Fourier Series and Trigonometric Series

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What type of function must be used for sine series expansion according to the given content?

A rational function

An even function

An odd function (correct)

A discontinuous function

What is the primary purpose of restricting results to a specific interval after applying Dirichlet to the extended function?

To limit the analysis only to the required interval. (correct)

To ensure continuity across all intervals.

To make the function periodic.

To ensure that the function remains odd.

In the context of cosine series expansion, what characteristic must the function possess?

It must be even and 2b periodic. (correct)

It needs to be defined only on odd intervals.

It must be odd and discontinuous.

It should not be locally integrable.

Why might a sine series expansion be preferred over a Fourier series expansion in some cases?

Sine series expand only for odd functions. Signup and view all the answers

What is the value of $F_f(x)$ for a piecewise continuous function f on $[l; l]$?

$\frac{f(x) + f(x)}{2}$ Signup and view all the answers

If f is continuous on E R, what is the relationship between $F_f(x)$ and f?

$F_f(x) = f(x)$ for all $x \in E$ Signup and view all the answers

For the function $f(x) = x$ defined on $[l; l]$, what is the nature of f?

f is periodic with period 2 Signup and view all the answers

What is the result of $b_n$ when calculating Fourier coefficients for an odd function?

$b_n = 0$ for all n Signup and view all the answers

In the context of Fourier series, which of the following statements is true regarding the periodic function described?

The function must be integrable over its period. Signup and view all the answers

What is the nature of the series represented by $\sum_{n=1}^{\infty} \frac{cos(nx) \sin(nx)}{n^2} + 5$?

It is a trigonometric series. Signup and view all the answers

What is the form of the points of discontinuity for the function f(x)?

2k, k ∈ Z Signup and view all the answers

What is the value of $a_0$ in the trigonometric series given?

0 Signup and view all the answers

What is the period of the function $f(x)$ if the trigonometric series is convergent?

Interval of length 2 Signup and view all the answers

On which interval is the function f(x) considered to be odd and periodic?

[0, 1] Signup and view all the answers

Which of the following statements is true about the coefficient $b_n$ from the trigonometric series?

$b_n$ can be expressed as $\frac{1}{2} \int_0^2 f(x) \sin(nx)dx$. Signup and view all the answers

What is the limit of f'(x) as x approaches 0 from the right?

2 Signup and view all the answers

What is the formula used for Ff(x) for x in the interval (0, ∞)?

$\sum_{n=1}^{\infty} \frac{sin(nx)}{n}$ Signup and view all the answers

What does uniform convergence of the trigonometric series imply about its coefficients?

The coefficients can be derived directly from the function f over the interval. Signup and view all the answers

How is the series $\sum_{n=1}^{\infty} a_n \cos(nx) + b_n \sin(nx)$ structured in the context of its coefficients?

The coefficients are determined from the integrals of the function f. Signup and view all the answers

What does the Dirichlet corollary imply about the behavior of function f(x) at the point 2?

f(2) is discontinuous. Signup and view all the answers

In the context of the series, what does the notation $\sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx))$ signify?

An infinite series representing a periodic function. Signup and view all the answers

What is necessary to apply Dirichlet's conditions for a Fourier series expansion of a function defined on an interval [0; b]?

An extended function must be defined on R and periodic. Signup and view all the answers

For a function that is locally integrable and odd, what property of its Fourier coefficients can be inferred?

Only sine coefficients are non-zero. Signup and view all the answers

What does S2 represent in the context of the presented Fourier series expansion?

The sum of all even Fourier coefficients. Signup and view all the answers

What is indicated by the periodicity of the function fe mentioned in the context?

fe shares the same values as f within the interval [0; b]. Signup and view all the answers

When calculating S2 for an even function, what does the expression $\int_0^1 f^2(x)dx$ represent?

The sum of squared Fourier coefficients. Signup and view all the answers

What is the significance of defining fe as even and 2b periodic for a cosine series expansion?

It facilitates the application of Dirichlet's theorem. Signup and view all the answers

Which condition is NOT necessary for the function f when making a cosine series expansion?

f must be differentiable on [0; b]. Signup and view all the answers

In the context of the Fourier series, what does the property of local integrability imply for the function f?

f may possess discontinuities but has finite integrals over bounded intervals. Signup and view all the answers

What characterizes function f being of C1 piecewise on an interval [a, b]?

The function can be expressed as a finite union of intervals. Signup and view all the answers

Which of the following series represents S1 when x is replaced with 0?

$S1 = 0$ Signup and view all the answers

What is the condition for applying Dirichlet’s theorem as mentioned?

f must be an even periodic function only in the interval [0, π]. Signup and view all the answers

What type of function behavior is suggested if both $\lim_{x \to x_i^+} f'(x)$ and $\lim_{x \to x_i^-} f'(x)$ exist and are finite?

The function is likely continuous at x_i. Signup and view all the answers

What condition must hold for the function f to be considered integrable on any closed bounded interval of R?

The function must have a finite number of discontinuities. Signup and view all the answers

In the Fourier series expansion, what series is used to express f(x)?

$f(x) = \sum_{n=0}^{\infty} \frac{4 \cos((2n + 1)x)}{(2n + 1)}$ Signup and view all the answers

What does the term 2l periodicity imply about the function f?

f is periodic with period 2. Signup and view all the answers

What is the significance of the expression $S2$ derived in the context?

It connects two separate series into one expression. Signup and view all the answers

Study Notes

Fourier Series

Fourier series represent periodic functions as an infinite sum of sine and cosine terms.
A function f is periodic if there exists a positive value T such that f(x + T) = f(x) for all x.
The smallest positive T is the fundamental period.

Trigonometric Series (Period 2π)

A trigonometric series is a series of functions of the form:
u₀(x) = α₀/2
uₙ(x) = aₙ cos(nx) + bₙ sin(nx) for n ≥ 1
Coefficients aₙ and bₙ are real numbers.

Convergence of Trigonometric Series

If both Σ|aₙ| and Σ|bₙ| converge, the trigonometric series converges normally (and thus uniformly) on ℝ to a continuous function.

Periodicity of Convergent Series

If a trigonometric series converges to a function f, then f is periodic with period 2π.

Coefficients on [0, 2π]

If a trigonometric series converges uniformly to a function f, its coefficients are given by:
α₀ = (1/π) ∫₀^(2π) f(x) dx
aₙ = (1/π) ∫₀^(2π) f(x) cos(nx) dx for n ≥ 1
bₙ = (1/π) ∫₀^(2π) f(x) sin(nx) dx for n ≥ 1

Fourier Series Expansion (Period 2π)

Let f be a real-valued function defined on ℝ with period 2π, integrable on any closed interval.
The Fourier series of f is given by:
Ff(x) = α₀/2 + Σ[aₙcos(nx) + bₙsin(nx)] for n ≥ 1
where α₀, aₙ, and bₙ are Fourier coefficients.

Parity and Fourier Coefficients

If f is an even function (f(-x)=f(x)), its Fourier coefficients satisfy aₙ = 0 and bₙ = (2/π)∫₀^π f(x)sin(nx) dx for any n ≥1.
If f is an odd function (f(-x)=-f(x)), its Fourier coefficients satisfy α₀ = 0 and aₙ = 0, bₙ=(2/π)∫₀^π f(x)sin(nx) dx for any n ≥1.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

Explore the concepts of Fourier series and trigonometric series, focusing on periodic functions and their representation as infinite sums of sine and cosine terms. Understand the conditions for convergence and determine the coefficients of the series within a given interval.