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Re: How to interpret this integral?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg112616] Re: How to interpret this integral?
*From*: "Sjoerd C. de Vries" <sjoerd.c.devries at gmail.com>
*Date*: Thu, 23 Sep 2010 04:21:35 -0400 (EDT)
Hi Julian,
The problem for aW==0 can be solved by using the following expression when aW equals zero:
In[129]:== Limit[result, aW -> 0]
Out[129]== (1/(w0^2))(-aV Cos[th0] + aV Cos[th0 + Ts w0] +
w0 (-v0 Sin[th0] + aV Ts Sin[th0 + Ts w0] + v0 Sin[th0 + Ts w0]))
For aW <0 I think you'll be able to solve the problem if you realize that FresnelC[a I] == I FresnelC[a] and FresnelS[a I] == -I FresnelS[a]. So replacing Sqrt[wA] with I Sqrt[-wA] and canceling the common factors of I (there's a wA^(3/2) in front) will yield your expression for wA < 0.
Cheers -- Sjoerd
-----Original Message-----
From: Julian Stoev / =D0=AE=D0=BB=D0=B8=D0=B0=D0=BD =D0=A1=D1=82=D0=BE=D0=
=B5=D0=B2 [mailto:julian.stoev at gmail.com]
Sent: Wednesday 22 September 2010 16:02
To: Sjoerd C. de Vries
Subject: [mg112616] Re: How to interpret this integral?
Hello Sjoerd,
I have no problems with Fresnel integrals. There is free code
available, which can evaluate them quite well for real arguments.
http://mymathlib.webtrellis.net/functions/fresnel_sin_cos_integrals.html
However Sqrt[negative] and 1/0 should be avoided for any possible
numerical values.
best!
--JS
On Wed, Sep 22, 2010 at 3:55 PM, Sjoerd C. de Vries
<sjoerd.c.devries at gmail.com> wrote:
> 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 problematic
>> 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
>
>
--
Julian Stoev, PhD.
Control Researcher
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