Jason Cawley
Wolfram Science Group
Phoenix, AZ USA
Registered: Aug 2003
Posts: 712 
First, this is not the right place for general Mathematica questions like this. You want the MathGroup list, which you can find here 
http://forums.wolfram.com/mathgroup/
To start with you can use Integrate, and do NMaximize on the resulting function of 9 variables  m, and your One, Two, etc. But I don't think it is particularly bounded, unless there are constraints on One, Two, etc that you haven't explained.
funct[m_, {One_, Two_, Three_, Four_, Five_, Six_, Seven_, Eight_}] := N[Sqrt[(Integrate[ Which[0 <= s < 3Pi/18, One, 3Pi/18 <= s < 7Pi/18, Two, 7Pi/18 <= s < 10Pi/18, Three, 10Pi/18 <= s < 18Pi/18, Four, 18Pi/18 <= s < 21Pi/18, Five, 21Pi/18 <= s < 25Pi/18, Six, 25Pi/18 <= s < 28Pi/18, Seven, 28Pi/18 <= s < 36Pi/18, Eight]* Sin[m*s], {s, 0, 2Pi}]^2 + Integrate[ Which[0 <= s < 3Pi/18, One, 3Pi/18 <= s < 7Pi/18, Two, 7Pi/18 <= s < 10Pi/18, Three, 10Pi/18 <= s < 18Pi/18, Four, 18Pi/18 <= s < 21Pi/18, Five, 21Pi/18 <= s < 25Pi/18, Six, 25Pi/18 <= s < 28Pi/18, Seven, 28Pi/18 <= s < 36Pi/18, Eight]* Cos[m*s], {s, 0, 2Pi}]^2)]]
That is your function, and it gives numerical values for any arguments just fine. When you set m = 4 you still have eight more degrees of freedom though, and unless you have constraints on them...
See if that helps. If you need more, please ask at the link above rather than here  this is a forum for New Kind of Science discussions, rather than general Mathematica howto's.
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