FOURIER SERIES LECTURE 2 | STUDY OF FORMULAS OF FOURIER SERIES AND PERIODIC FUNCTION

TIKLE'S ACADEMY OF MATHS

9 Aug 202311:29

Summary

TLDRThis educational video from the Akaademy YouTube channel focuses on the definition of periodic functions and the formula for Fourier series. The instructor explains the criteria for a function to be periodic, using the example of f(x) and its period T. They also cover the necessary conditions for a function to have a Fourier series, known as Dirichlet's conditions. The video then delves into the standard formula for Fourier series, explaining how to calculate the coefficients a_n and b_n. The instructor emphasizes the importance of understanding these concepts to solve problems involving Fourier series effectively.

Takeaways

😀 The video is part of a series on the topic of Fourier Series, specifically focusing on the definition of periodic functions and the formula for Fourier Series.
📚 The audience is expected to have a basic understanding of the previous lectures on the topic, including the definition of Fourier Series and its basic points.
🔢 A function f(x) is defined as periodic if f(x + T) = f(x) for every x in its domain, where T is a positive constant called the period.
🌐 The concept of the period T is crucial, and functions like sin(x) and cos(x) are given as examples of periodic functions with a period of 2π.
📐 The video emphasizes the importance of satisfying certain conditions, known as Dirichlet's conditions, for a function to have a Fourier Series.
🔑 Dirichlet's conditions include the function being single-valued, having a finite number of discontinuities, and a finite number of extrema within any period.
📘 The formula for the Fourier Series is provided, which includes terms for a0/2, and the sum of an and bn coefficients multiplied by cos(nπx/L) and sin(nπx/L) respectively.
🔍 The video explains how to find the coefficients an and bn through integration over the interval [c, c + 2L], where c and L are specific values related to the period.
📊 The script provides an example to illustrate how changes in the period affect the formula for the Fourier Series coefficients.
🎓 The video concludes by summarizing the key points covered, including the definition of periodic functions and the Fourier Series formula, and encourages the audience to note these down for future reference.

Q & A

What is the main topic of the video?
-The main topic of the video is the study of periodic functions and the Fourier series formula.
What is a periodic function?
-A periodic function is defined as a function f(x) that satisfies the condition f(x + T) = f(x) for some positive constant T, called the period.
What are the conditions for a function to be considered periodic?
-A function is considered periodic if it satisfies the condition f(x + T) = f(x) for all x in its domain, where T is a positive constant.
What is the period T in the context of the video?
-In the video, the period T is a positive constant that defines the repeating interval of a periodic function.
What are the Dirichlet conditions mentioned in the video?
-The Dirichlet conditions are three criteria that a function must satisfy to have a Fourier series. They include: 1) The function must be single-valued, 2) The function must have a finite number of discontinuities, and 3) The function must have a finite number of extrema in any interval.
What are the two important functions discussed in the video that are periodic?
-The two important periodic functions discussed are sin(x) and cos(x).
What is the standard formula for the Fourier series?
-The standard formula for the Fourier series of a function f(x) is given by f(x) = a0/2 + Σ[an * cos(nπx/L) + bn * sin(nπx/L)], where an and bn are the Fourier coefficients.
How are the Fourier coefficients an and bn calculated?
-The Fourier coefficients an and bn are calculated using integration over one period of the function. Specifically, an = (1/L) * ∫[f(x) * cos(nπx/L)] dx from 0 to L, and bn = (1/L) * ∫[f(x) * sin(nπx/L)] dx from 0 to L.
What is the significance of the lower and upper limits of integration in the Fourier coefficients formula?
-The lower and upper limits of integration in the Fourier coefficients formula represent one period of the function, which is crucial for calculating the coefficients accurately.
How does the period of a function affect the calculation of its Fourier series?
-The period of a function directly affects the calculation of its Fourier series because the coefficients an and bn are calculated over one period of the function, and the formula changes accordingly if the period is different from the standard 2π.
What is the next topic to be covered in the series after the Fourier series?
-The next topic to be covered in the series is even functions and odd functions.

Outlines

00:00

📚 Introduction to Periodic Functions and Fourier Series

This section welcomes viewers to the video, emphasizing the continuation of the study on Fourier series, specifically focusing on periodic functions. The video aims to explain the definition of periodic functions and introduce the Fourier series formula. It is highlighted that prior knowledge from earlier lectures is essential, particularly the definition and basics of Fourier series. The section explains that a function f(x) is periodic if it satisfies the condition f(x + nT) = f(x), where n is a positive integer, and T is a positive constant called the period. Examples of periodic functions like sin(x) and cos(x), both with period T = 2π, are given to illustrate the concept. The importance of these definitions in checking whether a function is periodic is underscored before moving on to the Fourier series formula.

05:04

📝 Dirichlet’s Conditions for Fourier Series

This paragraph delves into Dirichlet's conditions, which a function must satisfy to form a Fourier series. Three main conditions are outlined: 1) The function must be defined in the interval c ≤ x ≤ c + 2L, making it periodic with a period of 2L. 2) The function must have a finite number of discontinuities within this interval. 3) The function must have a finite number of maxima and minima within the interval. These conditions are crucial for determining if a Fourier series can be developed for a given function. If a function meets these conditions, it can be analyzed using the Fourier series, making these criteria foundational in understanding the mathematical formulation.

10:04

📏 Standard Formula of Fourier Series

This section presents the standard Fourier series formula, which is used when a function satisfies Dirichlet’s conditions. The formula is given as: f(x) = a0 / 2 + Σ [an cos(nπx / L) + bn sin(nπx / L)]. The coefficients a0, an, and bn are known as Fourier coefficients, which need to be calculated for each specific problem using their respective formulas. The process involves integrating over defined intervals, adjusting constants according to the given period, and modifying the limits based on whether the function period is 2π, π, or any other interval. The paragraph emphasizes that understanding the standard formula and the coefficient formulas is essential to solving problems involving Fourier series, setting the stage for more complex examples to be tackled in future videos.

Mindmap

Keywords

💡Periodic Function

A periodic function is a type of function that repeats its values at regular intervals or periods. In the video, the definition of a periodic function is explored, which is crucial for understanding the behavior of the functions discussed, particularly in the context of Fourier series. The script mentions that a function f(x) is periodic if f(x + T) = f(x) for all x, where T is a constant, referred to as the period. This concept is foundational for the study of Fourier series, as these series are used to represent periodic functions.

💡Fourier Series

Fourier Series is a mathematical tool used to decompose a function into a sum of simple trigonometric functions, which are sine and cosine waves. The script delves into the definition and formula of Fourier series, which is central to the video's educational content. It's used to analyze periodic functions by expressing them as an infinite sum of sines and cosines. The script mentions that the Fourier series formula involves coefficients a_n and b_n, which are calculated using integrals of the function over its period.

💡Coefficients

In the context of Fourier series, coefficients refer to the values a_n and b_n that multiply the sine and cosine terms, respectively. These coefficients are crucial as they determine the amplitude and phase of the sine and cosine waves that make up the periodic function. The script explains that these coefficients are found through integration and are essential for reconstructing the original function from its Fourier series representation.

💡Integration

Integration is a mathematical operation used to find the accumulated value of a function over an interval. In the script, integration is used to calculate the Fourier coefficients a_n and b_n. The process involves integrating the product of the function with either cosine or sine over one period and is a key step in applying Fourier series to a function.

💡Period

The period of a function is the length of the interval over which the function repeats its values. The script mentions that understanding the period of a function is essential for defining it as periodic and for correctly applying the Fourier series formula. The period is often denoted by T or 2π in the context of trigonometric functions.

💡Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in the study of Fourier series because these series are expressed in terms of these functions. The script discusses how these functions are used in the Fourier series formula to represent periodic functions. Sine and cosine functions have the property of being periodic, which makes them suitable for this purpose.

💡Dirichlet Conditions

Dirichlet Conditions are a set of criteria that a function must satisfy to have a convergent Fourier series. The script explains three such conditions: the function must be single-valued, have a finite number of discontinuities, and have a finite number of extrema (maxima and minima) within any period. These conditions are crucial for determining whether a given function can be accurately represented by a Fourier series.

💡Convergence

In the context of Fourier series, convergence refers to the property that the series of trigonometric functions approaches the function it represents as more terms are added. The script touches on the importance of convergence, as a Fourier series is only useful if it converges to the original function. The Dirichlet Conditions help ensure that the series will converge.

💡Extrema

Extrema, or maximum and minimum values of a function, are mentioned in the context of Dirichlet Conditions. The script specifies that a function must have a finite number of extrema within any period to satisfy the conditions for a Fourier series. This is important for ensuring that the function does not have wild oscillations that would prevent the series from converging.

💡Discontinuities

Discontinuities refer to points where a function is not continuous, such as jumps or breaks in the graph of the function. The script mentions that a function must have a finite number of discontinuities within any period to meet the Dirichlet Conditions. This is to ensure that the Fourier series can still converge despite these points of non-smoothness.

💡Interval

An interval is a set of continuous values on the number line. In the script, the interval is discussed in relation to defining the period of a function and the domain over which the Fourier coefficients are calculated. The interval is crucial for determining the limits of integration when finding the coefficients.

Highlights

Introduction to the topic of the video: Studying the definition of periodic functions and the formula for Fourier series.

Emphasis on the importance of understanding the basics from previous lectures to comprehend today's lecture.

Definition of a periodic function: A function f(x) is periodic if f(x + T) = f(x) for all x in its domain.

Explanation of the period T: T is a positive constant, referred to as the period of the function.

Introduction to the concept of fundamental period and how it relates to the period T.

Examples of periodic functions: sin(x) and cos(x) with a period of 2π.

Explanation of the Dirichlet conditions that a function must satisfy to have a Fourier series.

First Dirichlet condition: The function must be single-valued.

Second Dirichlet condition: The function must have a finite number of discontinuities in any given interval.

Third Dirichlet condition: The function must have a finite number of maxima and minima in any given interval.

Formula for Fourier series: Summation of terms involving sines and cosines with coefficients a_n and b_n.

Explanation of how to find the Fourier coefficients a_n and b_n using integration.

Formula for a_0, the constant term in the Fourier series.

Formula for a_n, the coefficient for cosine terms in the Fourier series.

Formula for b_n, the coefficient for sine terms in the Fourier series.

Note on the importance of understanding the limits of integration when calculating coefficients.

Example illustrating how changes in the period affect the formulas for a_n and b_n.

Summary of the video's content: Definition of periodic functions and the formula for Fourier series.

Anticipation of future lectures focusing on even functions and odd functions.