Sine series for cos(x) (FOURIER SERIES)

In summary, the conversation discusses the process of finding the sine series for cos(x) on the interval [0,2pi]. The question of the interval of validity is raised, specifically whether it is limited to [0,pi] or extends to all of R. It is clarified that the infinite sum of sines converges to cos(x) everywhere if it converges on an interval with a length of at least one period. Additionally, the formula used represents the 4pi periodic odd extension of cos(x).
  • #1
konradz
1
0
I was finally able to figure out how to find the sine series for cos(x), but only for [0,2pi]. A question i have though is what is the interval of validity? is it only [0,pi]?
Ie if I actually had to sketch the graph of the sum of the series, on all of R, would I have cosine or just a periodic extension of cosine from [0,2pi]?
 
Physics news on Phys.org
  • #2
Hey Konrad, welcome to PF.
I am afraid I don't entirely understand your question. You say that you have managed to write cos(x) as a(n infinite) sum of sines on the interval [0, 2pi].
But both cos(x) and the sines you used are periodic with period 2pi, aren't they? So if the infinite sum converges to cos(x) on an interval with a length of at least one period, then it converges to cos(x) everywhere, doesn't it?
 
  • #3
If you have expanded cos(x) in a sine series using [itex]p = 2\pi[/itex] in the formula
[tex] b_n = \frac 2 p \int_0^p \cos(x) \sin{\frac{n\pi x}{p}}\,dx[/tex]
what you are representing is the [itex]4\pi[/itex] periodic odd extension of cos(x).

[edit - corrected typo: bn not an]
 
Last edited:

FAQ: Sine series for cos(x) (FOURIER SERIES)

1. What is a Sine series for cos(x) (Fourier series)?

Sine series for cos(x) (Fourier series) is a mathematical representation of a periodic function, such as cos(x), as an infinite sum of sine functions with different frequencies and amplitudes. It is named after the French mathematician Joseph Fourier who first introduced the concept in the early 19th century.

2. How is a Sine series for cos(x) (Fourier series) calculated?

The coefficients of the Sine series for cos(x) (Fourier series) can be calculated using the Fourier series formula, which involves integrating the function over one period and solving for the coefficients. Alternatively, the coefficients can also be calculated using trigonometric identities and the orthogonality of sine functions.

3. What is the significance of using a Sine series for cos(x) (Fourier series)?

The Sine series for cos(x) (Fourier series) allows us to approximate any periodic function, including non-sinusoidal functions, as an infinite sum of simple sine functions. This can be useful in many applications, such as signal processing, data compression, and solving differential equations.

4. Are there any limitations to using a Sine series for cos(x) (Fourier series)?

One limitation of using a Sine series for cos(x) (Fourier series) is that it only accurately represents periodic functions. Non-periodic functions can also be approximated using a Fourier series, but the accuracy decreases as the function becomes more complex. Additionally, the convergence of the series may be slow or not exist for some functions.

5. How is a Sine series for cos(x) (Fourier series) different from a Taylor series?

A Taylor series represents a function as an infinite sum of monomial terms (powers of x). In contrast, a Sine series for cos(x) (Fourier series) represents a function as an infinite sum of sine functions. A Taylor series is valid for all values of x, while a Fourier series is only valid for periodic functions. Additionally, a Fourier series can provide better approximations for some non-periodic functions compared to a Taylor series.

Back
Top