Fourier series expansion in cosines

How to insert mathematical formulas to the website?

If you ever need to add one or two mathematical formulas to a web page, then the easiest way to do this is as described in the article: mathematical formulas are easily inserted onto the site in the form of pictures that are automatically generated by Wolfram Alpha. In addition to simplicity, this universal method will help improve the visibility of the site in search engines. It has been working for a long time (and, I think, will work forever), but is already morally outdated.

If you constantly use mathematical formulas on your site, then I recommend that you use MathJax - a special JavaScript library that displays mathematical notation in web browsers using MathML, LaTeX or ASCIIMathML markup.

There are two ways to start using MathJax: (1) using a simple code, you can quickly connect a MathJax script to your website, which will be automatically loaded from a remote server at the right time (list of servers); (2) download the MathJax script from a remote server to your server and connect it to all pages of your site. The second method - more complex and time-consuming - will speed up the loading of your site's pages, and if the parent MathJax server becomes temporarily unavailable for some reason, this will not affect your own site in any way. Despite these advantages, I chose the first method as it is simpler, faster and does not require technical skills. Follow my example, and in just 5 minutes you will be able to use all the features of MathJax on your site.

You can connect the MathJax library script from a remote server using two code options taken from the main MathJax website or on the documentation page:

One of these code options needs to be copied and pasted into the code of your web page, preferably between tags and or immediately after the tag. According to the first option, MathJax loads faster and slows down the page less. But the second option automatically monitors and loads the latest versions of MathJax. If you insert the first code, it will need to be updated periodically. If you insert the second code, the pages will load more slowly, but you will not need to constantly monitor MathJax updates.

The easiest way to connect MathJax is in Blogger or WordPress: in the site control panel, add a widget designed to insert third-party JavaScript code, copy the first or second version of the download code presented above into it, and place the widget closer to the beginning of the template (by the way, this is not at all necessary , since the MathJax script is loaded asynchronously). That's all. Now learn the markup syntax of MathML, LaTeX, and ASCIIMathML, and you are ready to insert mathematical formulas into your site's web pages.

Any fractal is constructed according to a certain rule, which is consistently applied an unlimited number of times. Each such time is called an iteration.

The iterative algorithm for constructing a Menger sponge is quite simple: the original cube with side 1 is divided by planes parallel to its faces into 27 equal cubes. One central cube and 6 cubes adjacent to it along the faces are removed from it. The result is a set consisting of the remaining 20 smaller cubes. Doing the same with each of these cubes, we get a set consisting of 400 smaller cubes. Continuing this process endlessly, we get a Menger sponge.

Fourier series of periodic functions with period 2π.

The Fourier series allows us to study periodic functions by decomposing them into components. Alternating currents and voltages, displacements, speed and acceleration of crank mechanisms and acoustic waves are typical practical examples of the use of periodic functions in engineering calculations.

The Fourier series expansion is based on the assumption that all functions of practical significance in the interval -π ≤x≤ π can be expressed in the form of convergent trigonometric series (a series is considered convergent if the sequence of partial sums composed of its terms converges):

Standard (=ordinary) notation through the sum of sinx and cosx

f(x)=a o + a 1 cosx+a 2 cos2x+a 3 cos3x+...+b 1 sinx+b 2 sin2x+b 3 sin3x+...,

where a o, a 1,a 2,...,b 1,b 2,.. are real constants, i.e.

Where for the range from -π to π the coefficients Fourier series are calculated using the formulas:

The coefficients a o , a n and b n are called Fourier coefficients, and if they can be found, then series (1) is called near Fourier, corresponding to the function f(x). For series (1), the term (a 1 cosx+b 1 sinx) is called the first or fundamental harmonic,

Another way to write a series is to use the relation acosx+bsinx=csin(x+α)

f(x)=a o +c 1 sin(x+α 1)+c 2 sin(2x+α 2)+...+c n sin(nx+α n)

Where a o is a constant, c 1 =(a 1 2 +b 1 2) 1/2, c n =(a n 2 +b n 2) 1/2 are the amplitudes of the various components, and is equal to a n =arctg a n /b n.

For series (1), the term (a 1 cosx+b 1 sinx) or c 1 sin(x+α 1) is called the first or fundamental harmonic, (a 2 cos2x+b 2 sin2x) or c 2 sin(2x+α 2) called the second harmonic and so on.

To accurately represent a complex signal typically requires an infinite number of terms. However, in many practical problems it is enough to consider only the first few terms.

Fourier series of non-periodic functions with period 2π.

Expansion of non-periodic functions.

If the function f(x) is non-periodic, it means that it cannot be expanded into a Fourier series for all values ​​of x. However, it is possible to define a Fourier series representing a function over any range of width 2π.

Given a non-periodic function, a new function can be constructed by selecting values ​​of f(x) within a certain range and repeating them outside that range at 2π intervals. Since the new function is periodic with period 2π, it can be expanded into a Fourier series for all values ​​of x. For example, the function f(x)=x is not periodic. However, if it is necessary to expand it into a Fourier series in the interval from o to 2π, then outside this interval a periodic function with a period of 2π is constructed (as shown in the figure below).

For non-periodic functions such as f(x)=x, the sum of the Fourier series is equal to the value of f(x) at all points in a given range, but it is not equal to f(x) for points outside the range. To find the Fourier series of a non-periodic function in the 2π range, the same formula of Fourier coefficients is used.

Even and odd functions.

They say a function y=f(x) is even if f(-x)=f(x) for all values ​​of x. Graphs of even functions are always symmetrical about the y-axis (that is, they are mirror images). Two examples of even functions: y=x2 and y=cosx.

A function y=f(x) is said to be odd if f(-x)=-f(x) for all values ​​of x. Graphs of odd functions are always symmetrical about the origin.

Many functions are neither even nor odd.

Fourier series expansion in cosines.

The Fourier series of an even periodic function f(x) with period 2π contains only cosine terms (i.e., no sine terms) and may include a constant term. Hence,

where are the coefficients of the Fourier series,

The Fourier series of an odd periodic function f(x) with period 2π contains only terms with sines (that is, it does not contain terms with cosines).

Hence,

where are the coefficients of the Fourier series,

Fourier series at half cycle.

If a function is defined for a range, say from 0 to π, and not just from 0 to 2π, it can be expanded in a series only in sines or only in cosines. The resulting Fourier series is called the half-cycle Fourier series.

If you want to obtain a half-cycle Fourier expansion of the cosines of the function f(x) in the range from 0 to π, then you need to construct an even periodic function. In Fig. Below is the function f(x)=x, built on the interval from x=0 to x=π. Since the even function is symmetrical about the f(x) axis, we draw line AB, as shown in Fig. below. If we assume that outside the considered interval the resulting triangular shape is periodic with a period of 2π, then the final graph looks like this: in Fig. below. Since we need to obtain the Fourier expansion in cosines, as before, we calculate the Fourier coefficients a o and a n

If you want to obtain a half-cycle Fourier expansion in terms of the sines of the function f(x) in the range from 0 to π, then you need to construct an odd periodic function. In Fig. Below is the function f(x)=x, built on the interval from x=0 to x=π. Because the odd function symmetrical about the origin, we construct line CD, as shown in Fig. If we assume that outside the considered interval the resulting sawtooth signal is periodic with a period of 2π, then the final graph has the form shown in Fig. Since we need to obtain the Fourier expansion of the half-cycle in terms of sines, as before, we calculate the Fourier coefficient. b

Fourier series for an arbitrary interval.

Expansion of a periodic function with period L.

The periodic function f(x) repeats as x increases by L, i.e. f(x+L)=f(x). The transition from the previously considered functions with a period of 2π to functions with a period of L is quite simple, since it can be done using a change of variable.

To find the Fourier series of the function f(x) in the range -L/2≤x≤L/2, we introduce a new variable u so that the function f(x) has a period of 2π relative to u. If u=2πx/L, then x=-L/2 for u=-π and x=L/2 for u=π. Also let f(x)=f(Lu/2π)=F(u). The Fourier series F(u) has the form

(The limits of integration can be replaced by any interval of length L, for example, from 0 to L)

Fourier series on a half-cycle for functions specified in the interval L≠2π.

For the substitution u=πх/L, the interval from x=0 to x=L corresponds to the interval from u=0 to u=π. Consequently, the function can be expanded into a series only in cosines or only in sines, i.e. into a Fourier series at half cycle.

The cosine expansion in the range from 0 to L has the form

Ministry of General and Vocational Education

Sochi State University tourism

and resort business

Pedagogical Institute

Faculty of Mathematics

Department of General Mathematics

GRADUATE WORK

Fourier series and their applications

In mathematical physics.

Completed by: 5th year student

signature of full-time education

Specialty 010100

"Mathematics"

Kasperova N.S.

Student ID No. 95471

Scientific supervisor: associate professor, candidate.

technical signature sciences

Pozin P.A.

Sochi, 2000

1. Introduction.

2. The concept of a Fourier series.

2.1. Determination of Fourier series coefficients.

2.2. Integrals of periodic functions.

3. Signs of convergence of Fourier series.

3.1. Examples of expansion of functions in Fourier series.

4. A note on the Fourier series expansion of a periodic function

5. Fourier series for even and odd functions.

6. Fourier series for functions with period 2 l .

7. Fourier series expansion of a non-periodic function.

Introduction.

Jean Baptiste Joseph Fourier - French mathematician, member of the Paris Academy of Sciences (1817).

Fourier's first works related to algebra. Already in lectures of 1796 he presented a theorem on the number of real roots algebraic equation lying between these borders (published 1820), named after him; complete solution the number of real roots of an algebraic equation was obtained in 1829 by J.S.F. By assault. In 1818, Fourier investigated the question of the conditions for the applicability of the method of numerical solution of equations developed by Newton, not knowing about similar results obtained in 1768 by the French mathematician J.R. Murailem. The result of Fourier's work on numerical methods for solving equations is “Analysis of Definite Equations,” published posthumously in 1831.

Fourier's main area of ​​study was mathematical physics. In 1807 and 1811 he presented to the Paris Academy of Sciences his first discoveries on the theory of heat propagation in solid body, and in 1822 published famous work“Analytical Theory of Heat”, which played a major role in the subsequent history of mathematics. This - mathematical theory thermal conductivity. Due to the generality of the method, this book became the source of all modern methods mathematical physics. In this work, Fourier derived differential equation thermal conductivity and developed ideas in the most general outline outlined earlier by D. Bernoulli, developed a method for separating variables (Fourier method) to solve the heat equation under certain given boundary conditions, which he applied to a number of special cases (cube, cylinder, etc.). This method is based on the representation of functions by trigonometric Fourier series.

Fourier series have now become a well-developed tool in the theory of partial differential equations for solving boundary value problems.

1. The concept of a Fourier series. (p. 94, Uvarenkov)

Fourier series play an important role in mathematical physics, elasticity theory, electrical engineering, and especially their special case– trigonometric Fourier series.

A trigonometric series is a series of the form

or, symbolically:

(1)

where ω, a 0 , a 1 , …, a n , …, b 0 , b 1 , …, b n , …- constant numbers (ω>0) .

Historically, certain problems in physics have led to the study of such series, for example, the problem of string vibrations (18th century), the problem of regularities in the phenomena of heat conduction, etc. In applications, consideration of trigonometric series , is primarily associated with the task of representing a given movement, described by the equation y = ƒ(χ), in

as the sum of the simplest harmonic vibrations, often taken in an infinitely large number, i.e., as the sum of a series of the form (1).

Thus, we come to the following problem: to find out whether for a given function ƒ(x) on a given interval there exists a series (1) that would converge on this interval to this function. If this is possible, then they say that on this interval the function ƒ(x) is expanded into a trigonometric series.

Series (1) converges at some point x 0, due to the periodicity of the functions

(n=1,2,..), it will turn out to be convergent at all points of the form (m is any integer), and thus its sum S(x) will be (in the region of convergence of the series) a periodic function: if S n ( x) – nth partial the sum of this series, then we have

and therefore

, i.e. S(x 0 +T)=S(x 0). Therefore, speaking about the expansion of some function ƒ(x) into a series of the form (1), we will assume ƒ(x) to be a periodic function.

2. Determination of series coefficients using Fourier formulas.

Let a periodic function ƒ(x) with period 2π be such that it is represented by a trigonometric series converging to a given function in the interval (-π, π), i.e., is the sum of this series:

. (2)

Let us assume that the integral of the function on the left side of this equality is equal to the sum of the integrals of the terms of this series. This will be true if we assume that the number series composed of the coefficients of a given trigonometric series is absolutely convergent, i.e., the positive number series converges

(3)

Series (1) is majorizable and can be integrated term by term in the interval (-π, π). Let's integrate both sides of equality (2):

.

Let us separately evaluate each integral appearing on the right-hand side:

, , .

Thus,

, where . (4)

Estimation of Fourier coefficients. (Bugrov)

Theorem 1. Let the function ƒ(x) of period 2π have a continuous derivative ƒ ( s) (x) order s, satisfying the inequality on the entire real axis:

│ ƒ (s) (x)│≤ M s ; (5)

then the Fourier coefficients of the function ƒ satisfy the inequality

(6)

Proof. Integrating by parts and taking into account that

ƒ(-π) = ƒ(π), we have

Integrating the right-hand side of (7) sequentially, taking into account that the derivatives ƒ ΄ , …, ƒ (s-1) are continuous and take same values at points t = -π and t = π, as well as estimate (5), we obtain the first estimate (6).

The second estimate (6) is obtained in a similar way.

Theorem 2. For the Fourier coefficients ƒ(x) the following inequality holds:

(8)

Proof. We have

The Fourier series of an even periodic function f(x) with period 2p contains only terms with cosines (i.e., does not contain terms with sines) and may include a constant term. Hence,

where are the coefficients of the Fourier series,

Fourier series expansion in sines

The Fourier series of an odd periodic function f (x) with period 2p contains only terms with sines (that is, it does not contain terms with cosines).

Hence,

where are the coefficients of the Fourier series,

Fourier series at half cycle

If a function is defined for a range, say from 0 to p, and not just from 0 to 2p, it can be expanded into a series only in sines or only in cosines. The resulting Fourier series is called the half-cycle Fourier series.

If you want to obtain a half-cycle Fourier expansion of the cosines of the function f (x) in the range from 0 to p, then you need to construct an even periodic function. In Fig. Below is the function f (x) = x, built on the interval from x = 0 to x = p. Since the even function is symmetrical about the f (x) axis, we draw line AB, as shown in Fig. below. If we assume that outside the considered interval the resulting triangular shape is periodic with a period of 2p, then the final graph looks like this: in Fig. below. Since we need to obtain the Fourier expansion in cosines, as before, we calculate the Fourier coefficients a o and a n

If you want to obtain the Fourier expansion on a half-cycle in terms of the sines of the function f (x) in the range from 0 to p, then you need to construct an odd periodic function. In Fig. Below is the function f (x) =x, built on the interval from x=0 to x=p. Since the odd function is symmetrical about the origin, we construct the line CD, as shown in Fig.

If we assume that outside the considered interval the resulting sawtooth signal is periodic with a period of 2p, then the final graph has the form shown in Fig. Since we need to obtain the Fourier expansion of the half-cycle in terms of sines, as before, we calculate the Fourier coefficient. b

Fourier series of periodic functions with period 2π.

The Fourier series allows us to study periodic functions by decomposing them into components. Alternating currents and voltages, displacements, speed and acceleration of crank mechanisms and acoustic waves are typical practical examples of the use of periodic functions in engineering calculations.

The Fourier series expansion is based on the assumption that all functions of practical significance in the interval -π ≤x≤ π can be expressed in the form of convergent trigonometric series (a series is considered convergent if the sequence of partial sums composed of its terms converges):

Standard (=ordinary) notation through the sum of sinx and cosx

f(x)=a o + a 1 cosx+a 2 cos2x+a 3 cos3x+...+b 1 sinx+b 2 sin2x+b 3 sin3x+...,

where a o, a 1,a 2,...,b 1,b 2,.. are real constants, i.e.

Where, for the range from -π to π, the coefficients of the Fourier series are calculated using the formulas:

The coefficients a o , a n and b n are called Fourier coefficients, and if they can be found, then series (1) is called the Fourier series corresponding to the function f (x). For series (1), the term (a 1 cosx+b 1 sinx) is called the first or fundamental harmonic,

Another way to write a series is to use the relation acosx+bsinx=csin(x+α)

f(x)=a o +c 1 sin(x+α 1)+c 2 sin(2x+α 2)+...+c n sin(nx+α n)

Where a o is a constant, c 1 =(a 1 2 +b 1 2) 1/2, c n =(a n 2 +b n 2) 1/2 are the amplitudes of the various components, and is equal to a n =arctg a n /b n.

For series (1), the term (a 1 cosx+b 1 sinx) or c 1 sin(x+α 1) is called the first or fundamental harmonic, (a 2 cos2x+b 2 sin2x) or c 2 sin(2x+α 2) called the second harmonic and so on.

To accurately represent a complex signal typically requires an infinite number of terms. However, in many practical problems it is sufficient to consider only the first few terms.

Fourier series of non-periodic functions with period 2π.

Expansion of non-periodic functions.

If the function f(x) is non-periodic, it means that it cannot be expanded into a Fourier series for all values ​​of x. However, it is possible to define a Fourier series representing a function over any range of width 2π.

Given a non-periodic function, a new function can be constructed by selecting values ​​of f(x) within a certain range and repeating them outside that range at 2π intervals. Since the new function is periodic with period 2π, it can be expanded into a Fourier series for all values ​​of x. For example, the function f(x)=x is not periodic. However, if it is necessary to expand it into a Fourier series in the interval from o to 2π, then outside this interval a periodic function with a period of 2π is constructed (as shown in the figure below).

For non-periodic functions such as f(x)=x, the sum of the Fourier series is equal to the value of f(x) at all points in a given range, but it is not equal to f(x) for points outside the range. To find the Fourier series of a non-periodic function in the 2π range, the same formula of Fourier coefficients is used.

Even and odd functions.

They say a function y=f(x) is even if f(-x)=f(x) for all values ​​of x. Graphs of even functions are always symmetrical about the y-axis (that is, they are mirror images). Two examples of even functions: y=x2 and y=cosx.

A function y=f(x) is said to be odd if f(-x)=-f(x) for all values ​​of x. Graphs of odd functions are always symmetrical about the origin.

Many functions are neither even nor odd.

Fourier series expansion in cosines.

The Fourier series of an even periodic function f(x) with period 2π contains only cosine terms (i.e., no sine terms) and may include a constant term. Hence,

where are the coefficients of the Fourier series,

The Fourier series of an odd periodic function f(x) with period 2π contains only terms with sines (that is, it does not contain terms with cosines).

Hence,

where are the coefficients of the Fourier series,

Fourier series at half cycle.

If a function is defined for a range, say from 0 to π, and not just from 0 to 2π, it can be expanded in a series only in sines or only in cosines. The resulting Fourier series is called the half-cycle Fourier series.

If you want to obtain a half-cycle Fourier expansion of the cosines of the function f(x) in the range from 0 to π, then you need to construct an even periodic function. In Fig. Below is the function f(x)=x, built on the interval from x=0 to x=π. Since the even function is symmetrical about the f(x) axis, we draw line AB, as shown in Fig. below. If we assume that outside the considered interval the resulting triangular shape is periodic with a period of 2π, then the final graph looks like this: in Fig. below. Since we need to obtain the Fourier expansion in cosines, as before, we calculate the Fourier coefficients a o and a n

If you want to obtain a half-cycle Fourier expansion in terms of the sines of the function f(x) in the range from 0 to π, then you need to construct an odd periodic function. In Fig. Below is the function f(x)=x, built on the interval from x=0 to x=π. Since the odd function is symmetrical about the origin, we construct the line CD, as shown in Fig. If we assume that outside the considered interval the resulting sawtooth signal is periodic with a period of 2π, then the final graph has the form shown in Fig. Since we need to obtain the Fourier expansion of the half-cycle in terms of sines, as before, we calculate the Fourier coefficient. b

Fourier series for an arbitrary interval.

Expansion of a periodic function with period L.

The periodic function f(x) repeats as x increases by L, i.e. f(x+L)=f(x). The transition from the previously considered functions with a period of 2π to functions with a period of L is quite simple, since it can be done using a change of variable.

To find the Fourier series of the function f(x) in the range -L/2≤x≤L/2, we introduce a new variable u so that the function f(x) has a period of 2π relative to u. If u=2πx/L, then x=-L/2 for u=-π and x=L/2 for u=π. Also let f(x)=f(Lu/2π)=F(u). The Fourier series F(u) has the form

(The limits of integration can be replaced by any interval of length L, for example, from 0 to L)

Fourier series on a half-cycle for functions specified in the interval L≠2π.

For the substitution u=πх/L, the interval from x=0 to x=L corresponds to the interval from u=0 to u=π. Consequently, the function can be expanded into a series only in cosines or only in sines, i.e. into a Fourier series at half cycle.

The cosine expansion in the range from 0 to L has the form