Tinhoi
Lv 4
Tinhoi asked in 科學及數學數學 · 1 decade ago

Fourier square curve

I have seen a function

f(x)=lim(n->inf) Σ(r=0 to n)sinnx/n

f '(x)=lim(n-> inf) Σ(r=0 to n)cosnx

f(mpi/2)= 0+....+0=0 , but

It is very strange to see f(x) remains constant when x \\m(pi)/2 ( m is integer),

I cant put in any values , nor prove f is a constant function ( ie f'(x)=0)

Moreover, to be the worse, I cant devive the expression of f'(x) converges for x is not mpi/2 , can you explain plz .

1 Answer

Rating
  • ?
    Lv 5
    1 decade ago
    Favorite Answer

    The problem is that the formula

    f '(x)=lim(n-> inf) Σ(r=0 to n)cosnx

    is WRONG!

    What you're doing here is differentiate a series of function term by term, but this is not valid for every series of functions. In fact, for term by term differentiation to be valid, there are strict conditions to be satisfied, and the series f(x) you gave here does not qualify for such an operation.

    One theorem for term by term differentiation is as follows (you can see how difficult it is!)

    Assume that each f_n is a real-valued function defined on (a,b) such that the derivative

    d(f_n (x))/dx exists for each x in (a,b). Assume that, for at least one point c in (a,b), the series Σf_n(c) converges. Assume further that there exists a function g such that

    Σd(f_n (x))/dx = g(x) uniformly on (a,b). Then the series can be differentiated term by term.

    This theorem can be found in theorem 9.14 in the book "Mathematical Analysis" by Apostol.

    You can see that the final assumption of the theorem is not satisfied for your series.

    2008-02-03 19:45:20 補充:

    You can see more details for the series in the web sitehttp://mathworld.wolfram.com/FourierSeries.htmlTha... for 貓朋 for providing the information.

    Source(s): Tom. M. Apostol, Mathematical Analysis.
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