Fourier Sine Series Of Sinx

Decomposing sin(x): Unveiling the Fourier Sine Series of sin(x)

The Fourier series, a powerful tool in mathematical analysis, allows us to represent periodic functions as an infinite sum of sine and cosine functions. This decomposition is crucial in various fields, from signal processing and image analysis to solving partial differential equations. While the concept might seem abstract, understanding how to find the Fourier series of even seemingly simple functions like sin(x) is fundamental to mastering this powerful technique. This article will delve into the specific case of the Fourier sine series of sin(x), addressing common challenges and providing a step-by-step approach. We will discover intriguing results that highlight important aspects of Fourier analysis.

1. Understanding the Fourier Sine Series

Before diving into the specifics of sin(x), let's briefly recall the definition of a Fourier sine series. A function f(x) defined on the interval [0, L] can be represented by its Fourier sine series as:

f(x) ≈ Σn=1∞ bn sin(nπx/L)

where the coefficients bn are given by:

bn = (2/L) ∫0L f(x) sin(nπx/L) dx

This series utilizes only sine functions, making it particularly useful for representing odd functions or functions defined on an interval where odd extension is appropriate. The interval [0, L] is crucial; the series is tailored to this specific interval.

2. The Case of f(x) = sin(x) on [0, π]

Let's consider the function f(x) = sin(x) on the interval [0, π]. This choice simplifies the calculation significantly. Notice that we've chosen the interval [0, π], which is a natural choice given the periodicity of sin(x). In this case, L = π. The formula for the coefficients bn becomes:

bn = (2/π) ∫0π sin(x) sin(nx) dx

This integral can be solved using various techniques, including integration by parts or trigonometric identities. A useful trigonometric identity is:

sin(A)sin(B) = (1/2)[cos(A-B) - cos(A+B)]

Applying this identity, we get:

bn = (1/π) ∫0π [cos((1-n)x) - cos((1+n)x)] dx

Now, let's evaluate the integral for different values of 'n':

For n = 1: b1 = (1/π) ∫0π [1 - cos(2x)] dx = 1

For n ≠ 1: bn = (1/π) [sin((1-n)x)/(1-n) - sin((1+n)x)/(1+n)] evaluated from 0 to π. This evaluates to 0 because sin(kπ) = 0 for any integer k.

Therefore, the Fourier sine series of sin(x) on the interval [0, π] is simply:

sin(x) = sin(x)

3. Implications and Interpretations

The result that the Fourier sine series of sin(x) on [0, π] is simply sin(x) itself might seem trivial at first. However, it highlights several important points:

Orthogonality of sine functions: The orthogonality of sine functions over the interval [0, π] is crucial. The integral of the product of two different sine functions (with integer multiples of π/L as arguments) over this interval is zero. This property is the foundation of the Fourier series.

Choice of interval: The choice of the interval [0, π] was deliberate. If we had chosen a different interval, the result would have been different, potentially leading to a more complex series.

Function properties: The fact that we obtain a simple representation stems from the fact that sin(x) is already a sine function and is defined over a suitable interval for its fundamental period.

4. Addressing Common Challenges

A common challenge arises when students attempt to find the Fourier sine series of functions that aren't already expressed as sine functions. The key is to carefully evaluate the integral for the coefficients bn using appropriate integration techniques and trigonometric identities. Another challenge lies in choosing the appropriate interval [0, L]. This choice depends on the function's behaviour and the desired representation.

5. Summary

This exploration of the Fourier sine series of sin(x) on the interval [0, π] demonstrates the power and elegance of Fourier analysis. While the result might appear straightforward, it reveals the underlying principles of orthogonality and the importance of choosing the appropriate interval for the series. The simplicity of the result underscores the inherent compatibility between sin(x) and its sine series representation over this specific interval. This understanding provides a solid foundation for tackling more complex Fourier series problems involving other functions and intervals.

Frequently Asked Questions (FAQs)

1. Why use a Fourier sine series instead of a full Fourier series (with cosines)? A Fourier sine series is advantageous when dealing with odd functions or when boundary conditions dictate the use of only sine terms, particularly in solving partial differential equations.

2. What if the interval is different from [0, π]? If the interval is [0, L], the series becomes Σn=1∞ bn sin(nπx/L), and the coefficients bn will change accordingly. The integral limits and the argument within the sine functions will adjust based on the new interval.

3. Can we find the Fourier sine series of any function? Yes, in theory, any function that satisfies certain conditions (e.g., piecewise smoothness) can be represented by a Fourier sine series on a specified interval [0, L].

4. How does the convergence of the Fourier sine series behave? The convergence depends on the function's smoothness. For a continuous and differentiable function, the convergence is generally good. Discontinuities can lead to Gibbs phenomenon near the discontinuities.

5. What are some applications of the Fourier sine series? Applications are abundant: solving heat equations with specific boundary conditions, analyzing periodic signals with odd symmetry, image processing (particularly for odd-symmetric images), and modeling physical phenomena with odd-symmetric properties.

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Fourier Sine Series - Mathphysics.com The Fourier sine series, in contrast, is well adapted to functions which are zero at x=0 and x=L, since all the functions sin(n \pi x/L) have this property. A related but less obvious property that begs for the use of sine functions is an odd symmetry: f(-x) = - f(x).

Fourier Analysis Workshop 1: Fourier Series - Royal Observatory, … Using a trigonometric identity, or otherwise, compute the Fourier Series for f(x) = x sin x for ⇡ < x < ⇡, and hence show that. 3 ⇥ 1 2 4 3 ⇥ 5 5 ⇥ 7 . . . What is its fundamental = cos period? | x|. cos(2mx). 2. Let f(x) = 1 + cos2(⇡x). Sketch f(x) and determine its fundamental period.

Differential Equations - Fourier Sine Series - Pauls Online Math … 16 Nov 2022 · In this section we define the Fourier Sine Series, i.e. representing a function with a series in the form Sum( B_n sin(n pi x / L) ) from n=1 to n=infinity. We will also define the odd extension for a function and work several examples finding the Fourier Sine Series for a function.

fourier series of $|\\sin x|$ - Mathematics Stack Exchange The function $x\mapsto f(x):=|\sin x|$ is even and $\pi$-periodic; therefore $f$ has a Fourier series of the form $$f(x)={a_0\over2}+\sum_{k=1}^\infty a_k \cos(2kx)$$ with $$a_k={2\over\pi}\int_0^\pi f(x)\cos(2k x)\ dx={2\over\pi}\int_0^\pi \sin x\cos(2k x)\ dx\ .$$ It follows that $$\eqalign{a_k&={1\over\pi}\int_0^\pi\left(\sin\bigl((1+2k)x ...

CHAPTER 4 FOURIER SERIES AND INTEGRALS This section explains three Fourier series: sines, cosines, and exponentials eikx. Square waves (1 or 0 or −1) are great examples, with delta functions in the derivative. We look at a spike, a step function, and a ramp—and smoother functions too.

Compute the fourier coefficients, and series for $\\log(\\sin(x))$ 16 Dec 2014 · The goal is to compute the Fourier series of $g(x)=\log\sin x$ over $[0,\pi]$, or the Fourier series of $h(x)=\log\sin\frac{x}{2}$ over $[0,2\pi]$, or the Fourier series of $f(x)=\log\cos\frac{x}{2}$ over $[-\pi,\pi]$.

Fourier Sine Series -- from Wolfram MathWorld 20 Jan 2025 · If f (x) is an odd function, then a_n=0 and the Fourier series collapses to f (x)=sum_ (n=1)^inftyb_nsin (nx), (1) where b_n = 1/piint_ (-pi)^pif (x)sin (nx)dx (2) = 2/piint_0^pif (x)sin (nx)dx (3) for n=1, 2, 3, .... The last equality is true because f (x)sin (nx) = [ …

10.4 Fourier Cosine and Sine Series - University of California, … The Fourier sine series of f(x) on [0;L] is X1 n=1 b nsin nˇx L; (6) where b n= 2 L Z L 0 f(x)sin nˇx L dx; n= 1;2;:::: (7) The trigonometric series in (4) is the Fourier series for f e(x), the even 2L-periodic extension of f(x). The trigonometric series in (6) is the Fourier series for f o(x), the odd 2L-periodic extension of f(x). These are

Fourier Trigonometric Series: Definition, Examples, and Applications 1 Aug 2024 · 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 .

Fourier sine series for sine with non-integer frequency - John D. 7 Apr 2020 · The Fourier series of an odd function only has sine terms—all the cosine coefficients are zero—and so the Fourier series is a sine series. What is the sine series for a sine function? If the frequency is an integer, then the sine series is just the function itself.

Fourier sine and cosine series - Wikipedia In mathematics, particularly the field of calculus and Fourier analysis, the Fourier sine and cosine series are two mathematical series named after Joseph Fourier. In this article, f denotes a real -valued function on which is periodic with period 2 L.

CHAPTER 4 FOURIER SERIES AND INTEGRALS - MIT … This section explains three Fourier series: sines, cosines, and exponentials eikx. Square waves (1 or 0 or −1) are great examples, with delta functions in the derivative. We look at a spike, a step function, and a ramp—and smoother functions too.

Fourier Series of The Sine Function - Mathematics Stack Exchange I'm going to give the short answer, here. Use orthogonality. The function sin $\left(\frac{\pi x}{L}\right)$ is orthogonal to the function sin $\left(\frac{n\pi x}{L}\right)$, for all $n\neq 1$, on the interval $(-L,L)$. Where $n=1$, the product of these functions is sin $^2\left(\frac{\pi x}{L}\right)$.

Trigonometric Fourier Series - GeeksforGeeks 13 Aug 2024 · Types of Trigonometric Fourier Series. 1. Fourier Sine Series: In order to find Fourier sine series for a odd function defined over time interval (0,\text{T}) we extend the time period to \text{(-T,T)} and since it is odd function it follows the property f(-t) = -f(t), the Fourier transform of f(t) can be transformed into following equation:

16 Fourier Analysis – Foundations of Computer Vision 16.3 Fourier Series. In 1822, French mathematician and engineer Joseph Fourier, as part of his work on the study on heat propagation, showed that any periodic signal could be written as an infinite sum of trigonometric functions (cosine and sine functions). ... One of Fourier’s original examples of sine series is the expansion of the ramp ...

10.5 Fourier Series: Linear Algebra for Functions - MIT Mathematics There are two different answers, both good: 1. The vector is infinitely long: v = (v1, v2, v3, . . .). It could be (1, 1 2, 14, . . .). 2. The vector is a function f(x). It could be v = sin x. We will go both ways. Then the idea of a Fourier series will connect them. After vectors come dot products.

Fourier series of sin(x) - Mathematics Stack Exchange 12 Sep 2016 · Fourier Sine series uses the orthogonal set $\{\sin(nx)\}^\infty_{n = 1}$ on $0 \le x \le \pi$

AE2 Mathematics Solutions to Example Sheet 2: Fourier Series f(x) | sin x| on (−π, π) with L = π: f(x) is an even function so bn = 0. On [0, π] we have = | sin x| = sin x. where cos nπ = (−1)n. = 1 & the function is neither odd nor even. thus giving the answer. The odd extension of f(x) originally defined on 0. x) sin(nπx) dx.

Fourier Series $\sin(\sin(x))$ - Mathematics Stack Exchange Can anyone find the Fourier Series of $ \sin(\sin(x))$? I have tried evaluating the integrals to determine the coefficients of each of the coefficients of the sine waves, but have no idea where to start computing the integrals.

Lecture 10: Fourier Sine Series - University of British Columbia This is known as a Fourier Series. This lecture deals with the procedure to determine the Fourier coe–cients bn. Our approach is motivated by the process introduced in Linear Algebra for projecting a vector onto a set of basis vectors. Key Concepts: Fourier Sine Series; Vector Projection; functions as inﬂnite dimensional vectors; orthogonality;

9.4: Fourier Sine and Cosine Series - Mathematics LibreTexts 18 Nov 2021 · The Fourier series simplifies if $f(x)$ is an even function such that $f(−x) = f(x)$, or an odd function such that $f(−x) = −f(x)$. Use will be made of the following facts. The function $\cos (n\pi x/L)$ is an even function and $\sin (n\pi x/L)$ is an odd function. The product of two even functions is an even function.

Fourier Sine Series Of Sinx

Decomposing sin(x): Unveiling the Fourier Sine Series of sin(x)

1. Understanding the Fourier Sine Series

2. The Case of f(x) = sin(x) on [0, π]

3. Implications and Interpretations

4. Addressing Common Challenges

5. Summary

Frequently Asked Questions (FAQs)

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