Fourier series problem solving for an,bn

In summary, the conversation discusses finding the values of a0, an, and bn for a given function f(x) with a period of 4. The equations for these values are given and the individual attempts at solving for them are discussed. The summary concludes by noting that there were two mistakes made in the calculations, with the correct formula for F(0) being 4cos(0)/n2pi2 instead of -1.
  • #1
dp182
22
0

Homework Statement


let f(x)={0;-2[itex]\leq[/itex]x[itex]\leq[/itex]0.
x;0[itex]\leq[/itex]x[itex]\leq[/itex]2
find a0
an
bn
given the period is 4

Homework Equations


a0=1/L[itex]\int[/itex]f(x)dx
an=1/L[itex]\int[/itex]f(x)cos(n[itex]\pi[/itex]x/L)
bn=1/L[itex]\int[/itex]f(x)sin(n[itex]\pi[/itex]x/L)

The Attempt at a Solution


so I can get a0 = 1 but I run into trouble with an. so I plug
an=1/2[itex]\int[/itex]xcos(n[itex]\pi[/itex]x/L) for the interval 0[itex]\leq[/itex]x[itex]\leq[/itex]2 and I get the solution
=(1/n2[itex]\pi[/itex]2)2xn[itex]\pi[/itex]sin(n[itex]\pi[/itex]x/2)+4cos(n[itex]\pi[/itex]x/2) then subbing in for x I get (1/n2[itex]\pi[/itex]2)(4n[itex]\pi[/itex]sin(n[itex]\pi[/itex])+4cos(n[itex]\pi[/itex])-1) can anyone tell me what I am doing wrong here?
 
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  • #2
Hi dp182!

dp182 said:

Homework Statement


let f(x)={0;-2[itex]\leq[/itex]x[itex]\leq[/itex]0.
x;0[itex]\leq[/itex]x[itex]\leq[/itex]2
find a0
an
bn
given the period is 4

Homework Equations


a0=1/L[itex]\int[/itex]f(x)dx
an=1/L[itex]\int[/itex]f(x)cos(n[itex]\pi[/itex]x/L)
bn=1/L[itex]\int[/itex]f(x)sin(n[itex]\pi[/itex]x/L)

The Attempt at a Solution


so I can get a0 = 1 but I run into trouble with an. so I plug
an=1/2[itex]\int[/itex]xcos(n[itex]\pi[/itex]x/L) for the interval 0[itex]\leq[/itex]x[itex]\leq[/itex]2 and I get the solution
=(1/n2[itex]\pi[/itex]2)2xn[itex]\pi[/itex]sin(n[itex]\pi[/itex]x/2)+4cos(n[itex]\pi[/itex]x/2) then subbing in for x I get (1/n2[itex]\pi[/itex]2)(4n[itex]\pi[/itex]sin(n[itex]\pi[/itex])+4cos(n[itex]\pi[/itex])-1) can anyone tell me what I am doing wrong here?

You made two mistakes: you forgot the term 1/L, so you still need to divide by 2. Secondly, when you "subbed for x", you made a mistake in "subbing for 0". That is, the correct formula is

[tex]\int_0^2{f(x)dx}=F(2)-F(0)[/tex]

But somehow, you calculated F(0) wrong.
 
  • #3
but wouldn't F(0) be 4cos(0)/n2pi2 which is just 4/n2pi2
 
  • #4
dp182 said:
but wouldn't F(0) be 4cos(0)/n2pi2 which is just 4/n2pi2

Indeed, but you wrote -1 instead of -4 in the end.
 
  • #5
ya but I brought out a 4/n2pi2 so its not -1 its -4/n2pi2
 

Related to Fourier series problem solving for an,bn

1. What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a sum of sine and cosine functions. This series is used to analyze and solve problems in various fields such as engineering, physics, and mathematics.

2. What do an,bn represent in a Fourier series?

The coefficients an and bn represent the amplitudes of the sine and cosine functions, respectively, in the Fourier series. These coefficients are used to determine the shape and frequency of the periodic function being analyzed.

3. How do you solve a Fourier series problem?

To solve a Fourier series problem, you need to first identify the function you want to represent as a Fourier series. Then, you can use various mathematical techniques such as integration, differentiation, and trigonometric identities to find the values of an and bn coefficients. Finally, you can plug these values into the Fourier series formula to obtain the complete series.

4. What are some applications of Fourier series?

Fourier series has various applications in fields such as signal processing, image reconstruction, and heat transfer. It is also used in solving differential equations, analyzing musical tones, and studying the behavior of waves.

5. Are there any limitations of using Fourier series?

One limitation of using Fourier series is that it can only represent periodic functions, which means it cannot be used for non-periodic functions. Additionally, it may not accurately represent functions with sharp corners or discontinuities. In such cases, other mathematical techniques may be more suitable for solving the problem.

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