Bug: dummy variable appears in result
- To: mathgroup at smc.vnet.net
- Subject: [mg84859] Bug: dummy variable appears in result
- From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
- Date: Wed, 16 Jan 2008 23:06:48 -0500 (EST)
In an effort to answer a question recently posed in sci.math.research, I
encountered a strange bug.
In[6]:=
2^(k - 1)*Integrate[Product[1 - Cos[2*i*Pi*x], {i, 1, k}], {x, 0, 1}]
Out[6]= (1/(i^2*Pi*Gamma[1 + k]))*(2^(-2 +
2*k)*(Sqrt[i^2]*Sqrt[Pi]*Gamma[1/2 + k] - 2*i*Cot[i*Pi]*Gamma[1 +
k]*Hypergeometric2F1[1/2, 1/2 - k, 3/2, Cos[i*Pi]^2]*Sqrt[Sin[i*Pi]^2]))
The above is nonsense because i, being the index of the product and hence
merely a "dummy variable", appears in the result! What's going on?
But anyway, Mathematica can successfully calculate the result if specific,
fairly small, positive integers k are given. For example,
In[7]:= Table[
2^(k - 1)*Integrate[Product[1 - Cos[2*i*Pi*x], {i, 1, k}], {x, 0, 1}],
{k, 1, 7}]
Out[7]= {1, 2, 3, 5, 6, 10, 12}
BTW, if anyone can find a closed form for the terms of the sequence a_k, I
and Tom Ward (the OP in sci.math.research) would be interested.
David W. Cantrell