Convergence of Fourier Series for f(t) = 1 + t with Only Cosine Terms in [0,pi]

In summary, the Fourier series for f(t) = 1 + t with only cosine terms in the interval [0,pi] converges to 1+pi/2 at t=0 and t=pi, and converges to f for all other values of t in the interval.
  • #1
malawi_glenn
Science Advisor
Homework Helper
6,735
2,458

Homework Statement



Write f(t) = 1 + t as Fourier series, with only cosine terms in the interval [0,pi]

For which values of t does the series converge to f ?



The Attempt at a Solution



Expand f = 1+t as an even function about t=0; so it will be a zig-zag with non continuous points at -pi,0,pi, 2pi etc

the first part is simple, but the second; I was thinking that the Fourier series is converging to f, exept where f is not continous. So the answer is [tex] t \in ]0,\pi[ [/tex]

Is that correct?
 
Physics news on Phys.org
  • #2
Yes. In fact it converges to 1+pi/2 at 0 and pi which is not the same as f.
 

FAQ: Convergence of Fourier Series for f(t) = 1 + t with Only Cosine Terms in [0,pi]

What is the convergence of a Fourier series?

The convergence of a Fourier series refers to the mathematical property of a series of periodic functions that approaches a limit as the number of terms used in the series approaches infinity. In other words, it is the property that allows us to approximate a periodic function with a finite number of sine and cosine functions.

How do you determine if a Fourier series is convergent?

A Fourier series is considered convergent if it satisfies the Dirichlet conditions, which state that the function must be periodic, have a finite number of discontinuities within a period, and have a finite number of maxima and minima within a period. Additionally, the function must also be square integrable, meaning that its integral over one period is finite.

What is the difference between pointwise and uniform convergence?

Pointwise convergence of a Fourier series refers to the convergence of the series at each individual point. In other words, as the number of terms in the series increases, the value of the series at a specific point approaches the value of the function at that point. Uniform convergence, on the other hand, refers to the convergence of the series as a whole. This means that the difference between the value of the series and the value of the function becomes smaller and smaller at every point as the number of terms increases.

Can a Fourier series converge to a non-periodic function?

No, a Fourier series can only converge to a periodic function. This is because the basis functions of a Fourier series (sine and cosine functions) are themselves periodic, and thus the combination of these functions can only result in a periodic function.

What is the Gibbs phenomenon in Fourier series?

The Gibbs phenomenon is a phenomenon that occurs when approximating a function using a Fourier series. It refers to the oscillations or overshoots that occur near a jump discontinuity in the function. These oscillations do not disappear as the number of terms in the series increases, but their amplitude decreases. This phenomenon is a result of the finite number of terms used in the series and is not a failure of convergence.

Similar threads

A few questions to make sure I understand Fourier series
Replies
3
Views
1K
Fourier series of this scale and shift of triangle wave
Replies
1
Views
777
On pointwise convergence of Fourier series
Replies
3
Views
1K
Show uniqueness in Dirichlet problem in unit disk
Replies
6
Views
1K
Fourier series for trigonometric absolute value function
Replies
8
Views
2K
Show function series involving arctan is not differentiable at x=0
Replies
26
Views
2K
How should I show the following by using the signum function?
Replies
14
Views
507
Find the Fourier Series of the function
Replies
12
Views
2K
Pointwise, uniform convergence of fourier series
Replies
1
Views
2K
Complex Fourier Series for cos(t/2)
Replies
3
Views
1K

Hot Threads

Recent Insights

Back
Top