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Representations of periodic functions 
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#1
Jan2714, 09:02 PM

P: 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: AČ = aČ + bČ 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
Jan2714, 09:42 PM

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P: 833

g(x) = 2 if x is an integer, g(x) = 1 if x is noninteger rational, g(x) = 0 if x is irrational. This is a periodic function with fundamental period equal to 1. 


#3
Jan2714, 11:07 PM

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P: 12,870

Also see: http://www.physicsforums.com/showthread.php?t=432185 #6. ... to understand how a function can be periodic with no fundamental period. 


#4
Jan2814, 09:55 AM

P: 686

Representations of periodic functions



#5
Jan2814, 07:29 PM

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P: 12,870

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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