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Re: How to interpret this integral?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg112608] Re: How to interpret this integral?
*From*: "Sjoerd C. de Vries" <sjoerd.c.devries at gmail.com>
*Date*: Thu, 23 Sep 2010 04:20:09 -0400 (EDT)
*References*: <i7c5sf$5es$1@smc.vnet.net>
Hi Julian,
I don't know how to get rid of the Fresnel integrals, but for use in C
programs couldn't you just generate a (large) table of Fresnel
integral values and interpolate where needed? They both converge to a
fixed value rather rapidly, so that should be doable.
For instance, you could sample the integrals at their local maxima at
+/- Sqrt[2/\[Pi]] Sqrt[\[Pi] + 2 \[Pi] k] with k integer and minima at
2 Sqrt[k].
Cheers -- Sjoerd
On Sep 22, 7:57 am, Julian <julian.st... at gmail.com> wrote:
> Hello All,
>
> After long interruption with symbolic computing, I am struggling how
> to make a useful result out of this integral.
>
> Integrate[(aV*t + v0)*Cos[(aW*t^2)/2 + th0 + t*w0, {t, 0, Ts}]
>
> It comes from the differential equations describing the motion of a
> wheeled robot. aV and aW are accelerations and v0, w0, th0 are initial
> conditions. The numerical solution clearly exists for different real
> accelerations, including positive, negative and zero.
>
> However the symbolic solution of Mathematica is:
> (Sqrt[Pi]*(-(aW*v0) + aV*w0)*Cos[th0 - w0^2/(2*aW)]*
> FresnelC[w0/(Sqrt[aW]*Sqrt[Pi])] + Sqrt[Pi]*(aW*v0 - aV*w0)*
> Cos[th0 - w0^2/(2*aW)]*FresnelC[(aW*Ts + w0)/(Sqrt[aW]*Sqrt[Pi])]
> +
> aV*Sqrt[aW]*(-Sin[th0] + Sin[th0 + (aW*Ts^2)/2 + Ts*w0]) +
> Sqrt[Pi]*(aW*v0 - aV*w0)*(FresnelS[w0/(Sqrt[aW]*Sqrt[Pi])] -
> FresnelS[(aW*Ts + w0)/(Sqrt[aW]*Sqrt[Pi])])*Sin[th0 - w0^2/
> (2*aW)])/
> aW^(3/2)
>
> Note that aW can be found inside a Sqrt function and also in the
> denominator. While I can see that FresnelS[Infinity] is well defined,
> so aW==0 should not be a real problem, it is still very problemati=
c
> how to use this result at this particular point. The case of negative
> aW is also interesting, because it will require a complex FresnelS and
> FresnelC. I somehow managed to remove this problem using
> ComplexExpand, but I am not sure this is the good solution.
>
> I tried to give Assumptions -> Element[{aW, aV, w0, v0, Ts, th0},
> Reals] to the integral, but there is no change in the solution.
>
> While I understand that the solution of Mathematica is correct in the
> strict mathematical sense, I have to use the result to generate a C-
> code, which will be evaluated numerically and the solution I have now
> is not working for this.
>
> Can some experienced user give a good advice how use get a solution of
> the problem?
>
> Thank you in advance!
>
> --JS
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