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Re: Re: Write an expression in a specific form

• To: mathgroup at smc.vnet.net
• Subject: [mg108039] Re: [mg108025] Re: Write an expression in a specific form
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Sun, 7 Mar 2010 04:02:43 -0500 (EST)
• References: <hmo293$qbh$1@smc.vnet.net> <201003050935.EAA29559@smc.vnet.net>
• Reply-to: drmajorbob at yahoo.com

I missed where c4 is defined in that... but starting out, we know it has  
to be equal to

G[m_,s_,x_]:=(1/(s*Sqrt[2*Pi]))*Exp[-(x-m)^2/(2*s^2)];
Integrate[G[m1,s1,x]  
G[m2,s2,x],{x,-Infinity,Infinity},Assumptions->{s1>0,s2>0}]

E^(-((m1-m2)^2/(2 (s1^2+s2^2))))/(Sqrt[2 \[Pi]] Sqrt[s1^2+s2^2])

No?

I think c4 was supposed to be c3, and sure enough, retrieving the RHS of  
the rule for c3, I have:

(E^(-((m1-m2)^2/(2 (s1^2+s2^2)))) Sqrt[(s1^2 s2^2)/(s1^2+s2^2)])/(Sqrt[2  
\[Pi]] s1 s2)//PowerExpand

E^(-((m1-m2)^2/(2 (s1^2+s2^2))))/(Sqrt[2 \[Pi]] Sqrt[s1^2+s2^2])

Bobby

On Fri, 05 Mar 2010 03:35:13 -0600, dh <dh at metrohm.com> wrote:

> Hi Ares,
> Mathematica really seems to have difficulties (that is I do not have the  
> nerves
> to wait so long) with a calculation that can done by hand. It look like
> it needs some human help.
> We may get an equation for the exponents with only 2 unknowns.
> We denot by m1,s1 and m2,s2 the parameters of the given Gaussian. m3,s3
> and c3 denote the searched Gaussian: c4 G[m3,s3]. tmp is an intermediate
> result we are not interested in:
> sol1 = Reduce[{ForAll[
>      x, (x - m1)^2/(2 s1^2) + (x - m2)^2/(2 s2^2) ==
>       tmp + (x - m3)^2/(2 s3^2)], s1 > 0, s2 > 0, s3 > 0}, {m3, s3, tmp},
>     Reals]
> this gives us m3,s3 and an intermediate result tmp. With this we may
> solve the equation for the pre factor. Toward this aim we change sol1 to
> rules and use it in the equation:
> G[m1, s1, x] G[m2, s2, x] == c3  G[m3, s3, x] /. sol1 /. sol1
> Note that in sol1 tmp is given in term of m3,s3. We must therefore apply
> /.sol1 twice.
> I also delete some superfluous info from the result. Here is the whole  
> code:
> ================
> G[m_, s_, x_] := (1/(s*Sqrt[2*Pi]))*Exp[-(x - m)^2/(2*s^2)];
> sol1 = Reduce[{ForAll[
>       x, -((x - m1)^2/(2 s1^2)) - (x - m2)^2/(2 s2^2) ==
>        tmp - (x - m3)^2/(2 s3^2)], s1 > 0, s2 > 0, s3 > 0}, {m3, s3,
>      tmp}, Reals];
> sol1 = Drop[sol1, 2] // ToRules
> sol2 = Reduce[{G[m1, s1, x] G[m2, s2, x] == c3  G[m3, s3, x] /.
>         sol1 /. sol1, s1 > 0, s2 > 0, s3 > 0, c3 > 0}, {c3},
>      Reals][[4]] // ToRules
> ===============
>
> Finally we may test if the calculation is correct:
> G[m1, s1, x] G[m2, s2, x] == c4  G[m3, s3, x] /. sol2 /. sol1 /.
>    sol1 // Simplify
>
> Daniel
>
> On 04.03.2010 11:33, Ares Lagae wrote:
>> Hi all,
>>
>> I am a beginner in Mathematica, and I have the following "problem": How  
>> can
>> I write an expression in a specific form?
>>
>> For example:
>>
>> - Define a Gaussian:
>>
>> G[m_, s_, x_] := (1/(s*Sqrt[2*Pi]))*Exp[-(x - m)^2/(2*s^2)];
>>
>> - Product of two Gaussians:
>>
>> G[m1, s1, x] * G[m2, s2, x]
>>
>> - How can I get Mathematica to write the result in terms of c * G[m_,  
>> s_,
>> x_]? I.e., get the values for c, m and s.
>>
>> Thanks,
>>
>> Ares Lagae
>>
>>
>
>


-- 
DrMajorBob at yahoo.com

• References:
  • Re: Write an expression in a specific form
    • From: dh <dh@metrohm.com>