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I am trying to compute the following integral:

Integrate[Exp[Sum[-((cw λ - b[i])^2/(2 σ^2)), {i, 1, n}]], {cw, 0, 1}]

And currently Mathematica outputs

(Sqrt[π/2] σ (Erf[(Sqrt[n] (λ - b[i]))/(Sqrt[2] σ)] + 
     Erf[(Sqrt[n] b[i])/(Sqrt[2] σ)]))/(Sqrt[n] λ)

Which is not only blatantly incorrect (there can be no dependence on i for example), but also has little connection with my input. If I replace n with an integer in the first expression, the output is the correct result for the integration, but I want the general result with n summands. What am I doing wrong?

Edit: I am willing to make assumptions, such as that λ, σ and all b[i]'s are real. Nonetheless, that does not seem to matter here.

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I'm pretty sure this has been asked before, but I can't find it. Anyone? –  belisarius Sep 29 '13 at 21:01
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1 Answer

up vote 10 down vote accepted

In this particular case Mathematica for some reason considers b[i] as a constant. Compare:

Integrate[Exp[Sum[-((cw λ - b)^2/(2 σ^2)), {i, 1, n}]], {cw, 0, 1}]
(Sqrt[π/2] σ (Erf[(b Sqrt[n])/(Sqrt[2] σ)]-Erf[(Sqrt[n] (b-λ))/(Sqrt[2] σ)]))/(Sqrt[n] λ)

A possible workaround consists in the manually expanding the sum

Integrate[Exp[(-n cw^2 λ^2 + 2 cw λ Sum[b[i], {i, 1, n}] - 
  Sum[b[i]^2, {i, 1, n}])/(2 σ^2)], {cw, 0, 1}]

enter image description here

There is another example of the strange behavior of the sum of b[i]

Sum[cw b[i], {i, 1, n}] - cw Sum[b[i], {i, 1, n}]

enter image description here

It is not simplified to 0 (even with Simplify and FullSimplify).

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I have expanded the sum already, but this is quite a quirk... –  emaster70 Sep 29 '13 at 22:30
    
Would you not consider this a bug ? –  b.gatessucks Sep 30 '13 at 7:12
    
@b.gatessucks Yes, it looks like a bug. I usually look for my own mistakes before considering something as a bug. –  ybeltukov Oct 2 '13 at 14:39
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