Mathematics Term Paper

LOVELYPROFESSIONALUNIVERSITY

MATHEMATICS TERM PAPER

TOPIC:-What are the Euler’s formula for the fourier coefficient? By what ideas did we get them?

Submitted by:-

Abhishek Pandey

RB40903A20

B4903

10906516

Submitted to:-

Mr. Gurpreet Singh Bhatia

(Dept. of Mathematics)

ACKNOWLEDGEMENT

THIS REPORT HAS BEEN MADE VERY CAREFULLY IN GUIDENCE OF Mr.GURPREET SINGH. I WHOLE HEARTEDLY THANK MY MATHS PROFESSOR FOR HIS KIND CO-OPERATION

IN CLASS TO MAKE UNDERSTAND THE GIVEN TOPIC FOR THE TERM PAPER .I ALSO THANK LOVELYPROFESSIONALUNIVERSITY THAT MADE US TO NOURISH OUR TALENT AND BRAIN TOO. AT LAST I WOULD LIKE TO THANK ALL MY FRIENDS WHO HELPED ME A LOT TO MAKE MY REPORT WORTH GIVING

Introduction:-

Inmathematics, aFourier seriesdecomposes aperiodic functionor periodic signal into a sum of simple oscillating functions, namelysins and cosines(orcomplex exponentials). The study of Fourier series is a branch ofFourier analysis. Fourier series were introduced byJoseph Fourier(1768–1830) for the purpose of solving the heatin a metal plate.

The heat equation is apartial differential equation. Prior to Fourier's work, there was no known solution to the heat equation in a general situation, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was asineorcosinewave. These simple solutions are now sometimes calledexigent solutions. Fourier's idea was to model a complicated heat source as a superposition (orlinear combination) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding exigentsolutions. This superposition or linear combination is called the Fourier series.

Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems.

The Fourier series has many applications inelectrical engineering,vibrationanalysis,acoustics,optics,signal processing,image processing,mechanics, econometrics[1], etc.

Fourier series is named in honour of Joseph Fourier (1768-1830), who made important contributions to the study oftrigonometric series, after preliminary investigations byLeonard Euler,Jean le Ronda d'Alembert, andDaniel Bernoulli. He applied this technique to find the solution of the heat equation, publishing his initial results in his 1807Mémoire sur la propagation de la chaleur dans les corps solidesand 1811, and publishing hisThéorie analytique de la chaleurin 1822.

From a modern point of view, Fourier's results are somewhat informal, due to the lack of a precise notion offunctionandintegralin the early nineteenth century. Later,DirichletandRiemannexpressed Fourier's results with greater precision and formality.

History:-

In this section,ƒ(x) denotes a function of the real variablex. This function is usually taken to beperiodic, of period 2π, which is to say thatƒ(x+2π)=ƒ(x), for all real numbersx. We will attempt to write such a function as an infinite sum, orseriesof simpler 2π–periodic functions. We will start by using an infinite sum ofsineandcosinefunctions on the interval [−π,π], as Fourier did (see the quote above), and we will then discuss different formulations and generalizations.

Fourier's formula for 2π-periodic functions using sines and cosines

For a periodic functionƒ(x) that is integrable on [−π,π], the numbers

and

are called the Fourier coefficients ofƒ. One introduces thepartial sums of the Fourier seriesforƒ, often denoted by

The partial sums forƒaretrigonometric polynomials. One expects that the functionsSNƒapproximate the functionƒ, and that the approximation improves asNtends to infinity. Theinfinite sum

is called theFourier seriesofƒ.

The Fourier series does not always converge, and even when it does converge for a specific valuex0ofx, the sum of the series atx0may differ from the valueƒ(x0) of the function. It is one of the main questions inharmonic analysisto decide when Fourier series converge, and when the sum is equal to the original function. If a function issquare-integrableon the interval [−π,π], then the Fourier series converges to the function atalmost everypoint. Inengineeringapplications, the Fourier series is generally presumed to converge everywhere except at discontinuities, since the functions encountered in engineering are more well behaved than the ones that mathematicians can provide as counter-examples to thispresumption. In particular, the Fourier series converges absolutely and uniformly toƒ(x) whenever the derivative ofƒ(x) .

Euler’s formula:-

The development ofinfinitesimal calculuswas at the forefront of 18th century mathematical research, and theBernoulli’s—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards ofmathematical rigor,[27]his ideas led to many great advances. Euler is well-known inanalysisfor his frequent use and development ofpower series, the expression of functions as sums of infinitely many terms, such as

Notably, Euler directly proved the power series expansions foreand theinverse tangentfunction. (Indirect proof via the inverse power series technique was given by Newton and Leibniz between 1670 and 1680.) His daring use of power series enabled him to solve the famousBasel problemin 1735 (he provided a more elaborate argument in 1741):[27]

A geometric interpretation of Euler's formula

Euler introduced the use of theexponential functionandlogarithmsin analytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative andcomplex numbers, thus greatly expanding the scope of mathematical applications of logarithms.[25]He also defined the exponential function for complex numbers, and discovered its relation to thetrigonometric functions. For anyreal numberφ,formula states that thecomplex exponentialfunction satisfies

A special case of the above formula is known asEuler's identity,

Called "the most remarkable formula in mathematics" byRichard Feynman, for its single uses of the notions of addition, multiplication, exponentiation, and equality, and the single uses of the important constants 0, 1,e,iand π.In 1988, readers of theMathematical Intelligencervoted it "the Most Beautiful Mathematical Formula Ever".In total, Euler was responsible for three of the top five formulae in that poll.

De Moiré’s formulais a direct consequence ofEuler's formula.

In addition, Euler elaborated the theory of highertranscendental functionsby introducing thegamma functionand introduced a new method for solvingquadratic equations. He also found a way to calculate integrals with complex limits, foreshadowing the development of moderncomplex analysis, and invented the calculus of variationsincluding its best-known result, theEuler–Lagrange equation.

Euler also pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study,analytic number theory. In breaking ground for this new field, Euler created the theory ofhyper geometric series,series, hyperbolicand the analytic theory ofcontinued fractions. For example, he proved theinfinitude of primesusing the divergence of the harmonic, and he used analytic methods to gain some understanding of the wayprime numbersare distributed. Euler's work in this area led to the development of theprime number theorem.

Applications:-

We now use the formula above to give a Fourier series expansion of a very simple function. Consider a saw tooth wave

In this case, the Fourier coefficients are given by

It can be proved that the Fourier series converges toƒ(x) at every pointxwhereƒis differentiable, and therefore:

/
/ (Eq.1)

Whenx= π, the Fourier series converges to 0, which is the half-sum of the left- and right-limit ofƒatx= π. This is a particular instance of theDirichlet theoremfor Fourier series.

Notices that the Fourier series expansion of our function looks much less simple than the formulaƒ(x) =x, and so it is not immediately apparent why one would need this Fourier series. While there are many applications, we cite Fourier's motivation of solving the heat equation. For example, consider a metal plate in the shape of a square whose side measuresπmeters, with coordinates (x,y) ∈ [0,π]×[0,π]. If there is no heat source within the plate, and if three of the four sides are held at 0 degreesCelsius, while the fourth side, given byy= π, is maintained at the temperature gradientT(x,π) =xdegrees Celsius, forxin (0,π), then one can show that the stationary heat distribution (or the heat distribution after a long period of time has elapsed) is given by

Here, sinh is thehyperbolic sinefunction. This solution of the heat equation is obtained by multiplying each term of Eq.1by sinh(ny)/sinh(nπ). While our example functionf(x) seems to have a needlessly complicated Fourier series, the heat distributionT(x,y) is nontrivial. The functionTcannot be written as aclosed-form expression. This method of solving the heat problem was only made possible by Fourier's work.

Another application of this Fourier series is to solve theBasel problemby usingParseval's theorem. The example generalizes and one may computeζ(2n), for any positive integern.

Exponential Fourier series:-

We can useEuler's formula,

whereiis theimaginary unit, to give a more concise formula:

The Fourier coefficients are then given by:

The Fourier coefficientsan,bn,cnare related via

The notationcnis inadequate for discussing the Fourier coefficients of several different functions. Therefore it is customarily replaced by a modified form ofƒ(in this case), such asFor and functional notation often replaces subscripting. Thus:

In engineering, particularly when the variablexrepresents time, the coefficient sequence is called afrequency domainrepresentation. Square brackets are often used to emphasize that the domain of this function is a discrete set of frequencies

CalculatingFourierSeries:-

In this section we calculate several Fourier series. As we know, to find a Fourier series simply means calculating various integrals, which can often be done with software or with integral tables. Since performing integrals is not much more interesting in the modern age than long division, our goals in this section will be to get a visual and analytic impression of what to expect from Fourier series, and to understand the rôle of symmetry in the calculations.

Let's begin by evaluating the Fourier series for thefunctions:

f(x) = 1 for 0x < L/2, but 0 for L/2xL

and g(x) = x, 0x < L.

The functions have not been defined at the points of discontinuity, but as we know, the Fourier series will converge there to the average of the limit from the left and the limit from the right. The end point L is essentially a jump point, because the periodic extension of the functions make the values x=L and x=0 equivalent.

Here is a graph of the function f, called a "square pulse" or "square wave" (when

extended periodically):

The length L has been chosen as.

We want to represent these functions in the form beginning with f(x).

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