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[integral] How to integrate a fourier series ?
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 : ****
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.


This question seems somewhat deja-vu ...
Did anyone else here post similar ideas ?



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