• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 [integral] How to integrate a fourier series ? tommy1729 Ultimate Fellow Posts: 1,620 Threads: 364 Joined: Feb 2009 05/04/2014, 02:11 PM (This post was last modified: 05/04/2014, 03:17 PM by tommy1729.) 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/tetrationforu...hp?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 tommy1729 Ultimate Fellow Posts: 1,620 Threads: 364 Joined: Feb 2009 05/04/2014, 03:19 PM This question seems somewhat deja-vu ... Did anyone else here post similar ideas ? regards tommy1729 « Next Oldest | Next Newest »

 Possibly Related Threads… Thread Author Replies Views Last Post Tetration Asymptotic Series Catullus 14 441 Yesterday, 10:18 AM Last Post: Gottfried Formula for the Taylor Series for Tetration Catullus 8 417 06/12/2022, 07:32 AM Last Post: JmsNxn Calculating the residues of $$\beta$$; Laurent series; and Mittag-Leffler JmsNxn 0 574 10/29/2021, 11:44 PM Last Post: JmsNxn Trying to find a fast converging series of normalization constants; plus a recap JmsNxn 0 535 10/26/2021, 02:12 AM Last Post: JmsNxn Reducing beta tetration to an asymptotic series, and a pull back JmsNxn 2 1,324 07/22/2021, 03:37 AM Last Post: JmsNxn Perhaps a new series for log^0.5(x) Gottfried 3 5,244 03/21/2020, 08:28 AM Last Post: Daniel Where is the proof of a generalized integral for integer heights? Chenjesu 2 4,832 03/03/2019, 08:55 AM Last Post: Chenjesu Taylor series of i[x] Xorter 12 25,125 02/20/2018, 09:55 PM Last Post: Xorter An explicit series for the tetration of a complex height Vladimir Reshetnikov 13 26,613 01/14/2017, 09:09 PM Last Post: Vladimir Reshetnikov Complaining about MSE ; attitude against tetration and iteration series ! tommy1729 0 3,671 12/26/2016, 03:01 AM Last Post: tommy1729

Users browsing this thread: 1 Guest(s)