Fourier Series for ex and Parseval Identity | Simple Problem Solution

Also, the series you tried to plot is not the Fourier series of ex.In summary, the process for finding the Fourier series of ex is to first determine the coefficients using the formula c_n = \frac 1 {2p}\int_{-p}^p f(x) e^{\frac {i n \pi}{p}}dx and then plot the series \sum_{k=-\infty}^{\infty}c_{n}e^{i n x} using the given coefficients. Additionally, the Parseval identity can be used to verify the correctness of the series by comparing the integral of the squared function to the sum of the squared coefficients.
  • #1
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Homework Statement


Find the Fourier series for ex for x in (-1,1).
Find the Parseval identity.

Homework Equations



The Attempt at a Solution


[tex]c_{n}=\frac{1}{2}\int_{-1}^{1}e^{x}e^{-i n x}dx[/tex]
Where cn are the coefficients of the Fourier series.

I tried plotting
[tex]\sum_{k=-\infty}^{\infty}c_{n}e^{i n x}[/tex]
together with ex and it doesn't seem to be correct...

Thank you for your help!
 
Last edited:
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  • #2
For starters, your formula for cn is wrong. For period P = 2p:

[tex]c_n = \frac 1 {2p}\int_{-p}^p f(x) e^{\frac {i n \pi}{p}}dx[/tex]

In your problem p = 1.
 

1. What is a Simple Fourier series problem?

A Simple Fourier series problem involves finding the Fourier series representation of a periodic function with a finite number of terms. This series is a sum of sine and cosine functions that can approximate the original function.

2. How do you solve a Simple Fourier series problem?

To solve a Simple Fourier series problem, you need to first determine the period of the function and then calculate the coefficients using the given formulas. These coefficients can be used to construct the Fourier series representation of the function.

3. What is the purpose of using a Fourier series in this problem?

The purpose of using a Fourier series in this problem is to approximate a periodic function with a simpler representation. This makes it easier to analyze and manipulate the function, and can also provide insight into the behavior of the original function.

4. Can a Simple Fourier series problem have an infinite number of terms?

Yes, a Simple Fourier series problem can have an infinite number of terms. However, in practice, only a finite number of terms are used to approximate the function, as using an infinite number of terms would result in the exact representation of the function.

5. What are some applications of Simple Fourier series problems?

Simple Fourier series problems have many applications in fields such as signal processing, image compression, and data analysis. They are also used to solve differential equations and model physical phenomena in engineering and physics.

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