how to get the Fourier series coefficients (fourier series engineering mathematics)

Summary

This educational script introduces the concept of Fourier series, a mathematical tool for representing functions as infinite sums of sines and cosines, which are particularly useful for periodic phenomena. The video explains the difference between Taylor and Fourier series, outlines the general form of a Fourier series, and delves into the process of calculating the coefficients 'a_n' and 'b_n' through integration over a 2π interval. It simplifies the complex-looking expression into a more digestible format, aiming to demystify the subject for students.

Takeaways

• 📊 The goal of the Taylor series is to write a function as a power series, resembling an infinite polynomial.
• 🔢 In the Taylor series, the function is expressed in terms of (x - a) raised to the power of whole numbers.
• 📐 The Taylor series uses coefficients, denoted as C's, for its power series representation.
• 🧮 The Fourier series aims to represent a function as a trigonometric series using sine and cosine functions.
• 🔄 Sine and cosine functions are used in the Fourier series because they are periodic and useful for representing repeating phenomena.
• 🔍 The general form of the Fourier series includes a constant term (a0), cosine terms (an * cos(nx)), and sine terms (bn * sin(nx)).
• ➕ The Fourier series combines these terms into a single summation for simplicity.
• ✏️ To find the coefficients an and bn in the Fourier series, integration is used instead of differentiation as in the Taylor series.
• 🧩 For calculating the coefficients, functions are integrated over an interval of length 2π, typically from -π to π.
• 🔧 Special considerations are made for integrals involving sine and cosine products, leading to specific formulas for an and bn.

Q & A

What is the main goal of the Fourier series?

The main goal of the Fourier series is to represent a function as a trigonometric series, specifically using sine and cosine functions as the building blocks.

How is the Fourier series different from the Taylor series?

The Fourier series uses sine and cosine functions to represent a function, focusing on periodicity, while the Taylor series represents a function as a power series using polynomial terms.

What are the building blocks of the Fourier series?

The building blocks of the Fourier series are sine and cosine functions, with coefficients a_n for cosine terms and b_n for sine terms.

Why are sine and cosine functions used in the Fourier series?

Sine and cosine functions are used because they are periodic, which makes them suitable for representing functions that exhibit repetitive behavior.

What is the general form of the Fourier series?

The general form of the Fourier series is a sum of cosine terms (a_0 plus a_n times cosine of n times x) and sine terms (b_n times sine of n times x), where n ranges from 1 to infinity.

How is the coefficient a_0 in the Fourier series calculated?

The coefficient a_0 is calculated by integrating the function over the interval from negative pi to pi and dividing by 2 pi, with a_0 being equal to 1 over 2 pi times the integral of the function.

What happens to the integral of the cosine terms in the Fourier series when n is not equal to m?

When n is not equal to m, the integral of the cosine terms in the Fourier series evaluates to 0 due to the orthogonality of the sine and cosine functions.

How is the coefficient a_n in the Fourier series determined?

The coefficient a_n is determined by multiplying the function by the cosine of n times x, integrating over the interval from negative pi to pi, and then dividing by pi.

What is the integral result of the sine terms in the Fourier series when n equals m?

When n equals m, the integral result of the sine terms in the Fourier series is pi, which contributes to the calculation of the b_n coefficients.

How do you calculate the coefficient b_n in the Fourier series?

The coefficient b_n is calculated by multiplying the function by the sine of n times x, integrating over the interval from negative pi to pi, and then dividing by pi.