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# A Fourier-type integral

a guest Oct 12th, 2011 25 Never
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1. (*  http://stackoverflow.com/q/7743774/421225  *)
2.
3. In[1]:= \$Version
4.
5. Out[1]= "8.0 for Linux x86 (64-bit) (February 23, 2011)"
6.
7. In[2]:= res = Integrate[
8.   Cos[(Pi x)/2]^2 Cos[((2 n + 1) Pi x)/2] Cos[((2 m + 1) Pi x)/2], {x, -1, 1},
9.    Assumptions -> Element[{n, m}, Integers]]
10.
11. Out[2]= ((-(1 + 2*n))*(3*m*(1 + m) + n + n^2)*Cos[n*Pi]*
12.     Sin[m*Pi] + (1 + 2*m)*(m + m^2 + 3*n*(1 + n))*Cos[m*Pi]*Sin[n*Pi])/
13.    (2*(-1 + m - n)*(m - n)*(1 + m - n)*(m + n)*(1 + m + n)*(2 + m + n)*Pi)
14.
15. In[3]:= nonzero = Flatten@Solve[Denominator[res] == 0, m]
16.
17. Out[3]= {m -> -2 - n, m -> -1 - n, m -> -1 + n, m -> -n, m -> n, m -> 1 + n}
18.
19. In[4]:= Table[Simplify[Limit[res, lim], Element[{m, n}, Integers]], {lim, nonzero}]
20.
21. Out[4]= {1/4, 1/2, 1/4, 1/4, 1/2, 1/4}
22.
23. The documentation claims that
24. "The result of Reduce[expr,vars] always describes exactly the same mathematical set as expr."
25. However:
26.
27. In[5]:= Reduce[res == 0, {m, n}, Integers]
28.
29. Out[5]= Element[m | n, Integers]
30.
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