[integral] How to integrate a fourier series ? - Printable Version +- Tetration Forum ( https://math.eretrandre.org/tetrationforum)+-- Forum: Tetration and Related Topics ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1)+--- Forum: Mathematical and General Discussion ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3)+--- Thread: [integral] How to integrate a fourier series ? ( /showthread.php?tid=859) |

[integral] How to integrate a fourier series ? - tommy1729 - 05/04/2014
Let f(z) be a real fourier series with period 1. Let A be a real number. How to find the integral from 0 to sin^2(A) of the function f(z) in closed form ? A closed form here allows an infinite sum or product. (or even an infinite power tower if you wish) Term by term integration of a fourier series fails. And the coefficients provide the values of certain integrals but only taken over its period. Numerical methods and riemann sums can fail ! So, I do not know how to proceed in the general case. --------- Related : when is this integral ... 1) C^2 2) C^oo 3) "tommy-integrable" (if that exists) see thread : ** http://math.eretrandre.org/tetrationforum/showthread.php?tid=861** 4) analytic ( all with respect to the real A ) --------- Also related : f(z) repeats by the rule f(z+1) = f(z). Now assume a " period shift " ; g(z) = f(z) for 0 < z < 0.5 but g(z+0.5) = g(z). Now what is the four. series of g(z) ? Sure I know the formula for the coefficients, but that includes integrals such as above ... Hence why this is related. Lets call going from f to g " period shift -0.5 ". I was fascinated by the idea to " extend " : doing a period shift +0.5. Afterall if we have a method to do period shift -0.5 , then by inverting that we should be able to do other period shifts. (probably +0.5 or +2) ---- Note : I consider also using an averaged continuum sum such as : CS ( f(A) dA ) going from A = 0 to A = +oo and divided by its lenght (A). However just as term by term integration can fail , this probably holds for continuum sums too. Hence probably a failure in most cases, and not a general solution. ---- Your thoughts are appreciated. regards tommy1729 RE: [integral] How to integrate a fourier series ? - tommy1729 - 05/04/2014
This question seems somewhat deja-vu ... Did anyone else here post similar ideas ? regards tommy1729 |