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Representations of periodic functions

  1. Jan 27, 2014 #1
    Correct me if I'm wrong, but exist 3 forms for represent periodic functions, by sin/cos, by exp and by abs/arg.

    I know that given an expression like a cos(θ) + b sin(θ), I can to corvert it in A cos(θ - φ) or A sin(θ + ψ) through of the formulas:

    A² = a² + b²

    tan(φ) = b/a
    sin(φ) = b/A
    cos(φ) = a/A

    tan(ψ) = a/b
    sin(ψ) = a/A
    cos(ψ) = b/A

    The serie fourier have other conversion, this time between exponential form and amplitude/phase
    [tex]f(t)=\gamma_0+2\sum_{n=1}^{\infty } \gamma_n cos\left ( \frac{2 \pi n t}{T}+\varphi_n \right )[/tex]
    ##\gamma_0 = c_0##
    ##\gamma_n = abs(c_n)##
    ##\varphi_n = arg(c_n)##

    I think that exist a triangular relation. Correct? If yes, could give me the general formulas for convert an form in other?
  2. jcsd
  3. Jan 27, 2014 #2


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    f(x) = 0 if x is rational, f(x) = 1 if x is irrational. This is a periodic function without a fundamental period.

    g(x) = 2 if x is an integer, g(x) = 1 if x is non-integer rational, g(x) = 0 if x is irrational. This is a periodic function with fundamental period equal to 1.
  4. Jan 27, 2014 #3

    Simon Bridge

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    Don't know what that means.

    The three forms you talk about are related via a phasor diagram and the euler relations.

    Also see:
    https://www.physicsforums.com/showthread.php?t=432185 #6.
    ... to understand how a function can be periodic with no fundamental period.
  5. Jan 28, 2014 #4
    See my book of math in annex... I have 3 distinct representations for fourier series. But I think that my relations in my book aren't very well connected. For example: given a expression like a cos(θ) + b sin(θ) how convert it in expression like c exp(θ ± φ)?

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  6. Jan 28, 2014 #5

    Simon Bridge

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    You can turn a trig expression to and from an exponential one using the Euler relations.
    $$\exp i\theta = \cos\theta + i\sin\theta = x+iy$$

    You can also get the relations between them by using one definition to expand the other one.
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