup at smc.vnet.net
• Subject: [mg9215] Re: Another Bug in Mathematica 3.0.0 definite integration
• From: weber at math.uni-bonn.de (Matthias Weber)
• Date: Fri, 24 Oct 1997 01:00:30 -0400
• Organization: RHRZ - University of Bonn (Germany)
• Sender: owner-wri-mathgroup at wolfram.com

```In article <62hfha\$m06\$4 at dragonfly.wolfram.com>, "Gregor Overney"
<overney at worldnet.att.net> wrote, commenting on :

> 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}]
> >

> Mathematica 3.0.1.1x would give you at least a warning, suggesting to
> carefully check the convergence.
>
> your input produces:
>
> Integrate::gener: Unable to check convergence
>
> and N[a] gives the obviously wrong value of -3.0123622967174799.
>
> GTO
>

Mathematica 3.0 gives the same messages already. This were also no
improvement, because for this integral there is no convergence to
check. And, in a previous post, somebody had an obviously finite
integral which Mathemetica declined to evaluate because of that
´convergence test´. A response by Wolfram recommended to turn this test
off (in general? in that case?). At least here, it is totally nonsense.

Concerning the actual problem: The primitive of the integrand
1/Sqrt[Sin[x]+Cos[x]
involves hypergeometric functions (after Simplify):

(Cot[(Pi - 4*x)/4]*Hypergeometric2F1[1/4, 1/2, 5/4,
(1 + Sin[2*x])/2]*Sqrt[2 - 2*Sin[2*x]])/
Sqrt[Cos[x] + Sin[x]]

and these being multivalued in general, it is conceivable that
Mathematica is ignoring the branching behaviour while computing the
definite integral. This is supported by the Plot of the above
expressiion which clearly has a jump at x=Pi/4. If you take this into
account by hand (no comments), you get the explicit value

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

which has the correct numerical value.

Is there anybody who has received information on this bug by Wolfram,
the impact of it on other integrals, possible workarounds and whether
they are willing to provide a fix?

Matthias Weber

```

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