Finding Fourier Series for (-π, π): Sketch Sum of Periods

In summary, the problem at hand is to find the Fourier series for a given function in the interval (-π, π) and to sketch its sum over several periods. The function is defined as f(x) = 0 for -π < x < 1/2π and f(x) = 1 for 1/2π < x < π. The equations needed are the sum of the coefficients multiplied by the cosine and sine functions, as well as the coefficients themselves for a0, an, and bn. After working out the a0, the integrals for an and bn can be limited to {π/2,π}. It may be helpful to separate the function into symmetric and anti-symmetric parts and use a
  • #1
struggles
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0

Homework Statement


Find the Fourier series defined in the interval (-π,π) and sketch its sum over several periods.
i) f(x) = 0 (-π < x < 1/2π) f(x) = 1 (1/2π < x < π)

2. Homework Equations

ao/2 + ∑(ancos(nx) + bnsin(nx))
a0= 1/π∫f(x)dx
an = 1/π ∫f(x)cos(nx) dx
bn = 1/π ∫f(x) sin(nx)

The Attempt at a Solution


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I've worked out the a0 by splitting the limits and integrating individually for -π < x < 1/2π and 1/2π < x < π. When I did this i got a0 = ½

for an = 1/π ∫(upper limit π, lower limit 1/2π) cos(nx) dx
= 1/π[1/n sin(nx)]
= 1/π ((1/n. sin(πn) - 1/n.sin(πn/2))
= 1/nπ(0- sin(πn/2)

Here is where i get stuck as sin(πn/2) is 0 for even values of n and alternates between 1, -1 for odd values.
Can i leave this written in sin form of the Fourier series as every other example I've changed the value of sin/cos to either 0 or (-1)n.

Thanks for any help!
 
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  • #2
Hi struggles:

You have already noticed that the integrals for the coefficients can be limited to {π/2,π}. That's a good start.

I think it might be helpful to next separate the function f(x) into two parts:
f(x) = S(x) + A(x)
where S(x) is symmetric, S(x) = S(-x)
and
A(x) is anti-symmertric, A(x) = -A(-x).​
Then think about the coefficients for A(x) and S(x) separately.

You may also want to think about the general form of the integrals and try a substitution
y = nx, dx = dy/n,​
including the lower and upper bounds of the integral.

Hope this helps.

Regards,
Buzz
 

Related to Finding Fourier Series for (-π, π): Sketch Sum of Periods

1. What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function using an infinite sum of trigonometric functions. It can be used to approximate any periodic function.

2. What does it mean to find a Fourier series for a function?

Finding a Fourier series for a function means expressing the function as an infinite sum of sine and cosine functions. This allows us to analyze and approximate the behavior of the function over a specific interval.

3. What is the significance of finding a Fourier series for (-π, π)?

The interval (-π, π) is known as the fundamental period for many trigonometric functions. Finding a Fourier series for this interval allows us to represent any periodic function within this interval with a high degree of accuracy using a finite number of terms.

4. How do you sketch the sum of periods for a Fourier series?

To sketch the sum of periods for a Fourier series, you would graph the individual terms of the series and then add them together. The resulting graph would show the periodic behavior of the function over the given interval.

5. Can a Fourier series accurately represent any function within the interval (-π, π)?

No, a Fourier series can only accurately represent functions that are periodic within the given interval. If a function is not periodic or has discontinuities within the interval, a Fourier series would not be an accurate representation.

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