gmane.comp.mathematics.maxima.general

Dieter Kaiser | 16 May 00:58

Problems with the function $specint

Because I have only time in the evening hours and I use the time to look at the
code, I can not answer so fast.

I will open a bug report at the weekend.

Perhaps further results I have got are interessting for you:

1. 

I have found a bug in the routine lt-exp and f35p147. This bug prevents the
calculation of integrals with e.g. sin(2*sqrt(a*t)). SPECINT gives the result 0.
Here the output of Maxima after correction:

(%i6) radcan(specint(%e^(-s*t)*sin(2*sqrt(a*t)),t));
(%o6) sqrt(%pi)*sqrt(a)*%e^-(a/s)/s^(3/2)

That is perfectly the tabulated expression and SPECINT now works for a lot of
other integrals too.

2. 

To show how we can extend the algorithm of SPECINT to calculate further
integrals, I have added code to calculate integrals of the form
t^-1*(%e^(-a*t)-%e^(-b*t)). The code works also for integrals like
t^-1*sin(a*t). Here an example

(%i4) specint(%e^(-s*t)*t^-1*sin(a*t),t);
(%o4) %i*(log(s-%i*a)-log(s+%i*a))/2

That's equivalent with the tabulated answer: atan(a/s)

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Robert Dodier | 17 May 19:59

Re: Problems with the function $specint

On 5/15/08, Dieter Kaiser <drdieterkaiser <at> web.de> wrote:

> Because I have only time in the evening hours and I use the time to look at the
>  code, I can not answer so fast.

No problem, Dieter. Thanks a lot for looking at specint, I appreciate
your help very much. I look forward to seeing the patches when you
can post them.

Thanks again for your help,

Robert Dodier

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Dieter Kaiser | 19 May 00:01

Re: Problems with the function $specint

I had a further look at the code of $specint.

The main idea of $SPECINT is to transform the special functions including log
and erf to a Hypergeometric function and use the known Laplace transform of a
Hypergeometric function F[p,q]([a1,...,ap],[b1,...,bq],z).

The function REF get the representation as a Hypergeometric function and after
some scaling and tests we try to get the Laplace transform in the routine
F19P220SIMP. There are about 16 functions implemented which try to use this
algorithm to get their Laplace transform.

The bad news is, that for more than a half of the functions implemented,
including log and erf, this algorithm don't work. The reason is that F19P220SIMP
only works for Hypergeometric functions F[p,q]([],[],t^m) with p+m-1<=q. 

So it works for bessel_j with a Hypergeometric function 0F1, or Whittaker M
function with the representation 1F1. But not for log with a Hypergeometric
function 2F1 or the erf function. Here we have a transformation
1F1(1/2;3/2;-z^2), and because of the quadratic exponent the condition p+m-1<=q
is not true.

That's the reason why we don't get the known Laplace transforms for a lot of
functions.

Next steps:

1. 

I have collected the formulas for the Laplace transforms of more specialized
Hypergeometric functions: the Hypergeometric 1F1 function, Hypergeometric 2F1

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Raymond Toy (RT/EUS | 19 May 16:12

Re: Problems with the function $specint

Dieter Kaiser wrote:
> 1. 
> 
> I have collected the formulas for the Laplace transforms of more specialized
> Hypergeometric functions: the Hypergeometric 1F1 function, Hypergeometric 2F1
> function, Whittaker M function, Whittaker W function and the Hypergeometric U
> function. (The algorithm of the Laplace transform for the Whittaker W function
> is implemented in the routine F16P217TEST. But the formula I have got looks more
> general.) 

All of the formulas used by specint come from Bateman Manuscript 
Project, Table of Integral Transforms.  F16p217 refers to formula 16 on 
page 217.

> To get more insight in the algorithm I have implemented some code for a noun
> form of a function hypergeometric and the Laplace transform for this function.
> Additionally, I have implemented the functions whittaker_m and whittaker_w. It
> is much more expressive to use such nouns instead of %f, %w or %m. 

[snip]
> There are more functions $SPECINT uses which should be represented by expressive
> noun forms, e.g.:
> 
> %l  -> laguerre, 
> %he -> hermite, 
> %d  -> parabolic_cylinder_d,
> %h  -> hankel_1 or hankel_2, 
> %e  -> elleptic_ec

These were not changed because there were no implementations of these

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Andrej Vodopivec | 19 May 21:31

Re: Problems with the function $specint

008/5/19 Raymond Toy (RT/EUS) <raymond.toy <at> ericsson.com>:
> The whole hypergeometric function package (hgfred and specint) needs
> lots of work, but there hasn't been many (any?) users, so I've been
> doing stuff whenever I feel like it.

The function simplify_sum uses hgfred:

(%i2) sum(n/((2*n+1)*2^n),n,0,inf)$
(%i3) simplify_sum(%);
(%o3) -(sqrt(2)*atanh(1/sqrt(2))-2)/2

I think this was added in version 5.15.

--

-- 
Andrej