Representations of periodic functions

[Jhenrique]

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?

 

[pwsnafu]

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

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.

 

[Simon Bridge]

I think that exist a triangular relation. Correct?

Don't know what that means.

If yes, could give me the general formulas for convert an form in other?

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.

 

[Jhenrique]

Don't know what that means.

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)?

 

[Simon Bridge]

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.