Re: Simple Trigonometric Integrals

*To*: mathgroup at smc.vnet.net*Subject*: [mg32347] Re: Simple Trigonometric Integrals*From*: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>*Date*: Wed, 16 Jan 2002 03:29:56 -0500 (EST)*Organization*: Universitaet Leipzig*References*: <a20n0a$4i4$1@smc.vnet.net>*Reply-to*: kuska at informatik.uni-leipzig.de*Sender*: owner-wri-mathgroup at wolfram.com

Hi, Mathematica does not split integrals of sums to sums of integrals. I dont use the linearity of the integration because some integrals over sums may be convergent, while the sum of the individual integrals is divergent. But *if* you know you have a flat sum of terms that should be easy to integrate you can do Integrate[#,{t,0,2Pi}] & /@ yourSumOfSimpleTerms In your example 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[#, {t, 0, 2 Pi}] & /@ Joe works fine and need 0.36 seconds. Regards Jens Joe Helfand wrote: > > Hey, > > 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: > > In[687]:= > 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); > > In[688]:= > 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. > > Joe