Re: Question

• To: mathgroup at christensen.cybernetics.net
• Subject: [mg465] Re: [mg450] Question
• From: bob Hanlon <hanlon at pafosu2.hq.af.mil>
• Date: Mon, 13 Feb 1995 23:23:49

```A further simplification can be made by converting the incomplete gamma
function to a confluent hypergeometric function.  The form of the function
is then

h[a_Real, b_Real, c_Real] := Hypergeometric1F1[1, 1 + c/b, -a]/c;
(* Abramowitz & Stegun, eqn. 6.5.12 *)

On Sun, 12 Feb 1995, bob Hanlon wrote:

> >  Message-Id: <9502080805.AA13790 at christensen.cybernetics.net.>
> >  Date: Tue, 7 Feb 1995 12:08:01 -0500
> >  To: mathgroup at christensen.cybernetics.net
> >  From: deburm at tiac.net (David E. Burmaster)
> >  Subject: [mg450] Question
> >
> >  +++++++++++++++
> >
> >  Dear MathGroup:
> >
> >  Any suggestions on how to integrate this function from zero to +Infinity??
> >
> >  a, b, and c are positive constants
> >
> >  a is approx 0.75
> >  b is approx 4
> >  c is approx 0.1
> >
> >  Here is the function that I wish to integrate
> >
> >          S = Exp[ -1 ( a (1 - Exp[ -b t]) + c t   ) ]
> >
> >  ++++++
> >
> >  NIntegrate should do the job, but I am hoping to find a closed-form
> solution
> >
>
> I couldn't find any simple way of doing it in Mathematica, so I looked it
> up in a table.
>
> Bob Hanlon
> hanlon at pafosu2.hq.af.mil
>
> ____________________
>
> Clear[f, g];
>
> f[a_Real, b_Real, c_Real] :=
> 	NIntegrate[Exp[-1 (a (1 - Exp[-b t]) + c t)],
> 	{t, 0, Infinity}];
>
> g[a_Real, b_Real, c_Real /; c/b > 0] :=
> 	Exp[-a] (-a)^(-c/b) Gamma[c/b, 0, -a]/b // Chop
> (* Gradshteyn & Ryzhik, 4th Edition, 1965, Eqn. 3.331.1 *)
>
> Table[{{a, b, c}, f[a, b, c], g[a, b, c]},
> 	{a, 0.5, 1.0, 0.25}, {b, 3.5, 4.5, 0.5}, {c, 0.08, 0.12, 0.02}]
>
> {{{{{0.5, 3.5, 0.08}, 7.67837, 7.67837},
>
>     {{0.5, 3.5, 0.1}, 6.16154, 6.16154},
>
>     {{0.5, 3.5, 0.12}, 5.15016, 5.15016}},
>
>    {{{0.5, 4., 0.08}, 7.6665, 7.6665},
>
>     {{0.5, 4., 0.1}, 6.14978, 6.14978},
>
>     {{0.5, 4., 0.12}, 5.13852, 5.13852}},
>
>    {{{0.5, 4.5, 0.08}, 7.65722, 7.65722},
>
>     {{0.5, 4.5, 0.1}, 6.14059, 6.14059},
>
>     {{0.5, 4.5, 0.12}, 5.1294, 5.1294}}},
>
>   {{{{0.75, 3.5, 0.08}, 6.02595, 6.02595},
>
>     {{0.75, 3.5, 0.1}, 4.84442, 4.84442},
>
>     {{0.75, 3.5, 0.12}, 4.05654, 4.05654}},
>
>    {{{0.75, 4., 0.08}, 6.01104, 6.01104},
>
>     {{0.75, 4., 0.1}, 4.82966, 4.82966},
>
>     {{0.75, 4., 0.12}, 4.04192, 4.04192}},
>
>    {{{0.75, 4.5, 0.08}, 5.9994, 5.9994},
>
>     {{0.75, 4.5, 0.1}, 4.81811, 4.81811},
>
>     {{0.75, 4.5, 0.12}, 4.03047, 4.03047}}},
>
>   {{{{1., 3.5, 0.08}, 4.73432, 4.73432},
>
>     {{1., 3.5, 0.1}, 3.81396, 3.81396},
>
>     {{1., 3.5, 0.12}, 3.20018, 3.20018}},
>
>    {{{1., 4., 0.08}, 4.71763, 4.71763},
>
>     {{1., 4., 0.1}, 3.79743, 3.79743},
>
>     {{1., 4., 0.12}, 3.18379, 3.18379}},
>
>    {{{1., 4.5, 0.08}, 4.70459, 4.70459},
>
>     {{1., 4.5, 0.1}, 3.78449, 3.78449},
>
>     {{1., 4.5, 0.12}, 3.17096, 3.17096}}}}
>
>

```

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