Half vs Full Range Fourier Series: Odd & Even Functions

[Dyls]

I'm a little confused about the difference between the half range Fourier series and the full range Fourier series. What is the difference between the two in an odd function like f(x)=x and an even function like f(x)=x^2 ? Maybe an example to clear things up. Thank you.

 

[matt grime]

Given a function from [0,1] to R where f(0)=0, there are several ways to make from it a periodic function on some interval. The simplest is to repeat the function by setting f(x+1)=f(x). However, there is another option, and one that cuts your work in half.

We extend the function to [-1.1] by

1) refelcting in the y axis, ie set f(-x) = f(x)

2) rotate 180 degrees about the origin by setting f(-x)=-f(x)

then repeat these to get a periodic function.

The first is even on the interval [-1,1] so it only has cosines in its Fourier series, the second is odd so only has sines.

So, take f(x)=x on [0,1] if we extend it to an even funciton on [-1,1] then we get |x|, if we extend to an odd function we just get x. The first has a Fourier series only using cosines, the second only using sines.

 

[M98Ranger]

The main difference between the half range Fourier series and the full range Fourier series lies in the range of the function being considered. In the half range Fourier series, the function is only considered over half of its domain, while in the full range Fourier series, the function is considered over its entire domain.

To understand this better, let's take the example of an odd function like f(x)=x. In this case, the function is symmetric about the origin, which means that f(x)=-f(-x). This property makes the function odd. When we apply the half range Fourier series to this function, we only consider the values of the function for x>0, as the values for x<0 can be obtained by reflecting the function about the y-axis. This is why it is called the half range Fourier series.

On the other hand, in the full range Fourier series, we consider the values of the function for the entire domain, i.e. both positive and negative values of x. In the case of an odd function, the full range Fourier series will have a cosine term with a coefficient of 0, as the function is symmetric about the origin and does not have any even components. This is because the cosine function is an even function, and when multiplied with an odd function, the resulting product is also odd, which is not present in an odd function.

Now, let's take the example of an even function like f(x)=x^2. In this case, the function is symmetric about the y-axis, which means that f(x)=f(-x). This property makes the function even. When we apply the half range Fourier series to this function, we only consider the values of the function for x≥0, as the values for x<0 can be obtained by reflecting the function about the y-axis. This is why it is called the half range Fourier series.

In the full range Fourier series, we consider the values of the function for the entire domain, i.e. both positive and negative values of x. In the case of an even function, the full range Fourier series will have a sine term with a coefficient of 0, as the function is symmetric about the y-axis and does not have any odd components. This is because the sine function is an odd function, and when multiplied with an even function, the resulting product is also odd, which is not present in an even function.

In summary, the main difference between the half