Register to reply

Representations of periodic functions

by Jhenrique
Tags: functions, periodic, representations
Share this thread:
Jhenrique
#1
Jan27-14, 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?
Phys.Org News Partner Mathematics news on Phys.org
'Moral victories' might spare you from losing again
Fair cake cutting gets its own algorithm
Effort to model Facebook yields key to famous math problem (and a prize)
pwsnafu
#2
Jan27-14, 09:42 PM
Sci Advisor
P: 820
Quote Quote by Jhenrique View Post
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
#3
Jan27-14, 11:07 PM
Homework
Sci Advisor
HW Helper
Thanks ∞
Simon Bridge's Avatar
P: 12,423
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:
http://www.physicsforums.com/showthread.php?t=432185 #6.
... to understand how a function can be periodic with no fundamental period.

Jhenrique
#4
Jan28-14, 09:55 AM
P: 686
Representations of periodic functions

Quote Quote by Simon Bridge View Post
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)?
Attached Thumbnails
dsa.jpg  
Simon Bridge
#5
Jan28-14, 07:29 PM
Homework
Sci Advisor
HW Helper
Thanks ∞
Simon Bridge's Avatar
P: 12,423
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.


Register to reply

Related Discussions
Is this result trivial? (improper integral representations of real functions)) Calculus 8
Periodic Functions Calculus & Beyond Homework 6
F(x) = x as a sum of periodic functions? General Math 10
Proving sums of periodic functions need not be periodic(almost periodic) Calculus & Beyond Homework 20
Plotting Phasor representations of functions... Introductory Physics Homework 1