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MathGroup Archive 2007

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Re: a suprising result from Integrate (Null appeared in the

  • To: mathgroup at smc.vnet.net
  • Subject: [mg74526] Re: a suprising result from Integrate (Null appeared in the
  • From: Peter Pein <petsie at dordos.net>
  • Date: Sat, 24 Mar 2007 05:19:01 -0500 (EST)
  • References: <eu1r2o$jd6$1@smc.vnet.net>

Bhuvanesh schrieb:
> Yes, this was reported and fixed quite a while back, and should be fine in the next version. Another example of the bad behavior was Integrate[Sqrt[Sin[x] + Cos[x]], x]. For your indefinite integrals I currently get, in the development build:
> 
until then, a simple linear shift of variables helps:

f[x_] = (1 - Sin[x])^(1/4);
approx = NIntegrate[f[x], {x, 0, Pi/2, 2*Pi}, WorkingPrecision -> 32]
inexact = Integrate[f[x], {x, 0, Pi/2, 2*Pi}]

5.699347567674386465367
0

exact =
  FullSimplify[Integrate[FullSimplify[f[x + Pi/2]], {x, -Pi/2, 0, 3*(Pi/2)}]]

8*2^(1/4)*EllipticE[Pi/4, 2]

Chop[N[exact] - approx]
0

as an alternative to FullSimplify, use TrigToExp:

F[z_] = Simplify[Integrate[TrigToExp[f[x + Pi/2]], {x, -Pi/2, z - Pi/2}]]

z*((1/z)*(4*I - 4*(1 + I)^(3/2)*Hypergeometric2F1[-(1/4), 1/2, 3/4, -I]) +
   (2*2^(3/4)*((I*(-I + E^(I*z))^2)/E^(I*z))^(1/4)*(-1 - I*E^(I*z) +
   2*Sqrt[1 + I*E^(I*z)]*Hypergeometric2F1[-(1/4), 1/2, 3/4, (-I)*E^(I*z)]))/
    ((-I + E^(I*z))*z))

Plot[Chop[F[z]], {z, 0, 2*Pi}]
[omitted]

alternateexact = FullSimplify[
  Subtract @@ (Limit[F[z], z -> Pi/2, Direction -> #1] & ) /@ {1, -1}]

(8*2^(1/4)*Sqrt[Pi]*Gamma[3/4])/Gamma[1/4]

Chop[N[alternateexact] - approx]
0

Peter


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