# Representations of periodic functions

1. ### Jhenrique

686
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
$$f(t)=\gamma_0+2\sum_{n=1}^{\infty } \gamma_n cos\left ( \frac{2 \pi n t}{T}+\varphi_n \right )$$
##\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. ### pwsnafu

938
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.

3. ### Simon Bridge

15,282
Don't know what that means.

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

Also see:
... to understand how a function can be periodic with no fundamental period.

4. ### Jhenrique

686
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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5. ### Simon Bridge

15,282
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.