- #1

- 868

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[tex]

\sum_{n=1}^{\inf}a_{n}cos(xn) + b_{n}sin(xn)

[/tex]

what is the FS of

[tex]

g(x) = (f(x) + f(-x))/2

[/tex]

I think that the answer is

[tex]

\sum_{n=1}^{\inf}a_{n}cos(xn)

[/tex]

am i right?

thanks.

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- Thread starter daniel_i_l
- Start date

In summary, when given a function f and a Fourier series, the resulting Fourier series for g(x) = (f(x) + f(-x))/2 will only contain cosines. This is because the sine expansion from the Fourier series for an even function disappears, as sine is an odd function. Similarly, if h(x) = (f(x) - f(-x))/2, the Fourier series for h will only contain sines, as cosine is an even function. This can be formally demonstrated by noting that g(x)sin(nx) and h(x)cos(nx) are both odd functions, with integrals from -a to a equaling 0.

- #1

- 868

- 0

[tex]

\sum_{n=1}^{\inf}a_{n}cos(xn) + b_{n}sin(xn)

[/tex]

what is the FS of

[tex]

g(x) = (f(x) + f(-x))/2

[/tex]

I think that the answer is

[tex]

\sum_{n=1}^{\inf}a_{n}cos(xn)

[/tex]

am i right?

thanks.

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- #2

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So,yes, the Fourier series will contain only cosines.

Daniel.

- #3

Science Advisor

Homework Helper

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If h(x)= (f(x)- f(-x))/2, the "odd part" of f, then the Fourier series for h would be

[tex] \sum_{n=1}^{\inf}a_{n}sin(xn) [/tex]

Since sine is an odd function and cosine is an even function they divide f into even and odd parts.

If you want a more "formal" demonstration of that just note that g(x)sin(nx) and h(x)cos(nx) are odd functions themselves. Their integrals from -a to a must be 0.

The Fourier Series is a mathematical representation of a periodic function in terms of sines and cosines. It allows us to break down a complex function into simpler components, making it easier to analyze and manipulate.

To find the FS of g(x), you will need to calculate the coefficients of the sines and cosines using the Fourier series formula. This involves integrating the function over one period and solving for the coefficients. Once you have the coefficients, you can write out the FS as an infinite sum of sines and cosines.

The FS of g(x) is important because it allows us to approximate a complex function with a simpler one. This can be useful in solving differential equations, analyzing signals, and understanding the behavior of periodic systems.

No, the FS of g(x) is an approximation of the original function. The accuracy depends on the number of terms included in the series. The more terms included, the closer the approximation will be to the original function.

The FS of g(x) can only be used to represent periodic functions. If a function is not periodic, then the FS cannot be applied. In this case, other mathematical methods may be used to analyze and manipulate the function.

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