Any Math wizards?

[Tiamat]

You start with:

I(a) = Integral [exp(-x^2)*sin(2ax)*(1/x)]dx (with bounds of 0 to positive infinity)

The text goes on to say "Clearly, the above equation can be manipulated to form the following useful result"

I(a) = 0.5*(PI)*erf(a)

Any insight as to how this is clear? I see an integral with 3 functions of x -- cant do integration by parts.

The exponential part of the integral looks similar to the error function, and the latter looks similar to the sine integral Si(infinity), but I have absolutely no idea how to derive the second equation.


Any hints would be appreciated. I have consulted 3 advanced engineering mathematics text books without finding any help :/

 

[Heisenberg]

Maple does it without any problem. Try integration by parts using u=exp(-x^2) and dv=sin(2ax)*(1/x). I think all those integrals are in the tables.

 

[Tiamat]

Originally posted by: Heisenberg
Maple does it without any problem. Try integration by parts using u=exp(-x^2) and dv=sin(2ax)*(1/x). I think all those integrals are in the tables.

Thanks for the insight. I dont have integral tables on me, so I was manually doing all the integrations. I don't have any computational software or a TI-89 either. Usually, just going through standard calculus techniques has done me well, but with the introduction of integral functions, i got overwhelmed.

 

[Heisenberg]

Originally posted by: Tiamat

Originally posted by: Heisenberg
Maple does it without any problem. Try integration by parts using u=exp(-x^2) and dv=sin(2ax)*(1/x). I think all those integrals are in the tables.

Thanks for the insight. I dont have integral tables on me, so I was manually doing all the integrations. I don't have any computational software or a TI-89 either. Usually, just going through standard calculus techniques has done me well, but with the introduction of integral functions, i got overwhelmed.

No problem. At the point where it sounds like you are in your coursework, you really begin to need at least a table of integrals. It simply isn't efficient to try and reinvent the wheel on every problem.

 

[dullard]

Originally posted by: Heisenberg
Maple does it without any problem. Try integration by parts using u=exp(-x^2) and dv=sin(2ax)*(1/x). I think all those integrals are in the tables.

They should be in the tables.

u=exp(-x^2) leads to the error function.

dv=sin(2ax)*(1/x) leads to the Si(x) function.

I would hope any good table would have both.

 

[JoeFahey]

Darn, Heisberg beat me to it......

 

[Tiamat]

Originally posted by: dullard

Originally posted by: Heisenberg
Maple does it without any problem. Try integration by parts using u=exp(-x^2) and dv=sin(2ax)*(1/x). I think all those integrals are in the tables.

They should be in the tables.

u=exp(-x^2) leads to the error function.

dv=sin(2ax)*(1/x) leads to the Si(x) function.

I would hope any good table would have both.

yeah, i already knew what they lead to, but i couldnt perform the step by step derivation.

I am getting some additional understanding with:
http://mathworld.wolfram.com/SineIntegral.html