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MathGroup Archive 2006

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Re: MeijerG evaluates an imaginary part, which does not exist

  • To: mathgroup at smc.vnet.net
  • Subject: [mg64232] Re: MeijerG evaluates an imaginary part, which does not exist
  • From: Bhuvanesh <lalu_bhatt at yahoo.com>
  • Date: Tue, 7 Feb 2006 03:36:02 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

This is what I get in the development kernel.

In[1]:= meijer = MeijerG[{{1/2}, {}}, {{1/2, 1/2}, {0}}, 1/4];

In[2]:= FunctionExpand[meijer] //InputForm

Out[2]//InputForm= 
(-EulerGamma + E^2*EulerGamma + CoshIntegral[1] - E^2*CoshIntegral[1] + Log[2] - 
  E^2*Log[2] - 2*E*EulerGamma*Sinh[1] + 2*E*Log[2]*Sinh[1] + SinhIntegral[1] + 
  E^2*SinhIntegral[1])/(E*Sqrt[Pi])

In[3]:= N[%, 20]

Out[3]= 0.72979177703127333466

In[4]:= Integrate[((1 + E^x*Cos[x] + E^x*Sin[x])/(2*E^x))/(1 + x^2), {x, 0, z}, 
         Assumptions -> z > 0] //InputForm

Out[4]//InputForm= 
(-2*E*Pi - 2*E^(1 + 2*I)*Pi + 2*E^I*CosIntegral[I] - 2*E^(2 + I)*CosIntegral[I] + 
  (1 - I)*E^I*(-I + E^2)*CosIntegral[I - z] + (1 + I)*E^I*(I + E^2)*CosIntegral[I + z] - 
  (2*I)*E*Gamma[0, -I] + (2*I)*E^(1 + 2*I)*Gamma[0, I] + (2*I)*E*Gamma[0, -I + z] - 
  (2*I)*E^(1 + 2*I)*Gamma[0, I + z] + (2*I)*E^(1 + 2*I)*Log[-I - z] - (2*I)*E*Log[I - z] + 
  (2*I)*E*Log[-I + z] - (2*I)*E^(1 + 2*I)*Log[I + z] - (2*I)*E^I*SinIntegral[I] - 
  (2*I)*E^(2 + I)*SinIntegral[I] - (1 - I)*E^I*SinIntegral[I - z] + 
  (1 + I)*E^(2 + I)*SinIntegral[I - z] + (1 + I)*E^I*SinIntegral[I + z] - 
  (1 - I)*E^(2 + I)*SinIntegral[I + z])/(8*E^(1 + I))

In[5]:= Limit[%, z -> Infinity] //FullSimplify //InputForm

Out[5]//InputForm= 
(Pi - E^2*ExpIntegralEi[-1] + ExpIntegralEi[1] + 2*E*CosIntegral[1]*Sin[1] + 
  E*Cos[1]*(Pi - 2*SinIntegral[1]))/(4*E)

In[6]:= N[%, 20]

Out[6]= 0.92303721095520214407

The Interval result from FunctionExpand[meijer] was, if I recall correctly, a Sum issue that was too late to fix for the 5.2 release. Sorry for the inconvenience.

Bhuvanesh,
Wolfram Research.


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