# Finding the FS of g(x): Is it Right?

• daniel_i_l
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.

#### daniel_i_l

Gold Member
Lets say that you have a function f and a FS:
$$\sum_{n=1}^{\inf}a_{n}cos(xn) + b_{n}sin(xn)$$
what is the FS of
$$g(x) = (f(x) + f(-x))/2$$
I think that the answer is
$$\sum_{n=1}^{\inf}a_{n}cos(xn)$$
am i right?
thanks.

Of course, the sine expansion from the Fourier series for an even function goes away, since sine is an odd function.

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

Daniel.

Yes, g(x) is the "even part" of f.

If h(x)= (f(x)- f(-x))/2, the "odd part" of f, then the Fourier series for h would be
$$\sum_{n=1}^{\inf}a_{n}sin(xn)$$

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.

## 1. What is the Fourier Series (FS) of g(x)?

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.

## 2. How do I find the FS of g(x)?

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.

## 3. Why is it important to find the FS of g(x)?

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.

## 4. Is the FS of g(x) always accurate?

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.

## 5. Can I use the FS of g(x) to solve any type of 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.