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
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
