Re: Question

• To: mathgroup at christensen.cybernetics.net
• Subject: [mg458] Re: [mg450] Question
• From: bob Hanlon <hanlon at pafosu2.hq.af.mil>
• Date: Sun, 12 Feb 1995 22:09:41

```>  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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