Complex Fourier series has a singular term

In summary, the conversation is about finding the complex Fourier series for the function f(t)=t(1-t) on the interval 0<t<1. The necessary equations and attempts at a solution are discussed, including the issue of the singular term for n=0 and how it affects the graph. The solution is found by plugging in n=0 before integrating instead of after.
  • #1
Ragnord
4
0

Homework Statement



Find the complex Fourier series for f(t)=t(1-t), 0<t<1

Homework Equations



[tex]\sum_{n=-\infty}^{\infty}c_{n}e^{2in\pi t}[/tex]

where [tex]c_{n}=\int_{0}^{1}f(t)e^{-2in\pi t}dt[/tex]


The Attempt at a Solution



I've worked out that c[tex]_{n}=-1/(2n^2 \pi^2)[/tex]. The problem is that for n=0, it is singular. Is there some way around this or does it mean that the complex Fourier series doesn't exist?
I tried using maple to graph the series with the n=0 term omitted and it comes out to the right shape, but is shifted vertically down some, leading me to believe that the singular term should be replaced by a constant or something.
 
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  • #2
[tex] c_0=\int_0^1 f(t)\,dt [/tex]
 
  • #3
Well that was easy, just like a real Fourier series. Thanks.
I'm interested in knowing why that's the case though, I haven't seen anything about doing anything special for [tex] c_{0} [/tex] in anything I've seen about complex Fourier series.
 
  • #4
Just plug in n=0 before integrating instead of after!
 

What is the definition of a singular term in a Complex Fourier series?

A singular term in a Complex Fourier series is a term that has a singularity or a point of discontinuity at a specific point in the series. This means that the function represented by the series is not continuous at that point, and the value of the term becomes undefined.

Why are singular terms important in Complex Fourier series?

Singular terms play a crucial role in understanding the behavior of functions represented by Complex Fourier series. They can indicate points of discontinuity or abrupt changes in the function, which can greatly affect its overall shape and behavior.

How are singular terms identified in a Complex Fourier series?

Singular terms can be identified by examining the coefficients of the series. If a coefficient is infinite or undefined, then that term is considered singular. Another way to identify singular terms is by graphing the function and looking for points of discontinuity.

Can singular terms be included in the representation of a function using Complex Fourier series?

Yes, singular terms can be included in the representation of a function using Complex Fourier series. In fact, including these terms can lead to a more accurate representation of the function, especially if it has points of discontinuity or sharp changes in behavior.

How do singular terms affect the convergence of a Complex Fourier series?

Singular terms can greatly affect the convergence of a Complex Fourier series. In some cases, they can cause the series to have a slower rate of convergence or even not converge at all. This is because the behavior of the function at these points is not continuous, making it difficult for the series to accurately represent the function.

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