Question

Let \(f \in E[0, \pi]\) and $$ f(x) \sim \sum_{n=1}^{\infty} b_{n} \sin n x $$ denote the sine series of \(f\), while $$ f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty} a_{n} \cos n x $$ denotes the cosine series of \(f\). What equals the function $$ g(x)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right] $$ at each point in \([-\pi, \pi]\) ?

Short answer

The function \( g(x) \) equals \( f(x) \) symmetrically extended over \([-\pi, \pi]\).

Step-by-step solution

Understanding the Series Representations

Given a function \( f \) defined on \([0, \pi]\), we have its sine series \( f(x) \sim \sum_{n=1}^{\infty} b_{n} \sin nx \) and cosine series \( f(x) \sim \frac{a_{0}}{2} + \sum_{n=1}^{\infty} a_{n} \cos nx \). These trigonometric series represent \( f \) over its domain with sine terms taking odd harmonics and cosine terms even harmonics in separate intervals.

Combining Sine and Cosine Series

The function \( g(x) = \frac{a_{0}}{2} + \sum_{n=1}^{\infty} \left[a_{n} \cos nx + b_{n} \sin nx \right] \) combines both sine and cosine terms. This represents the Fourier series of \( f \), which integrates both periodic components to create a generalized representation of \( f \) over a symmetric interval \([-\pi, \pi]\).

Identifying the Role of g(x)

The function \( g(x) \) at each point in \([-0, \pi] \) adds sine terms and cosine terms. The Fourier series coefficient expressions indicate that this sum finds \( f(x) \) universally valid in both periodic extensions: even for cosine and odd for sine, forming periodic \( g(x)\).

Conclusion on Function Equality

On the interval \([-0,0] \), the cosine series mirrors that of \([0,\u00bpi] \), but the sine series makes \( f \) odd. Thus, \( g(x) \) essentially extends \( f(x) \) symmetrically into \([-\pi, \pi] \) as a full Fourier series, essentially modeling an even extension and remaining periodic throughout \( [-\pi, \pi] \).

Key concepts

Sine Series

In mathematics, a sine series is a way of expressing a function as a sum of sine functions. Each function in the series corresponds to a specific harmonic component. In the context of Fourier series, the sine series involves only sine terms of the form \( b_n \sin(nx) \), where \( b_n \) are the coefficients that depend on the function being represented, and \( n \) denotes the harmonic number.

A sine series is particularly useful for representing odd functions over a given interval. Odd functions have the property that they are symmetric about the origin. This means that if you fold the graph along the y-axis, each half will be a reflection of the other. Thus, the sine series captures this symmetry effectively.

Because sine functions themselves are odd, they align perfectly with the characteristics of odd functions. This makes the sine series ideal for functions that are defined as odd within the given interval, contributing harmonics that enhance the accuracy of the representation.

Cosine Series

A cosine series functions similarly to a sine series but uses cosine terms instead of sine terms. The general form of a cosine series is either standalone or part of a Fourier expansion, expressed as \( \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(nx) \). Here, \( a_n \) are the coefficients determined by integrating the initial function with cosine terms over the interval of interest.

Cosine series are used to represent even functions. Even functions have a symmetry about the y-axis, which means they remain unchanged when mirrored over this axis. The cosine terms, being inherently even functions themselves, are well-suited to depict this symmetry.

In practical applications, cosine series help approximate functions that are evenly distributed and provide an efficient mathematical tool to represent evenly symmetric behaviors across an interval.

Trigonometric Representation

The trigonometric representation of a function f(x) refers to expressing the function in terms of sine and cosine components. This comprehensive representation is known as a Fourier series. It includes both sine and cosine series elements to create a complete view of periodic functions across a symmetrical interval such as \([-\pi, \pi]\).

With trigonometric representation, functions can be decomposed into their fundamental periodic elements, where both sine and cosine terms account for different aspects of the function's shape.

• The sine terms represent the vertical oscillations.
• Cosine terms reflect horizontal symmetries.

This combination allows for capturing all variances in a function, whether even or odd, providing a flexible and powerful model for periodic functions.

The use of both components in Fourier series enables representing almost any periodic function, no matter how complex it might be, as long as it satisfies the Dirichlet conditions.

Periodic Functions

Periodic functions are functions that repeat their values at regular intervals or periods. The concept of periodicity is integral in understanding both natural phenomena and mechanical systems that exhibit repeatable behavior.

In mathematics, a function \( f(x) \) is said to be periodic if there exists a positive number \( T \) such that \( f(x+T) = f(x) \) for all \( x \). Common examples include trigonometric functions like sine and cosine, each of which repeats with a period of \( 2\pi \).

• These functions form the building blocks of Fourier series.
• They provide ways to simplify complex waveforms into recognizable patterns.

The study and application of periodic functions allow us to analyze wave patterns, vibrations, sounds, and various recurring phenomena in a structured way.

Understanding periodicity is crucial because it describes how functions behave over long periods, ensuring predictability and stability in various scientific and engineering contexts.