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Chapter 10: Problem 35

Give an example of: A function, \(f(x),\) with period \(2 \pi\) whose Fourier series has no cosine terms.

Short Answer

Expert verified
The function is \(f(x) = \sin(x)\) with no cosine terms in its Fourier series.

Step by step solution

01

Understanding the Problem

We need to find a function \(f(x)\) whose Fourier series consists only of sine terms and has a period of \(2 \pi\). This implies the function must be odd since cosine terms, which are even, appear due to the even parts of a signal.
02

Choosing the Function

An example of such a function is the sine function itself, \(f(x) = \sin(x)\). This function is periodic with a period of \(2 \pi\) and is odd. Hence, its Fourier series should contain only sine terms.
03

Writing the Fourier Series

For the function \(f(x) = \sin(x)\), the Fourier series is simply \(f(x) = \sin(x)\) because it already is a basic sine function. There are no cosine terms in this expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Periodic Functions
A periodic function is one that repeats its values at regular intervals or periods. In mathematical terms, a function \(f(x)\) is said to be periodic with period \(T\) if for all values of \(x\), \(f(x + T) = f(x)\).
Periodic functions are integral in Fourier analysis because they allow complex, repeating signals to be broken down into simpler components.
  • One of the simplest examples of a periodic function is the sine function, \(\sin(x)\), which repeats itself every \(2\pi\) radians.
  • Understanding periodicity is crucial when working with Fourier series, as they decompose periodic functions into a sum of simpler trigonometric functions.
In this context, if a function has a period of \(2\pi\), it repeats its behavior every \(2\pi\). This is an important property utilized when calculating Fourier series, ensuring the function's decomposition starts repeating every full cycle.
Exploring the Sine Function
The sine function is a quintessential example of a periodic function, showcasing a wave-like pattern that repeats every \(2\pi\) radians. It is defined as \(f(x) = \sin(x)\) and exhibits unique properties that make it valuable in mathematical and physical sciences.
  • The sine function is periodic with a period of \(2\pi\), meaning \(\sin(x + 2\pi) = \sin(x)\) for all \(x\).
  • It is an odd function, symmetric about the origin. This symmetry affects the function鈥檚 Fourier series representation.
In a Fourier series, sine functions are significant due to their orthogonality with cosine components. This property allows a clear differentiation between odd and even parts of a function when decomposing into a Fourier series.
Properties of Odd Functions
Odd functions are known for their unique symmetry with respect to the origin. For a function \(f(x)\) to be considered odd, it must satisfy the condition \(f(-x) = -f(x)\) for all \(x\) in its domain.
Odd functions are exclusively represented in Fourier series by sine terms. This is because sine functions themselves are odd.
  • An even function, like a cosine, does not change sign when \(x\) is negated, hence they contribute to the even part of a Fourier series.
  • When a function is entirely odd, like \(f(x) = \sin(x)\), its Fourier series has no cosine components.
This makes understanding odd functions critical when attempting to find functions whose Fourier series consist solely of sine terms, illustrating how symmetry properties influence function representation.

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Most popular questions from this chapter

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