MaplePrimes - answers and comments on Question, integrating [exp(-cx)]/[a^2+x^2]^1/2

limits?

PatrickT — Tue, 05 Jan 2010 02:57:54 Z

you haven't said what limits you were integrating within

expr := (exp(-c*x))/(a^2+x^2)^1/2;

                                    exp(-c x)
                        expr := 1/2 ---------
                                      2    2
                                     a  + x

int( expr, x=0..1 );

  1/4 I (exp(2 I c a) Ei(1, c a I) - Ei(1, -I c a)

         - exp(2 I c a) Ei(1, c + c a I) + Ei(1, c - c a I))

        exp(-I c a)/a

int( expr, x=0..infinity );

  {
  { undefined , 0 < -I a
  {

        undefined , 0 < a I

             lim       -1/4 I exp(-I c a) (
        x -> infinity

                  2
        exp(c a I)  Ei(1, c x + c a I)

         - exp(c a I) exp(-I c a) Ei(1, c x - c a I)

         - exp(2 I c a) Ei(1, c a I) + Ei(1, -I c a))/a , otherwise

I think you did the wrong function +

tsunamiBTP — Tue, 05 Jan 2010 04:05:02 Z

This is where I am confused. I do not want to define limits because I want a generic analytical expression, not to evaluate the integral. Should I define the limits from 0 to an arbitrary constant? Is this why MAPLE simply spits out the expression that I typed in?

MAPLE has been a bit of an aggravating sore spot for me & so I don't use it that much which is probably why I have so many issues with it.

The expression should be to integrate with respect to x and the variables a,b, & c are real +:

exp(-c*x)/((a^2+b*x^2)^0.5)

antiderivative

tsunamiBTP — Tue, 05 Jan 2010 09:41:12 Z

YEP, I cannot seem to find an analytical antideriv either. By appearance it appears there should be one. In fact, if 1 exists it would describe the radiation field emanating from a point source along a radial axis which I would think someone has solved. MAYBE NOT

Anyone has any other suggestions I welcome them!

jakubi is correct on the placement of the sq root

tsunamiBTP — Tue, 05 Jan 2010 21:56:32 Z

see the symbolic expression by Robert Israel.

All parameters, a, b, & c must be symbolic. I can say in most instances b -->1.0 for my specific application.

The difficulty arises when a is something other than ZERO.

I ended up solving the integral by hand but had to employ transformations to get it in the form of:

(exp(-cx))/x

It took me about an hour. Yesterday I spent the entire day attempting to get MAPLE to do the work. What use is MAPLE? Everytime I use it nothing is EVER straight forward. Software is supposed to speed things up & provide assistance. MAPLE never does that.

I wasted time & money on this package!!!!!

I will try to upload my solution, any comments if incorrect is welcome!!

I scanned in my solution but----

tsunamiBTP — Tue, 05 Jan 2010 22:08:38 Z

How do I attach a JPEG scan? I attempted to use the Insert/Edit Image icon at the top of the editing toolbox, but it is asking for a URL address.dd

My solution (Probably not 100% correct)

tsunamiBTP — Wed, 06 Jan 2010 04:04:54 Z

I was not successful at uploading the images so I converted everything to pdf, so the supposed link info is below. I do not think my solution is totally correct but it appears to be n the right track since I know from the text what the solution for the far field radiation distribution is supposed to be. Page 1 is the derivation of the integral of concern. Page 2 is the attempt to solve the integral. I am pretty cdonfident in page 1. Evaluation of the integral on p. 2 is where troubles reside & that is where I was hoping MAPLE would be of assistance.

Any feedback as to how to correctly evaluate that integral is welcome. Maybe it can only be done numerically?

<a href='http://www.mapleprimes.com/files/9288_piston radiation model.pdf'>Download 9288_piston radiation model.pdf</a><br/><a href='http://www.mapleprimes.com/viewfile/3949'>View file details</a>