RE: RE: Can you help me to solve this Integrate using Mathematica ?
- To: mathgroup at smc.vnet.net
- Subject: [mg36269] RE: [mg36248] RE: [mg36221] Can you help me to solve this Integrate using Mathematica ?
- From: "DrBob" <drbob at bigfoot.com>
- Date: Thu, 29 Aug 2002 01:37:53 -0400 (EDT)
- Reply-to: <drbob at bigfoot.com>
- Sender: owner-wri-mathgroup at wolfram.com
The integral doesn't converge when b is real, because of the second order singularity at Abs[b] in that case. Arg[b^2]!=0 is used to express this because the issue is whether that singularity is on the positive x-axis. Behavior at 0 and Infinity are fine if Re[a]>0, but otherwise those are problems too, so Mathematica has the right answer in those terms. I can't vouch for the formula it comes up with when the conditions are met, but it's not closed-form in the usual sense anyway; but it's closed-form in terms of functions MATHEMATICA is comfortable with! Bobby Treat -----Original Message----- From: David Park [mailto:djmp at earthlink.net] To: mathgroup at smc.vnet.net Subject: [mg36269] [mg36248] RE: [mg36221] Can you help me to solve this Integrate using Mathematica ? Valdeci, It appears that the integral will not converge if b is real. Mathematica gives... Integrate[Exp[-a/x]/(x^2 - b^2), {x, 0, Infinity}] If[Re[a] > 0 && Arg[b^2] != 0, (1/2)*Sqrt[-(1/b^2)]* (2*CosIntegral[a*Sqrt[-(1/b^2)]]* Sin[a*Sqrt[-(1/b^2)]] + Cos[a*Sqrt[-(1/b^2)]]* (Pi - 2*SinIntegral[a*Sqrt[-(1/b^2)]])), Integrate[1/(E^(a/x)*(-b^2 + x^2)), {x, 0, Infinity}]] If b is Real then Arg[b^2]==0 and Mathematica doesn't solve it. Let's define a function that allows us to test specific values of a and b. f[a_, b_][x_] = Exp[-a/x]/(x^2 - b^2); Integrate[f[2, 3][x], {x, 0, Infinity}] Integrate::"idiv" : "Integral of 1/(E^(2/x)*(-9 + x^2)) does not converge on \ {0, Infinity}." But if we use an imaginary value for b... Integrate[f[2, 3I][x], {x, 0, Infinity}] %//N (1/6)*(2*CosIntegral[2/3]*Sin[2/3] + Cos[2/3]*(Pi - 2*SinIntegral[2/3])) 0.254022 David Park djmp at earthlink.net http://home.earthlink.net/~djmp/ From: Valdeci Pereira Mariano de Souza [mailto:vpms at rc.unesp.br] To: mathgroup at smc.vnet.net Dear Group I would like to know, if is possible to solve Integrate[(exp[-a/x])/(x^2-b^2),{x,0,infinity}] with a and b constant ( Real ) using Mathematica. Thankfully, Valdeci Mariano ********************************************************************* Valdeci Mariano de Souza Master´s Degree of Applied Physics - Unesp/Rio Claro - State of São Paulo - Brazil Laboratory of Electrical Measurements email : vpms at rc.unesp.br or valdecimariano at bol.com.br phone : ( 0XX19 ) 526 - 2237 *********************************************************************