Explore Fourier Series & Jacobi Elliptic Functions: Hints for Sine Expansion

In summary, the conversation discusses finding the first few terms of a sine expansion of a function involving the jacobic elliptic cnoidal function and real parameters. It is suggested to start with a simpler problem involving only the cosine function, but the issue of the sinusoidal term in the denominator is raised. Eventually, it is concluded that the simpler problem can be solved, but the elliptic version is more challenging.
  • #1
ss.undefined
2
0
Hi all. I am currently trying to find the first few terms of a sine expansion of

q(t)= (A+B*cn(t,m))/(C+D*cn(t,m))

where m is the modulus and cn is the jacobic elliptic cnoidal function and A,B,C,D real and C>D implying no poles. I realize that I should start with a simpler problem. Do some of you have any hints for the sinusoidal version:

q(t) = (A+B cos(t))/(C+D cos(t))

How do I handle this sinusoid term in the denominator?
 
Physics news on Phys.org
  • #2
So, I think I prematurely jumped the gun in asking this question. I can do the sinusoidal version. And for the elliptic version, ugh. Yeah. Exactly.
 

What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a sum of sine and cosine waves with different amplitudes and frequencies. It is named after French mathematician Joseph Fourier, who first introduced the concept in the early 19th century.

Why are Fourier series important?

Fourier series have many applications in physics, engineering, and mathematics. They allow us to analyze and approximate periodic functions, which are common in real-world phenomena. They also provide a useful tool for solving differential equations and understanding the behavior of waves and vibrations.

What is the difference between a Fourier series and a Fourier transform?

A Fourier series represents a periodic function as a sum of sine and cosine waves with different amplitudes and frequencies, while a Fourier transform represents a non-periodic function as a sum of complex exponentials with different frequencies. In other words, a Fourier transform is a generalization of a Fourier series for non-periodic functions.

How do you calculate a Fourier series?

To calculate a Fourier series, you need to find the coefficients of the sine and cosine terms by evaluating integrals. The specific method for calculating these coefficients depends on the function being represented and the interval over which it is defined. Once the coefficients are determined, the Fourier series can be constructed by summing the individual sine and cosine terms.

What are some real-world applications of Fourier series?

Fourier series have numerous applications in various fields, including signal processing, image and sound compression, heat transfer analysis, and solving partial differential equations. They are also used in areas such as music theory, astronomy, and financial analysis.

Similar threads

I Orthogonal Basis of Periodic Functions: Beyond Sines and Cosines
    • Calculus
4
Replies
139
Views
4K
Compact summary of Fourier series equations
    • Calculus
Replies
11
Views
854
Fourier series of this scale and shift of triangle wave
    • Calculus and Beyond Homework Help
Replies
1
Views
203
A few questions to make sure I understand Fourier series
    • Calculus and Beyond Homework Help
Replies
3
Views
273
I Inverse Laplace to Fourier series
    • Calculus
Replies
1
Views
1K
Complex Fourier Series for cos(t/2)
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
I Fourier Series: I don't understand where I am wrong --
    • Calculus
Replies
1
Views
1K
Fourier series for trigonometric absolute value function
    • Calculus and Beyond Homework Help
Replies
8
Views
2K
How to Expand a Function in a Half-Range Fourier Series
    • Calculus
Replies
8
Views
3K
Fourier Series Expansion
    • Calculus and Beyond Homework Help
Replies
9
Views
1K

Hot Threads

Recent Insights

Back
Top