Fourier series problem

In summary: The factor of 2 comes from the fact that the odd extension of a function has a doubled period compared to the original function. In summary, the solution uses a sine Fourier expansion for ##\sin^2(x)## by taking its odd extension and using the half range formulas, which results in a factor of 2 in the integral.
  • #1
Nikitin
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Homework Statement


https://wiki.math.ntnu.no/_media/tma4120/2013h/tma4120_h11.pdf

Check out the solution to problem 4b)

My question is: Why do they set ##b_n = \frac{2}{\pi} \int_{0}^{\pi}(...)dx## instead of ##b_n = \frac{1}{\pi} \int_{0}^{\pi} (...)dx##?

Ie, why did they multiply the integral with 2? Did they find the Fourier integral to the odd expansion of ##sin(x)^2##? Because that's the only way for sin(x)^2 to be defined by solely a sine Fourier expansion?

EDIT: Ooops, posted in wrong section. Please move to calculus and beyond homework forum! sorry!
 
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  • #2
Nikitin said:

Homework Statement


https://wiki.math.ntnu.no/_media/tma4120/2013h/tma4120_h11.pdf

Check out the solution to problem 4b)

My question is: Why do they set ##b_n = \frac{2}{\pi} \int_{0}^{\pi}(...)dx## instead of ##b_n = \frac{1}{\pi} \int_{0}^{\pi} (...)dx##?

Ie, why did they multiply the integral with 2? Did they find the Fourier integral to the odd expansion of ##sin(x)^2##? Because that's the only way for sin(x)^2 to be defined by solely a sine Fourier expansion?

Yes, that's exactly it. Since they only have sine functions for the eigenfunction expansion, the FS must represent an odd function so they take the odd extension of ##\sin^2(x)## and use the half range formulas.
 
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1. What is a Fourier series problem?

A Fourier series problem is a mathematical problem that involves representing a periodic function as a sum of sines and cosines. It is named after French mathematician Joseph Fourier, who developed the concept in the early 19th century.

2. What is the purpose of solving a Fourier series problem?

The purpose of solving a Fourier series problem is to find a way to represent a periodic function in terms of simpler trigonometric functions. This can be useful in many applications, such as signal processing, image analysis, and solving differential equations.

3. How do you solve a Fourier series problem?

To solve a Fourier series problem, one must follow a series of steps including determining the period of the function, finding the coefficients of the sine and cosine terms, and then using these coefficients to write out the final Fourier series representation of the function.

4. What are some common applications of Fourier series?

Fourier series have many practical applications, including signal processing, image analysis, and solving differential equations. They are also used in fields such as physics, engineering, and mathematics to model and analyze periodic phenomena.

5. Are there any limitations to using Fourier series?

While Fourier series can be a powerful tool for representing and analyzing periodic functions, there are some limitations to their use. For example, they may not be suitable for representing non-periodic functions or functions with discontinuities. Additionally, the convergence of Fourier series may be slow for certain types of functions.

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