Fourier Series Even and Odd Functions: Quiz and Flashcards

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Questions and Answers

What is the definition of an odd function?

A function where f(−x) = -f(x) (correct)

A function where f(−x) = f(x)

A function that has symmetry around the origin

A function that has symmetry about the y-axis

Which of the following statements is true about even functions graphically?

They have no specific symmetry

They have symmetry about the x-axis

They have symmetry around the origin

They have symmetry about the y-axis (correct)

Which of the following functions is considered odd?

$5x^3 - 3x$ (correct)

$ ext{sin } x$

$ ext{cos } x$

$-x^6 + 4x^4 + x^2 - 3$

What can be said about the product of an even function and an odd function?

It is always odd Signup and view all the answers

When integrating an odd function over a symmetric domain, what is the result?

$0$ Signup and view all the answers

What is a characteristic of integrating even functions over symmetric domains?

The result is twice the integral over half of the domain Signup and view all the answers

What are the coefficients called in a Fourier series?

Fourier coefficients Signup and view all the answers

What is the big advantage of Fourier series compared to Taylor series?

Can handle discontinuities Signup and view all the answers

What is automatically concluded about the bn coefficients for even functions?

They are all zero Signup and view all the answers

Which identities are useful for Fourier series when n is an integer?

cos(nπ) = -1 and sin(nπ) = 0 Signup and view all the answers

What should be done to find a Fourier series?

Compute the integrals of f (x) Signup and view all the answers

What happens to bn if f(x) is an even function?

They are all zero Signup and view all the answers

What is the period of a periodic function?

The number T such that f(x + T) = f(x) for every x Signup and view all the answers

How are sine and cosine functions related to periodicity?

They are the most basic periodic functions Signup and view all the answers

If a function has period 2p, what interval is this function defined on?

(−p, p) Signup and view all the answers

How can a periodic function be visually explained?

As a function that repeats its behavior after a certain interval Signup and view all the answers

What does Fourier series help with in relation to periodic functions?

It simplifies complex functions into sums of simpler trigonometric functions Signup and view all the answers

What are the Fourier coefficients a0 and an when the function f(x) is odd?

a0 = 0, an = 0 Signup and view all the answers

If a function is neither even nor odd, which formulas should be used to compute Fourier coefficients?

Formulas from <code>equation (2.2)</code> Signup and view all the answers

In Example 1, how is the function f(x) defined on the interval [-1,0]?

f(x) = 1 Signup and view all the answers

What do the Fourier coefficients represent in a Fourier series?

The frequency components of the function Signup and view all the answers

If a function is periodic and defined on one period, what can be computed using Fourier series?

The decomposition of the function into a sum of sines and cosines Signup and view all the answers

Study Notes

Fourier Series

The Fourier series of a function f(x) is an infinite series involving sines and cosines, represented by the formula: ∞ f (x) = a0 + ∞ [an cos(nπx/p) + bn sin(nπx/p)]
The Fourier coefficients a0, an, and bn are calculated using the formulas: a0 = (1/p) ∫f(x)dx, an = (1/p) ∫f(x)cos(nπx/p)dx, and bn = (1/p) ∫f(x)sin(nπx/p)dx

Fourier Coefficients

The Fourier coefficients of an even function simplify to: bn = 0
The Fourier coefficients of an odd function simplify to: a0 = 0 and an = 0

Even and Odd Functions

An even function has symmetry about the y-axis, and satisfies the condition: f(-x) = f(x)
An odd function has symmetry about the origin, and satisfies the condition: f(-x) = -f(x)
Examples of even functions: sums of even powers of x, cos x
Examples of odd functions: sums of odd powers of x, sin x

Integrating Even and Odd Functions

If f(x) is an odd function, then ∫f(x)dx = 0 over a symmetric domain
If f(x) is an even function, then ∫f(x)dx = 2∫f(x)dx over a symmetric domain

Periodic Functions

A periodic function has repetitive behavior, and satisfies the condition: f(x + T) = f(x) for every x
The smallest period T is called the period of the function
Examples of periodic functions: sin x, cos x, sin(πx), cos(πx)

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Description

Learn the basics of Fourier series including the definitions of even and odd functions. Discover how to identify even and odd functions graphically and algebraically. Practice with examples involving sums of odd powers of x.

Introduction to Fourier Series: Even and Odd Functions