Fourier trigonometric series is a way to represent a periodic function as a sum of sine and cosine functions. This series was named after the French mathematician Joseph Fourier, this series allows us to represent complex periodic functions as an infinite sum of sine and cosine functions.

This article aims to provide a comprehensive set of practice problems on the Fourier Trigonometric Series, complete with solutions, to help reinforce understanding and proficiency in this fundamental topic.

Fourier Trigonometric Series

Fourier Trigonometric Series is a powerful tool for expressing a periodic function f(x) as a sum of sine and cosine functions. This representation is particularly useful because sines and cosines are the fundamental building blocks of periodic functions. For a function f(x) with period 2π, the Fourier series can be written as:

[Tex]f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin(nx) \right) [/Tex]

In this expression:​

• a₀ is Average or Constant Term
• a_n are Coefficients for Cosine terms, representing the even harmonics
• b_n are Coefficients for Sine terms, representing the odd harmonics

These coefficients a₀, a_n and b_n are determined by the function f(x) and are computer as follows:

[Tex]a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) dx[/Tex]

[Tex]a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) cos(nx) dx[/Tex]

[Tex]b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) sin(nx) dx[/Tex]

These formulas allow us to determine how much of each sine and cosine component is present in the original function f(x). Essentially, they break down any periodic function into its fundamental frequency components, which can be very helpful in understanding and analyzing the behavior of the function across its domain.

Fourier Trigonometric Series Formulas

Fourier Coefficients for 2π-Periodic Functions

To find the Fourier series of a function with period 2π, we calculate the following coefficients

• Constant Term (Average Value): Represents the average value of the function over one period.

[Tex]a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) dx[/Tex]

• Cosine Coefficients (Even Terms): Determine the amplitude of the cosine terms in the series.

[Tex]a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) cos(nx) dx[/Tex]

• Sine Coefficients (Odd Terms): Determine the amplitude of the sine terms in the series.

[Tex]b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) sin(nx) dx[/Tex]

These coefficients are obtained by integrating the product of the function f(x) with the corresponding sine or cosine term over one period. This process effectively “extracts” the contribution of each frequency component from the function.

Fourier Series for Functions with Period T

For a function with period T, the Fourier series representation adjusts to accommodate the different period. The general form of the series is:

[Tex]f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos(\frac{2\pi nx}{T}) + b_n \sin(\frac{2\pi nx}{T}) \right) [/Tex]

Coefficients for this series are given by:

• Constant Term [Tex]a_0[/Tex]:

[Tex]a_0 = \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} f(x) \, dx[/Tex]

• Cosine Coefficients [Tex]a_n[/Tex]:

[Tex]a_n = \frac{2}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} f(x) \cos\left(\frac{2\pi nx}{T}\right) \, dx[/Tex]

• Sine Coefficients [Tex]b_n[/Tex]:

[Tex]b_n = \frac{2}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} f(x) \sin\left(\frac{2\pi nx}{T}\right) \, dx[/Tex]

In this general form, the coefficients are scaled to match the period T of the function. This allows the series to accurately reflect the periodic nature of f(x) over its specific period.

Fourier Trigonometric Series Practice Questions

Question 1: Find the Fourier series of the function f(x)=x for −π < x < π.

Solution:

For f(x) = x,

Fourier coefficients are calculated as follows:

Constant Term [Tex]a_0[/Tex]:

[Tex]a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} x dx = 0[/Tex]

Since the function f(x)=x is an odd function, its average value over a symmetric interval around xero is zero.

Cosine Coefficients [Tex]a_n[/Tex]:

[Tex]a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x cos(nx) dx = 0[/Tex]

For odd functions multiplied by cosine (an even function), the integral over a symmetric interval around zero is zero.

Sine Coefficients [Tex]b_n[/Tex]:

[Tex]b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} xsin(nx) dx = \frac{2(-1)^{n+1}}{n}[/Tex]

This integral evaluates to the given formula for [Tex]b_n[/Tex]

Therefore, the Fourier series for f(x) = x is:

[Tex]f(x) = \sum_{n=1}^{\infty} \frac{2(-1)^{n+1}}{n} sin(nx)[/Tex]

Question 2: Determine the Fourier series for the function f(x) = sin(x) for 0 ≤ x ≤ 2π.

Solution:

Function f(x) = sin(x) has period [Tex]2\pi[/Tex]. The Fourier coefficients are:

Constant Term [Tex]a_0[/Tex]:

[Tex]a_0 = \frac{1}{2\pi} \int_{0}^{2\pi} sin(x) dx = 0[/Tex]

Integral of sin(x) over a full period is zero.

Cosine Coefficients [Tex]a_n[/Tex]:

[Tex]a_0 = \frac{1}{2\pi} \int_{0}^{2\pi} sin(x) dx = 0[/Tex]

The integral of sin(x)cos(nx) over a full period is zero for all n.

Sine Coefficients [Tex]b_n[/Tex]:

Since [Tex]a_0 = 0[/Tex], [Tex]a_n = 0[/Tex], and [Tex]b_n = 0[/Tex] for [Tex]n\neq1[/Tex], the only non-zero term is b1.

Fourier series is:

f(x) = sin(x) for [Tex]0\leq x\leq 2\pi[/Tex]

Question 3: Find the Fourier series of f(x) = |x| on the interval -π < x < π.

Solution:

For f(x) = |x|, which is an even function, all [Tex]b_n = 0[/Tex], Fourier series will contain only cosine terms:

[Tex]f(x) = a_0 + \sum_{n=1}^{\infty} a_n cos(nx)[/Tex]

Coefficients are:

Constant Term [Tex]a_0[/Tex]:

[Tex]a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} |x| dx = \pi[/Tex]

This is the average value of the absolute value function over one period.

Cosine Coefficients [Tex]a_n[/Tex]:

[Tex]a_n = \frac{2}{\pi} \int_{0}^{\pi} cos(nx) dx = \frac{2(-1)^{n+1}}{n^2}[/Tex]

These are coefficients for cosine terms, capturing even symmetry of |x|

Thus, Fourier series for f(x) = |x| is:

[Tex]f(x) = \frac{\pi}{2} + \sum_{n=1}^{\infty} \frac{2(-1)^{n+1}}{n^2} cos(nx)[/Tex]

Also Check,

• Taylor Series
• Fourier Series
• Maclaurin series

Conclusion

Fourier Trigonometric Series is a fundamental tool in analyzing periodic functions, breaking them down into simpler sine and cosine components. Mastering this concept is essential for students in mathematics and engineering. By understanding and applying the formulas and solving practice problems, we can gain deeper insights into the behavior of complex waveforms across various fields of study.

Understanding the Fourier Series is crucial for students pursuing studies in mathematics, engineering, and physical sciences. It helps in breaking down complex waveforms into simpler components, facilitating easier analysis and manipulation.

Fourier Trigonometric Series- FAQs

What is the Purpose of Fourier Series?

Fourier series helps to decompose complex periodic functions into simpler sine and cosine components, making it easier to analyze and understand their behavior in various applications.

Why are only sines and cosines used in Fourier series?

Sines and cosines are the natural functions that describe periodic phenomena. They form a complete orthogonal basis for representing periodic functions, which makes them ideal for decomposing any periodic function into its fundamental frequencies.

How do Coefficients a_n and b_n Relate to Function f(x)?

Coefficients [Tex]a_n[/Tex] and [Tex]b_n[/Tex] measure the contribution of the cosine and sine terms at different frequencies n in the function f(x). They determine the amplitude of each frequency component in the Fourier series.

What is Significance of Parseval’s Theorem?

Parseval’s Theorem connects the total energy or power of a function over a period to the sum of the squares of its Fourier coefficients. It provides a way to quantify the function’s energy distribution among its frequency components.

Can any Function be Represented by a Fourier series?

Any periodic function that is integrable over one period can be represented by a Fourier series. Even non-periodic functions can be approximated by Fourier series within a specified interval.