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Simple Trigonometric Integrals

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  • Subject: [mg32338] Simple Trigonometric Integrals
  • From: Joe Helfand <jhelfand at>
  • Date: Tue, 15 Jan 2002 02:30:21 -0500 (EST)
  • Organization: University of Maryland College Park
  • Sender: owner-wri-mathgroup at


    I have a thing about Mathematica.  Sometimes I have a real long
expression that involves the integral of the sum of lots of cosines and
sines of some variable let's say 't'.  But having done some fancy maths
on my own to reduce it and get into a simple integral from 0 to 2 Pi,
and the sines and cosines all involve some integer multiple of t, the
integration takes for ever, it basically hangs.  Now, although the
expression is long, and there are a lot of terms in it, it still just
becomes a simple periodic integral from zero to 2 pi, and all the
trigonometric terms involving t should just drop out.  Kind of like what
sometimes can happen if you are playing around with a Fourier series
expansion (by the way, does Mathematica have a built in Fourier Series
expansion?  I mean something like Series[], but returns fourier
coefficients?).  Uptill now, I have been able to get by with something
like using

periodicIntegral={Cos[t] -> 0, Cos[2 t] -> 0, Cos[3 t] -> 0, Cos[4 t] ->
0, Cos[5t] -> 0, Cos[6 t] -> 0, Cos[7 t] -> 0, Sin[t] -> 0, Sin[2 t] ->
0, Sin[3 t] -> 0, Sin[4 t] -> 0, Sin[5t] -> 0, Sin[6 t] -> 0, Sin[7 t]
-> 0};

and then doing a replace on the expresion, multiplying the result by 2
Pi.  But now I am in a bind where no amount of TrigReduce, TrigExpand,
TrigFactor, etc. will get this big ass expression into the desired form
where the above is approriate (because there are other sines and cosines
of other variables that get put into the terms and stand by
themselves).  Still, the expression should be easy to do for the
computer, even I can go through and set these terms to zero, but it will
just take me a long time.  An example of what I am talking about, just
try the following:

Joe = a c Cos[t]/(g s) + b q Cos[2 t]/(c f) + c Cos[3 t]/(d a) +
      d f Cos[4 t]/(h a n) + e q Cos[5t]/(g a) + f l Cos[6 t]/(w r m) +
      g b Cos[7 t]/(o n x) + h Sin[t]/(b c) + i Sin[2 t]/(h e r) +
      j y Sin[3 t]/(l p) + d k Sin[4 t]/(j c) + l m a Sin[5 t]/(f s b h)
      m p Sin[6 t]/(k j) + q n Sin[7 t]/(x c);

Integrate[Joe, {t, 0, 2 Pi}]

and you willl see it takes a long time to integrate.  (It will
eventually get done.)  I know this is just zero, but why does it take so
long for the computer to figure out?  It is true that my expression is
even longer than this one, so essentially it hangs, but basically it is
the same problem.  I do not what to be hunting through my equation from
hell setting all the relevant trigonometric terms to zero when the
computer should be able to do this.  Well, sorry for the harangue but I
greatly appreciate you reading down so far, really.  If you have any
suggestions or comments, point out I am an idiot there is some simple
thing in Mathematica, please send it.


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