Re: Re: an Integrate question

*To*: mathgroup at smc.vnet.net*Subject*: [mg58937] Re: [mg58918] Re: an Integrate question*From*: Pratik Desai <pdesai1 at umbc.edu>*Date*: Sun, 24 Jul 2005 01:21:56 -0400 (EDT)*References*: <dbie3r$brm$1@smc.vnet.net><dbkkiq$s49$1@smc.vnet.net> <dbninb$hlj$1@smc.vnet.net> <200507230932.FAA29106@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Drago Ganic wrote: >Antonio, >We have a more precise analytical solution in Mathematica 5.2 than in 5.1. >Here is the solution I got with 5.2: > >In[3]:= >Integrate[E^(I*h*x)/(E^(m*Sqrt[k2 + x2])* >(Abs[x] + Sqrt[k2 + x2])), {x, 0, Infinity}] > >Out[3]= >Piecewise[{{((1/2)*(2*Gamma[0, (-I)*h*Sqrt[k2 + x2]] - >2*Log[(-I)*h] + 2*Log[(-I)*h*Sqrt[k2 + x2]] - >Log[k2 + x2]))/E^((I*h + m)*Sqrt[k2 + x2]), >Im[h] > 0}, >{(Pi*MeijerG[{{0}, {1/2}}, {{0, 0}, {1/2}}, >I*h*Sqrt[k2 + x2]])/E^(m*Sqrt[k2 + x2]), >h \[Element] Reals && Im[h] <= 0}}, >Integrate[E^(I*h*x - m*Sqrt[k2 + x2])/ >(Abs[x] + Sqrt[k2 + x2]), {x, 0, Infinity}, >Assumptions -> h \[NotElement] Reals && Im[h] <= 0]] >and using assumptions we get just one solution: > >In[6]:= >Assuming[{h, m} \[Element] Reals && Re[k2] <= 0), >Integrate[E^(I*h*x)/(E^(m*Sqrt[x2 + k2])*(Abs[x] + Sqrt[x2 + k2])), >{x, 0, Infinity}]] > >Out[6]= >(Pi*MeijerG[{{0}, {1/2}}, {{0, 0}, {1/2}}, >I*h*Sqrt[k2 + x2]])/E^(m*Sqrt[k2 + x2]) > >I did not verify the solution. But I don't understand the condition: > > h \[Element] Reals && Im[h] <= 0 > >because h \[Element] => Im[h] == 0 ?!? Your case should be covered with >the solution given. > >Gretings, >Drago Ganic > >"Antonio Carlos Siqueira" <acsl at dee.ufrj.br> wrote in message >news:dbninb$hlj$1 at smc.vnet.net... > > >>Dear Mathgroup >>First of all thanks to Carl Woll and Andreas Dieckman for providing the >>answer to the integral. Digging my results a little bit I found >>myself with a little different integral. I am trying to calculate some >>electromagnetic field in conductive soil, the integral looks like this: >>Integrate[Exp[-m Sqrt[x2+k2]]*> Exp[I h >>x]/(Abs[x]+Sqrt[x2+k2]),{x,0,Infinity}] >>where h and m are real and k is either pure imaginary or has a negative >>real part. >>Any ideas on how to solve such integral analytically? >>After some tests I believe this integral is a job for NIntegrate, >>nevertheless it is worth asking if anyone knows any trick to transform >>this integral in something that can be solved analytically. >>Nice Regards >>Antonio >> >> >> > > > > Here is something that works a little faster Assuming[h Ïµ Reals && m Ïµ Reals, Integrate[E^(I*h*x)/(E^(m*Sqrt[ k2 + x2])*(Abs[x] + Sqrt[k2 + x2])), {x, 0, Infinity}]] // Timing >>\!\({8.122000000000014`\ Second, \[ExponentialE]\^\(\(-m\)\ \@\(k2 + x2\)\)\ \ \[Pi]\ MeijerG[{{0}, {1\/2}}, {{0, 0}, {1\/2}}, \[ImaginaryI]\ h\ \@\(k2 + x2\)]}\) TagSet[h, Im[h], 0] TagSet[h, Re[h], h] TagSet[m, Im[m], 0] TagSet[m, Re[m], m] Integrate[E^(I*h*x)/(E^(m*Sqrt[k2 + x2])*(Abs[x] + Sqrt[k2 + x2])), {x, 0, Infinity}] // Timing >>\!\({6.789999999999992`\ Second, \[ExponentialE]\^\(\(-m\)\ \@\(k2 + x2\)\)\ \ \[Pi]\ MeijerG[{{0}, {1\/2}}, {{0, 0}, {1\/2}}, \[ImaginaryI]\ h\ \@\(k2 + x2\)]}\) Now if you want to work by simplifying your integrand Clear[f, h, m] TagSet[h, Im[h], 0] TagSet[h, Re[h], h] TagSet[m, Im[m], 0] TagSet[m, Re[m], m] f[x_] = E^(I*h*x)/(E^(m*Sqrt[k2 + x2])*(Abs[x] + Sqrt[k2 + x2])) // ExpToTrig \ // Simplify Integrate[f[x], {x, 0, 1}] // Timing >>\!\({26.577999999999975`\ Second, \(-\[ExponentialE]\^\(\((\(-\[ImaginaryI]\)\ \ h - m)\)\ \@\(k2 + x2\)\)\)\ \((CosIntegral[h\ \@\(k2 + x2\)] - CosIntegral[ h\ \((1 + \@\(k2 + x2\))\)] + \[ImaginaryI]\ SinIntegral[ h\ \@\(k2 + x2\)] - \[ImaginaryI]\ SinIntegral[ h\ \((1 + \@\(k2 + x2\))\)])\)}\) The trick that I always use and makes the integration go faster is that if you simplify your expression and find certain cluster of constants just replace them by simple names like m*Sqrt[k2+x2] by say s, Mathematica finds the simplified version more easy to integrate and you can always replace your parameters by thier actual values (relationships) PS: This was all done on 5.1 -- Pratik Desai Graduate Student UMBC Department of Mechanical Engineering Phone: 410 455 8134

**References**:**Re: an Integrate question***From:*"Drago Ganic" <drago.ganic@in2.hr>