# Fourier series

In analysis, the Fourier series are a fundamental tool in the study of the periodic functions. It is starting from this concept that the branch of mathematics known under the name of developed analyzes harmonic.

The study of a periodic function by Fourier series comprises two aspects:

• the analyzes , which consists of the determination of the continuation of its coefficients of Fourier;
• the synthesis , which makes it possible to find, in a certain direction, the function using the continuation of its coefficients.

Beyond the problem of the decomposition, the theory of Fourier series establishes a correspondence between the periodic function and the coefficients of Fourier. So the analysis of Fourier can be regarded as a new way of describing the periodic functions. Operations such as derivation are written simply in terms of coefficients of Fourier. The construction of a periodic function solution of a functional equation can be reduced to construction corresponding coefficients of Fourier.

Fourier series were introduced by Joseph Fourier in 1822, but it was necessary one century so that the analysts give off the adapted tools of study: a theory of the integral fully satisfactory and the first concepts of the analyzes functional. They are still currently the subject of active research for themselves, and caused several new branches: analyzes harmonic, theory of the signal, Ondelette S, etc

Fourier series usually meet in the decomposition of periodic signals, the study of the electric currents, the cerebral waves, in the sound Synthèse, the Image processing, etc

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## Intuitive approach

A function F of a real variable is known as periodic of period T when it checks: for any reality X , $f \left(x+T\right) =f \left(X\right)$. The frequency F is the reverse of the period: F=1/T .

The periodic functions of period T easiest to study are the sinusoidal functions whose frequency is a multiple of F

$x \ mapsto \ cos \left(2 \ pi \ cdot N \ cdot F \ cdot X\right)$ and $x \ mapsto \ sin \left(2 \ pi \ cdot N \ cdot F \ cdot X\right)$

### Trigonometrical polynomials

A linear Combinaison of these elementary sinusoidal functions bears the trigonometrical name of Polynôme and constitutes also a periodic function of period T . It can rewrite like linear combination of functions $x \ mapsto e^ \left\{2in \ pi F X\right\}$, the use of the complex numbers and the function Exponentielle making it possible to simplify the notations.

A trigonometrical polynomial P is thus written in the form:

$P \left(X\right) = \ sum_ \left\{n=- \ infty\right\} ^ \left\{+ \ infty\right\} c_n \left(P\right) \ exp \ left \left(I \ frac \left\{2n \ pi\right\} \left\{T\right\} X \ right\right),$
where the coefficients cn (P) are almost all null, and can be obtained by the following formula:
$\ frac \left\{1\right\} \left\{T\right\} \ int_ \left\{- T/2\right\} ^ \left\{T/2\right\} P \left(T\right) e^ \left\{- I \ frac \left\{2n \ pi\right\} \left\{T\right\} T\right\} \, dt=c_n \left(P\right) \ qquad \left(1\right)$.

### Principle of Fourier series

For reasons of dimension, one cannot obtain all the periodic functions of period T like such a combination.

The idea subjacent with the introduction of Fourier series is to be able to obtain a function T - periodic, for example continues, like summons sinusoidal functions:

$f \left(X\right) = \ sum_ \left\{n=- \ infty\right\} ^ \left\{+ \ infty\right\} c_n \left(F\right) \ exp \ left \left(I \ frac \left\{2n \ pi\right\} \left\{T\right\} X \ right\right)$
with the coefficients C N (F), called coefficients of Fourier of F , defined by the formula:
$c_n \left(F\right) = \ frac \left\{1\right\} \left\{T\right\} \ int_ \left\{- T/2\right\} ^ \left\{T/2\right\} F \left(T\right) e^ \left\{- I \ frac \left\{2n \ pi\right\} \left\{T\right\} T\right\} \, dt$.

It is this time about a true infinite sum, i.e. of a limit of finished sum, which corresponds to the concept of nap of series.

Many calculations are translated in a very simple way on the coefficients of the trigonometrical polynomials, like the calculation of derivative. It is possible to generalize them on the level of the general coefficients of Fourier.

In a strict sense, the formula of decomposition is not correct in general. It is it, punctually, under good assumptions of regularity relating to F . Alternatively, one can give him direction while placing oneself in good functional spaces.

## Historical aspects

Fourier series constitute the oldest branch of the analyzes harmonic, but do not remain about it less one alive field, with the many open-ended questions. The study of their characteristics went hand in hand, during all the 19th century, with progress of the theory of the integration.

### Origins

The first considerations on the trigonometrical series appear about 1400 in India, at Madhava, leader of the school of Kerala. In Occident, one finds them at the 17th century at James Gregory, at the beginning of the 18th century at Brook Taylor. It is the work of the latter, Methodus Incrementorum Directa and Inversa , appeared in 1715, which gives the kickoff to the systematic study of the vibrating cords and the propagation of the sound, major research topic during all the century.

A controversy bursts in the years 1750 between of Alembert, Euler and Daniel Bernoulli on the problem of the vibrating cords. D' Alembert determines the equation of wave and its analytical solutions. Bernoulli also obtains them, in the form of decomposition in trigonometrical series. The controversy relates to the need for reconciling these points of view with the questions of regularity of the solutions. According to J. - P. Kahane, it will have an important role in the genesis of Fourier series.

Bernoulli had introduced trigonometrical series into the problem of the vibrating cords to superimpose elementary solutions.

Joseph Fourier (1768 - 1830) introduces the equation of heat into a first report in 1807 which it supplements and present in 1811 for the Grand Prix of Mathematics. This first work, discussed in the field of the analysis, was not published. In 1822, Fourier exposes the series and the transformation of Fourier in his treaty analytical Théorie of heat . It states that a function can be broken up in the form of trigonometrical series, and that it is easy to prove the convergence of this one. He judges even any assumption of useless continuity.

In 1829, Dirichlet (1805 - 1859) gives a first statement correct of convergence limited to the continuous periodic functions per pieces having only one finished number of extrema. Dirichlet considered that the other cases were brought back there; the error will be corrected by Jordan in 1881.

In 1848, Wilbraham is the first to highlight the phenomenon of Gibbs while being interested in the behavior of Fourier series in the vicinity of the points of discontinuity.

### Joint projection of Fourier series and the real analysis

The Memory on the trigonometrical series of Bernhard Riemann (1826 - 1866) , published in 1867, constitutes a decisive projection. The author raises a major hurdle by defining for the first time a theory of the satisfactory integration. He shows in particular that the coefficients of Fourier have a null limit ad infinitum, and a result of convergence known as the theorem of sommability of Riemann.

Georg Cantor (1845 - 1918) publishes a series of articles on the trigonometrical series between 1870 and 1872, where it shows its theorem of unicity. Cantor refines its results by seeking " whole of unicité" , for which its theorem remains checked. It is the origin of the introduction of the Set theory.

In 1873, Of Wood-Reymond (1831 - 1889) gives the first example of function continues periodic from which the Fourier series diverge in a point. The last quarter of the 19th century sees relatively few advanced in the field Fourier series or of the real analysis in general, whereas the Analyze complexes knows a fast progression.

In a note of 1900 and in an article of 1904, Fejér (1880 - 1959) shows its uniform theorem of convergence using the proceeded of summation of Cesàro (mean arithmetic of the sums partial of Fourier). Especially, it works out a new principle: systematic association between regularization by means of a “core” and Proceeded of summation for the Fourier series.

### New tools of study

Henri Lebesgue (1875 - 1941) gives to the theory Fourier series his final framework by introducing a news theory of integration. In a series of publications which are spread out 1902 with 1910, it extends the theorems of its predecessors, in particular the theorem of Riemann on the limit of Fourier series. It also proves several new theorems of convergence. The majority of its results appear in its Leçons on the trigonometrical series published in 1906.

In 1907, Pierre Fatou (1878 - 1929) shows the equality of Parseval within the general framework of the functions of summable square. The same year, Frigyes Riesz (1880 - 1926) and Ernst Fischer (1875 - 1954) , independently, proves the reciprocal one. These results take part in the birth of a new field, the analyzes functional.

Henceforth, the questions of convergence in functional spaces are considered through the study of the properties of the continuations of cores and the associated operators. Most of the results passes by questions of estimate of standards called " constants of Lebesgue" , which becomes a systematic object of study.

In parallel, the problem of the simple convergence of Fourier series gives place to several dramatic turns of events with the publication of results which knew a great repercussion and surprised the contemporaries. In 1926, Andreï Kolmogorov (1903 - 1987) builds an integrable example of function from which the Fourier series diverge everywhere. In 1966, Lennart Carleson (1928) establishes on the contrary that the Fourier series of a summable function of square converge almost everywhere towards this function. Other results (Kahane and Katznelson 1966, Hunt 1967) come to supplement the study. Research goes then on the convergence of Fourier series to several dimensions, still imperfectly known.

## Coefficients of Fourier

The definition of the coefficients of Fourier carries on the periodic functions integrable within the meaning of Lebesgue over one period. For a periodic function, being of class $L^p$ implies the integrability. This includes/understands in particular the continuous, or continuous functions by pieces, periodicals. Here the notations of the first paragraph are taken again.

### Complex coefficients

The coefficients of Fourier (complex) of F (for $n \ in \ Z$) are given by:

$c_n \left(F\right) = \ frac \left\{1\right\} \left\{T\right\} \ int_ \left\{- T/2\right\} ^ \left\{T/2\right\} F \left(T\right) e^ \left\{- I \ frac \left\{2n \ pi\right\} \left\{T\right\} T\right\} \, dt \,$.
By periodicity of the intégrande, these coefficients can also be calculated by considering the integral on any segment length T . For n=0, the coefficient $c_0 \left(F\right)$ is not other than the median value of F .

If n>0, one calls Harmonique row N of the function F the sinusoidal function of frequency N F obtained by taking account of the coefficients of Fourier of index N and - N, given by:

$x \ mapsto c_n \left(F\right) e^ \left\{I \ frac \left\{2n \ pi\right\} \left\{T\right\} X\right\} + c_ \left\{- N\right\} \left(F\right) e^ \left\{- I \ frac \left\{2n \ pi\right\} \left\{T\right\} X\right\}$.

The Fourier series of F are the series of functions obtained by summoning the successive harmonics, in other words the Série of functions:

$S_n \left(F\right) = \ sum_ \left\{k=-n\right\} ^ \left\{k=n\right\} c_k \left(F\right) e^ \left\{I \ frac \left\{2k \ pi\right\} \left\{T\right\} X\right\} \,$.
One of the questions which the theory of Fourier answers is to determine the mode of convergence of this series (specific convergence, uniform Convergence, quadratic Convergence,…).

### Real coefficients

If the function F is with actual values, it can be interesting to handle real coefficients, in particular in the case of even or odd functions. One thus defines the real coefficients of Fourier of F :

• $a_0 \left(F\right) = \ frac \left\{1\right\} \left\{T\right\} \ int_ \left\{- T/2\right\} ^ \left\{T/2\right\} F \left(T\right) \, dt=c_0$;
• $b_0 \left(F\right) =0$;
• For N >0, $a_n \left(F\right) = \ frac \left\{2\right\} \left\{T\right\} \ int_ \left\{- T/2\right\} ^ \left\{T/2\right\} F \left(T\right) \ cos \ left \left(NT \ frac \left\{2 \ pi\right\} \left\{T\right\} \ right\right) \, dt$;
• For N >0, $b_n \left(F\right) = \ frac \left\{2\right\} \left\{T\right\} \ int_ \left\{- T/2\right\} ^ \left\{T/2\right\} F \left(T\right) \ sin \ left \left(NT \ frac \left\{2 \ pi\right\} \left\{T\right\} \ right\right) \, dt$.

There still, the periodicity authorizes to change the interval of integration.

The Harmonique of row N is rewritten then like the function:

$x \ mapsto a_n \left(F\right) \ cos \ left \left(nx \ frac \left\{2 \ pi\right\} \left\{T\right\} \ right\right) + b_n \left(F\right) \ sin \ left \left(nx \ frac \left\{2 \ pi\right\} \left\{T\right\} \ right\right) = \ chi_n \ cos \ left \left(nx \ frac \left\{2 \ pi\right\} \left\{T\right\} + \ Phi_n \ right\right) \,$,
where $\ chi_n$ and $\ Phi_n$ modulo 2$\ pi$ explicitly depends on the $a_n \left(F\right)$ and $b_n \left(F\right)$.

Following convention can also be selected for a0 : $a_0 \left(F\right) = \ frac \left\{2\right\} \left\{T\right\} \ int_ \left\{- T/2\right\} ^ \left\{T/2\right\} F \left(T\right) \, dt$, what is not interpreted then any more like a median value, but is the double. This last convention harmonizes the definitions of the coefficients which then begin all with $2/T$.

The systems of coefficients ( has N , B N ), for positive N , and C N , for relative N whole are linearly dependant by the following relations:

• $\ forall N \ geq 0$, $c_ \left\{\ pm N\right\} \left(F\right) = \ frac12 \left(a_n \left(F\right) \ mp i.b_n \left(F\right)\right)$;
• For $n>0$, $a_n \left(F\right) =c_n \left(F\right) +c_ \left\{- N\right\} \left(F\right)$ and $i.b_n \left(F\right) =c_ \left\{- N\right\} \left(F\right) - c_n \left(F\right)$.
The last identities remain true for $n=0$ under the convention of the coefficient in $2/T$.

The parity of a function is translated on the coefficients of Fourier:

• a function F is even, if and only if c-n (F) =cn (F)   for all N . In the case of a real function F , this property becomes bn (F)   = 0 for all N .
• a function F is odd, if and only if c-n (F) =-cn (F)   for all N . In the case real that gives an (F)   = 0 for all N .

### Equality of Parseval

For a function T - periodic continuous per pieces, or more generally of integrable square over one period, the equality of Parseval affirms the convergence of the following series and the identity:

$\ sum_ \left\{n=- \ infty\right\} ^ \left\{+ \ infty\right\}|c_n \left(F\right)|^2= \ frac \left\{1\right\} \left\{T\right\} \ int_ \left\{- T/2\right\} ^ \left\{T/2\right\}|F \left(T\right)|^2dt = \ frac \left\{1\right\} \left\{T\right\} \ int_0^T|F \left(T\right)|^2dt= \|F \|^2 \;$.
This result is equivalent to a convergence on average quadratic of corresponding Fourier series (see below).

The equality of Parseval implies in particular that the coefficients of Fourier of F tend (sufficient quickly) towards 0 in the infinite one. According to the assumptions of regularity on F , the speed of convergence can be specified (see below).

### Effect of derivation on the coefficients

For a continuous function and $\ mathcal C^1$ per pieces, one establishes, by Intégration by parts:

$c_n \left(F \, \text{'}\right) =2i \ pi N c_n \left(F\right) /T$.
More generally, for a function of class $\ mathcal C^ \left\{K\right\}$ and $\ mathcal C^ \left\{k+1\right\}$ per pieces, one establishes:
$c_n \left(f^ \left\{\left(k+1\right)\right\}\right) = \left(2i \ pi n/T\right) ^ \left\{k+1\right\} c_n \left(F\right)$.

### Coefficients and regularity of the function

The coefficients of Fourier characterize the function: two functions having the same coefficients of Fourier are equal almost everywhere . In particular, in the continuous case per pieces, they coincide in all the points except a finished number.

A certain number of results connect regularity of the function and behavior ad infinitum coefficients of Fourier.

• the Théorème of Riemann-Lebesgue watch that the coefficients of Fourier of a function F integrable over one period tend towards 0 when N tends towards the infinite one.
• the identity of Parseval admits reciprocal: a function is of square summable over one period if and only if the series of the squares of the modules of the coefficients of Fourier converges. It is the Théorème of Riesz-Fischer.
• There exists little of similar characterizations for other functional spaces. One can affirm however that a periodic function is $\left\{\ mathcal C\right\} ^ \ infty$ if and only if its coefficients of Fourier are with fast decrease,
$\ forall p \ in \ mathbb \left\{NR\right\}, \ qquad c_n \left(F\right) =o \ left \left(\ frac1$

## Reconstitution of the functions

One of the key questions of the theory is that of the behavior of the Fourier series of a function and in the event of convergence of the equality of its sum with the function initially considered, this with an aim of being able to replace the study of the function itself by that of its Fourier series, who authorize analytical operations easily easy to handle. Under suitable assumptions of regularity, a periodic function can break up indeed like summons sinusoidal functions.

### Theorem of specific convergence (of Dirichlet)

For a Periodic function F of period T , continues in a reality X , and derivable on the right and on the left in X , the theorem of Dirichlet affirms the convergence of its Fourier series evaluated in X and gives the equality:

$f \left(X\right) = \ sum_ \left\{N = - \ infty\right\} ^ \left\{\ infty\right\} c_n \left(F\right) \ cdot e^ \left\{I nx \ frac \left\{2 \ pi\right\} \left\{T\right\}\right\}$.
If F is with actual values, the equality above is rewritten with the real coefficients of Fourier:
$f \left(X\right) = a_0 \left(F\right) + \ sum_ \left\{N = 1\right\} ^ \left\{\ infty\right\} \ left \left(a_n \left(F\right) \ cdot \ cos \ left \left(nx \ frac \left\{2 \ pi\right\} \left\{T\right\} \ right\right) + b_n \left(F\right) \ cdot \ sin \ left \left(nx \ frac \left\{2 \ pi\right\} \left\{T\right\} \ right\right) \ right\right)$.

The assumptions can be weakened. The function F can only be continuous on the left and on the right in X and with variation limited on a vicinity of X . In this case, F ( X ) must be replaced by the median value of F in X , that is to say thus the average between its limits on the right and on the left in X . The demonstration of the theorem is based on the fact that the Fourier series are calculated by Produit convolution with a trigonometrical polynomial with the remarkable properties: the Core of Dirichlet.

### Uniform theorem of convergence of Dirichlet

The uniform theorem of convergence of Dirichlet is a total version of the theorem of specific convergence. For a function T - periodic and continuously derivable in the vicinity of a segment I , the Fourier series of F converge uniformly towards F on I .

The demonstration consists in noting that the constants in the estimates of the proof of the theorem of specific convergence can be selected independently of the point of evaluation $x \ in I$.

In particular, the Fourier series of a continuously derivable function and T - periodic, converges uniformly on $\ mathbf R$ towards the function.

### Phenomenon of Gibbs

The phenomenon of Gibbs is an effect edge observed in the vicinity of a discontinuity of the function. To illustrate it, here the representation of the terms of order 10,50,250 of the Fourier series of the function " créneau".

The trigonometrical polynomial N ème term of the Fourier series, $S_n \left(F\right)$, is a continuous function, it is thus normal that it cannot approach uniformly the function crenel which, it, is not it. On one of the zones of " plateau" , apart from a vicinity of discontinuity, however, the Fourier series converge uniformly towards the function (it is indistinguishable on the last graph).

On the level of the point of discontinuity, $S_n$ undergoes strong a Oscillation, a kind of " sursaut". The images let suspect and calculation shows indeed that the amplitude of this start tends towards a constant. Precisely if the function has a discontinuity of amplitude $\ Delta y$, then $S_n$, while remaining continuous, knows a " saut" in ordinate being worth about 18% of more.

Convergence on average quadratic relates to convergence for the square standard:

$\|F \|^2= \ int_ \left\{0\right\} ^T|F \left(T\right)|^2dt$
This standard is defined for example on space E of the functions T - periodic and continuous, or on space F of the functions T - periodic measurable of integrable square identified modulo equality on a negligible Ensemble. The standard comes from the scalar Produit:
$\left(F \ cdot G\right) = \ frac1T \ int_ \left\{- T/2\right\} ^ \left\{T/2\right\} F \left(T\right) \ overline \left\{G \left(T\right)\right\} dt$.
Space E is dense in space F and spaces it normalized F is complete; it can be obtained like supplemented E .

Let us introduce the exponential function complexes index N

$e_n: X \ mapsto e^ \left\{in \ frac \left\{2 \ pi\right\} \left\{T\right\} X\right\}$.
The family $\left(e_n\right)$ form an orthonormal family. This family is in particular free. The space which it generates is the space of the trigonometrical polynomials, subspace of E . N ème coefficient of Fourier of F is the scalar product of F by E N :
In particular, trigonometrical N ème polynomial of F is the orthogonal projection of F on the space generated by $\left(e_k\right) _ \left\{- N \ Leq K \ Leq N\right\}$.
• the family $\left(e_n\right)$ is a Base of Hilbert: the subspace of the trigonometrical polynomials is dense in E and F .

• the Fourier series of a function T - periodic of integrable square over one period converges in names $L^2$ towards the function considered.

A consequence is the equality of Parseval.

### Theorem of Fejér

The theorem of Fejér consists in improving the convergence given by the theorem of uniform convergence of Dirichlet by carrying out a limit of Cesàro of the sums partial of the Fourier series. For a continuous function and T - periodic, one notes:

$S_n \left(F\right) = \ sum_ \left\{K = - N\right\} ^ \left\{N\right\} c_k \left(F\right) \ cdot e^ \left\{I kx \ frac \left\{2 \ pi\right\} \left\{T\right\}\right\}$ then $\ sigma_N \left(F\right) = \ frac1N \ sum_ \left\{K = 0\right\} ^ \left\{N-1\right\} S_N \left(F\right) = \ sum_ \left\{K = - N+1\right\} ^ \left\{N-1\right\} \ frac \left\{N-k\right\} \left\{NR\right\} c_k \left(F\right) \ cdot e^ \left\{I kx \ frac \left\{2 \ pi\right\} \left\{T\right\}\right\}$
The theorem of Fejér affirms that, under the only assumption of continuity, the continuation of the functions $\ sigma_N \left(F\right)$ converges uniformly towards F .

This theorem of Fejér constitutes a possible demonstration of the trigonometrical version of the Théorème of Stone-Weierstrass. It is shown by using the properties of a particular trigonometrical polynomial: the Noyau of Fejér of index N is positive and the continuation of these cores constitutes a Approximation of the identity.

The trigonometrical polynomial $\ sigma_N \left(F\right)$ admits frequencies being spread out $-nf$ with $nf$. For each frequency, the preceding coefficient is modified. The new coefficients tend to give more importance to the small frequencies and to deaden the terms of raised frequency, which makes it possible to smooth the too abrupt behaviors.

### Simple convergence

The positive tests obtained by considering other modes of convergence do not make lose its relevance being studied of simple convergence.

Within the framework of the continuous functions, the theorem of Fejér makes it possible to affirm that if the Fourier series of F converge simply, then she admits for limit the function F . On the other hand considerations of analyzes functional make it possible to prove that there exists a continuous function from which the Fourier series diverge in at least a point: precisely it is about an application of the Théorème of Banach-Steinhaus to the operator of convolution by the function Noyau of Dirichlet. It is also possible to give simple explicit examples of them. It is thus the case of the pi-periodic function $2 \ defined by:\ forall X \ in, \, \, F \left(X\right) = \ sum_ \left\{n=1\right\} ^ \left\{+ \ infty\right\} \ frac1 \left\{n^2\right\} \ sin \ left \left(\ left \left(2^ \left\{n^3\right\} +1 \ right\right) \ frac 2 \ right\right)$

The possible period ranges are known thanks to two complementary theorems.

• On the one hand, according to a theorem of Kahane and Katznelson, for any whole of null Measurement of Lebesgue, one can find a function continues of which the Fourier series diverge in any point from this unit.
• In addition, according to the theorem of Carleson, the Fourier series of a continuous function converges almost everywhere towards this function.

If one widens the framework with the integrable functions over one period,

• the theorem of Kolmogorov ensures that there exists an integrable function from which the Fourier series diverge in any point,

• on the other hand the theorem of Lennart Carleson referred to above was proven within the framework of the functions L2 and has even an extension to the spaces Lp for p>1 . For such functions, the Fourier series of F converge almost everywhere.

## Applications

### Calculations of series

The application of the theorems of Dirichlet and Parseval, previously stated, make it possible to calculate the exact value of the sum of numerical series remarkable, among which:
$\ frac \left\{\ pi^2\right\} \left\{6\right\} = \ frac \left\{1\right\} \left\{1^2\right\} + \ frac \left\{1\right\} \left\{2^2\right\} + \ frac \left\{1\right\} \left\{3^2\right\} + \ frac \left\{1\right\} \left\{4^2\right\} + \ cdots = \ sum_ \left\{n=1\right\} ^ \left\{\ infty\right\} \ frac \left\{1\right\} \left\{n^2\right\}$
$\ frac \left\{\ pi^2\right\} \left\{12\right\} = 1 - \ frac \left\{1\right\} \left\{2^2\right\} + \ cdots + \left(- 1\right) ^ \left\{n+1\right\} \ frac \left\{1\right\} \left\{n^2\right\} + \ cdots= \ sum_ \left\{n=1\right\} ^ \left\{\ infty\right\} \left(- 1\right) ^ \left\{n+1\right\} \ frac \left\{1\right\} \left\{n^2\right\}$
$\ frac \left\{\ pi^2\right\} \left\{8\right\} = \ frac \left\{1\right\} \left\{1^2\right\} + \ frac \left\{1\right\} \left\{3^2\right\} + \ frac \left\{1\right\} \left\{5^2\right\} + \ cdots + \ frac \left\{1\right\} \left\{\left(2n+1\right) ^2\right\} + \ cdots= \ sum_ \left\{k=0\right\} ^ \left\{\ infty\right\} \ frac \left\{1\right\} \left\{\left(2k+1\right) ^2\right\}$
$\ frac \left\{\ pi\right\} \left\{4\right\} = 1 - \ frac \left\{1\right\} \left\{3\right\} + \ frac \left\{1\right\} \left\{5\right\} - \ frac \left\{1\right\} \left\{7\right\} + \ cdots + \ frac \left\{\left(- 1\right) ^n\right\} \left\{2n+1\right\} + \ cdots= \ sum_ \left\{k=0\right\} ^ \left\{\ infty\right\} \ frac \left\{\left(- 1\right) ^k\right\} \left\{2k+1\right\}$ (formula of Leibniz)
$\ frac \left\{\ pi-1\right\} \left\{2\right\} = \ sum_ \left\{n=1\right\} ^ \left\{\ infty\right\} \ frac \left\{\ sin \left(N\right)\right\}\left\{N\right\} = \ sum_ \left\{n=1\right\} ^ \left\{\ infty\right\} \ left \left(\ frac \left\{\ sin \left(N\right)\right\}\left\{N\right\} \ right\right) ^ \left\{2\right\}$
The value of the numerical series $\ sum_ \left\{n=1\right\} ^ \left\{+ \ infty\right\} \ frac \left\{1\right\} \left\{n^ \left\{2p\right\}\right\}$, i.e. the value of the Function zeta of Riemann for the even entireties,

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