Fractional Calculus & Inverse Laplace Transforms

• To: mathgroup at smc.vnet.net
• Subject: [mg35138] Fractional Calculus & Inverse Laplace Transforms
• From: jonathan.landy at roche.com (Jonathan)
• Date: Tue, 25 Jun 2002 19:55:23 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Hi,
I've been working on an inverse Laplace transform, here it is:

(1)
w         1
L-1 [ --------- * ----- ] ,  -1<a<1, a!=0
S^2 + w^2    s^a

I solved this by extending the following identity to fractional
powers of a:
L^-1 [ f(s)/s^a ] = (I^a o f)(t) = a indefinite integrals of f(t);

Now,for my solution I got an infinite series, this is alright
because the other method of solving the problem gave some
hypergeometric solutions that were more complicated.
I plugged my solution into mathematica and took the Laplace
trasform and it gave (1)!  However if one gives a value for a, I get
a hypergeometric equation again.  So my questions are: is mathematica
assuming that a is an integer the first time I ran the thing?  Also
does anyone know if I can even make the assumption that I
did (that the identity for the inverse laplace works for fractional
a)?  It appears that I can, because I tested it on some cases one can
do by hand, but that is by no means a proof that it works in general.
However, most of the stuff in fractional calculus is just a definition
anyways, so maybe this is ok?  I don't know.  Thanks for the help.
Also, please can you respond through the message board, because my
email account is not working very well right now.
thanks again.

```

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