From: fc03@ns1.cc.lehigh.edu (Frederick W. Chapman)
Newsgroups: sci.math.symbolic
Subject: Re: Maple can diff(expr), but not int(diff(expr))?
Date: 14 Apr 92 19:03:27 GMT
Organization: Lehigh University

In article <1992Apr14.133340.16852@ux1.cso.uiuc.edu>, gerber@sirius.astro.uiuc.e
du (Richard A. Gerber) writes:
>
>I wrote that Maple could not seem to solve:
>int( diff(BesselJ(1,x),x), x);
>
>I received a nice way to solve this:


[DELETED:  Elegant use of 'laplace', 'normal', and 'invlaplace' for the
           simplification of definite integrals.]


Here is a slight variation on the use of Laplace transforms in computing
anti-derivatives.  An antiderivative F of a function f satisfies the initial
value problem F' = f, F(0) = C (arbitrary constant).  If L denotes the
Laplace transform, and s is the independent variable in the transform domain,
then

                        L(f)  =  L(F')  =  s L(F) - C

                            L(F)  =  L(f)/s  +  C/s

                                 -1
                          F  =  L   ( L(f)/s )  +  C

Note how nicely the arbitrary constant C (of integration) shows up in the
final answer!  This works very well in Maple (using 'laplace', 'normal', and
'invlaplace'), but it is *crucial* to simplify the Laplace transform the
right way before attempting to invert it.

Even simpler is to actually solve the initial value problem using 'dsolve'
in conjunction with the 'laplace' option:

> f := diff(BesselJ(1,x),x);

                                            BesselJ(1, x)
                       f := BesselJ(0, x) - -------------
                                                  x

> dsolve(diff(F(x),x) = f, F(x), laplace);

                          F(x) = F(0) + BesselJ(1, x)

----------------------------------------------------------------------------

>I also wrote that even though a simple closed form solution exists,
>I could not get Maple V to evaluate:
>
>>int(BesselJ(1,x)*exp(-x),x=0..infinity);
>
>I received responses stating that this was not as simple a problem as it
>appeared. This is an example of a definite integral that has a closed
>form solution, while the corresponding indefinite integral does not.
>I've been told that the people at Maple at actively working on these
>type of integrals.

This integral is just the definition of the Laplace transform of BesselJ(1,x),
evaluated at s=1. Maple *can* perform this computation:

> laplace(BesselJ(1,x), x, s);

                                  2     1/2
                                (s  + 1)    - s
                                ---------------
                                    2     1/2
                                  (s  + 1)

> subs(s=1, ");

                                    1/2       1/2
                              1/2 (2    - 1) 2


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