Olivier Bouillot

Gauss-Laguerre and related quadratures -Explicit formula of the error term

We investigate a new quadrature, namely the Gauss-Laguerre-like quadrature, close to the Gauss-Laguerre quadrature, those whose weight function is modified from z Þ ÝÑ e ´z to z Þ ÝÑ p1 `e´z q ´1.

Firstly, we prove an explicit error formula for the Gauss-Laguerre and the Gauss-Laguerre-like quadratures, which apply to a subclass of holomorphic functions with isolated singularities outside of s0; `8r.

Then, we explore the asymptotic of the respective error terms on two simple examples but sufficiently general to be meaningful for the reader. In order to control explicitly their relative error and to compute the related integrals, we also give explicit upper bounds of the error in these two examples.

Introduction

In 1905, Lindelöf (see [START_REF] Lindelöf | Le calcul des résidus et ses applications à la théorie des fonctions[END_REF], chapter 3) has shown, under precise hypothesis on functions, called the 1D-Lindelöf hypothesis in [START_REF] Bouillot | Phase Portraits of Bi-dimensional Zeta Values[END_REF], the following: In many cases, the first integral can be explicitly computed while the second can not. Thus, to numerically compute sums of functions evaluated at positive integers by Equation (1), we focus on computing integrals ż `8 0 f ptq dt e t `1 . To this end, we introduce and tabulate a new quadrature close to the Gauss-Laguerre one. Its weight function moves from t Þ ÝÑ e ´t as in the case of the Gauss-Laguerre quadrature, to t Þ ÝÑ 1 1 `et . The integral expressions seems to be similar. But, the main change concerns the apparition of singularities on the weight function: we have modified an entire function for a function with infinite simple poles in C . . . This change has a significant impact on the error term of the quadrature: automatically, the error term will contain an infinite number of terms.

In Section 2, we will first review the Gauss-Laguerre quadrature. In particular, we will focus on the state of the art on its error term e GL N . Then, we define the Gauss-Laguerre-like quadrature.

In particular, in this Section, we will introduce two main examples that will be continued later. The first example concerns the meromorphic function z Þ ÝÑ e ´z 1 `z2 integrated numerically by the Gauss-Laguerre quadrature. The second one deals with the meromorphic function z ÝÑ 1 1 `ez , integrated this time by the Gauss-Laguerre-like quadrature.

These examples are quite simple but turn out to be sufficiently general to be meaningful on other examples of applications of these quadratures when applied to meromorphic functions: each newly added term is of the form of the error term of the first example, i.e. easily depending on the quantity

U n pzq " ż `8 0 t n pt ´zq n`1 dt , (2)
for a complex number z, according to the presence of a new singularity on the integrand function in comparison with the two main examples.

The main goal of the paper is then to explore in detail the error term of this new quadrature, in particular, to provide explicit formula and upper bounds for this error term. For simplicity reasons, we will restrict ourselves to holomorphic functions with isolated singularities outside of s0; `8r. Another restriction is the growth rate of the integrand function: it must not be bigger than a polynomial on an infinite family of circles centered in 0 whose radius are growing to `8.

To this end, we will first prove the following result on the error of the Gauss-Laguerre quadrature before extending it to the Gauss-Laguerre-like quadrature.

Theorem 1.2. Let N be a positive integer. Let also f : C ÝÑ C be an holomorphic function on C ´Spf q such that:

• Spf q is a closed subset of C, called the set of the isolated singular points of f ;

• f has only isolated singularities on C and no singularities in s0; `8r;

• the integral

ż `8 0
f ptq e ´t dt is well-defined.

Let us also consider an increasing sequence pR n q nPN of positive real numbers growing to `8 such that:

' no singular point of f and no zero of L n are located on the circles Cp0, R k q, k " 0, 1, ¨¨¨; ' f does not grow faster than a polynomial on all the circles Cp0, R k q, k " 0, 1, ¨¨¨:

Dn 0 P N , DC ą 0 , @k P N , @z P C , |z| " R k ùñ |f pzq| ď C|z| n0 . (3)

If N ą n 0 , the error e GL N pf q of the N -point Gauss-Laguerre quadrature is then given by: e GL N pf q " ż `8

0 ÿ sPSpf q Res ˆz Þ ÝÑ f pzq pt ´zq N `1L N pzq , s ˙¨t N e ´t dt . (4)
In Section 3, we will first prove Theorem 1.2 and extends it to the Gauss-Laguerre-like quadrature (Theorem 3.3). Then, we apply these two theorems to our two main examples to find out the explicit expressions of the error term.

Let us precise here that the extension to the Gauss-Laguerre-like quadrature requires a precise study of a family of functions, postponed in Annex 6 to keep the focus on the main goal of the article.

The error obtained in the first example is essentially a term U n piπq, divided by an evaluation of a Laguerre polynomial. The second example is more complicated: the quadrature error expresses as the integral of a series. If we could term-by-term integrate it (which will be proved later, in Section 5), therefore, we obtain a series of terms similar to this of the first example.

To understand the asymptotics of an error term we now have to elucidate the behaviour, when n goes to infinity, of U n pzq for all complex numbers z, which is obtained in Section 4, using the Laplace's method. With Perron's formula giving the asymptotics of Laguerre's polynomials when n ÝÑ `8 (see Equation ( 125)), we also need to understand the behaviour on iR of these polynomials. Therefore, in Section 5, we give explicit upper bounds of all the terms involved in the error terms of our two main examples and prove their asymptotics. Therefore, we can predict numerically how these errors go to 0, and then compute the corresponding integral with a predefined number of exact digits. Moreover, we give Conjectures on sharp upper bounds concerning the error terms of the two main examples.

The Gauss-Laguerre-like quadrature

In this preamble Section, we review here known results on the Gauss-Laguerre quadrature. We also introduce the Gauss-Laguerre-like quadrature, as well as the two main example that will be continued during the whole article.

The Gauss-Laguerre quadrature

The N -points Gauss-Laguerre quadrature is defined for continuous functions f : R `ÝÑ C such that t Þ ÝÑ f ptqe ´t P L 1 pR `q by (see [START_REF] David | Methods of Numerical Integration[END_REF], §3.6): where e GL N pf q denotes the error of the quadrature, px GL k,N q kPrr1;N ss are the nodes of the quadrature, i.e. the zeros of the N -th Laguerre polynomial L N pxq, and finally w GL k,N are the associated weights defined by:

ż `8 0 f ptqe ´t dt " N ÿ k"1 w GL k,N f pt GL k,N q `eGL N pf q , (5)
w GL k,N " 1 L 1 n px GL k,N q ż `8 0 L N pxqe ´x x ´xGL k,N dx " x GL k,N pN `1q 2 ˆLN`1 px GL k,N q ¯2 (6)
(see [START_REF] Szegö | Orthogonal Polynomials[END_REF], Formula (3.4.3) and [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF], Formula 25.4.45).

In Table 1, we have tabulated the nodes and the weights of the 16-points Gauss-Laguerre quadrature, up to 20 significant digits. The numbers in the parentheses stand for the number of zeros between the decimal points and the first significant digit.

Example 1. Let us consider the function a defined over R `by apxq " 1 1 `x2 . If we denote respectively by Ci and Si the integral cosine and integral sine defined by

Cipxq " ż `8 x cos x x dx , Sipxq " ż `8 x sin x x dx , (7)
then, it is not difficult to see that:

ż `8 0 e ´x 1 `x2 dx " sinp1qCip1q `´π 2 ´Sip1q ¯cosp1q « 0.621449624235813... ( 8
)
We can use Gauss-Laguerre quadratures to experimentally compute this value: Let us mention that, for greater powers of two, we need to use a library for arbitrary-precision floating-point arithmetic to obtain more precise results.

We remark here that when N is quadrupled, the number of zeros before the first significative digits essentially doubles. So, we could conjecture that there exists C P R such that ln ˇˇe GL n paq ˇˇ"

N ÝÑ`8 C ? N .
More precisely, we can also conjecture that C " ´2? 2 (see Conjecture 1).

2.2 A Gauss-Laguerre-like quadrature.

Let us now define a quadrature associated with the weight function ω : R `ÝÑ R defined by: ωptq " 1 1 `et for all t P R `.

Therefore, we can successively write:

ż `8 0 f ptq 1 `et dt " ż `8 0 f ptq 1 `e´t e ´t dt " N ÿ k"1 w GL k,N 1 `e´t GL k,N f pt GL k,N q `eGL N ˆt Þ ÝÑ f ptq 1 `e´t " N ÿ k"1 w k,N f pt k,N q `EN pf q , (9)
where:

• the nodes t k,N are defined by the Gauss-Laguerre's nodes: t k,N " t GL k,N ;

• the weights w k,N are defined by w k,N " w GL k,N • the error E N pf q is defined by:

E N pf q " e GL N ˆt Þ ÝÑ f ptq 1 `e´t ˙. ( 10
)
In Table 2, we have tabulated the nodes and the weights of the 16-points Gauss-Laguerre-like quadrature, up to 20 significant digits. One more time, the numbers in the parentheses stand for the number of zeros between the decimal points and the first significant digit.

Example 2. It is not difficult to compute exactly

ż `8 0 dt 1 `et : ż `8 0 dt 1 `et " ÿ ką0 p´1q k ż `8 0 e ´kt dt " ÿ ką0 p´1q k k " ln 2 . ( 11
)
Consequently, let us test this quadrature to the constant function f " 1:

Approximation given by the N N ´points Gauss-Laguerre-Error -like Quadrature

Error formula and convergence properties.

The most well-known formula concerning the error e n pf q of the Gauss-Laguerre quadrature, but useless in a practical context, is due to Markov (see [START_REF] David | Methods of Numerical Integration[END_REF], Equation 3.6.3 or [START_REF] Gautschi | A Survey of Gauss-Christoffel Quadrature Formulae[END_REF], Equation 1.18):

@n P N ˚, Dξ P R ˚, e n pf q " n! 2 p2nq! f p2nq pξq , (12)
if the function f is C 2n pR `q. However, this error formula is mostly unusable... and can not be used to prove that the remainder e n pf q or E n pf q goes to 0 when n goes to `8. On one hand, it can be inextricable to compute explicitly the p2nq th -derivative, depending on the function f ; on the other hand, this derivative can take arbitrary large values even if the error is small. Let us remind the reader that the Gauss-Laguerre quadrature is unfortunately known to have poor convergence properties, even to be unstable when used to numerically integrate functions over r0; `8r by the very bad following rule:

ż `8 0 f ptq dt " ż `8 0 `f ptqe t ˘e´t dt « N ÿ k"1 w GL k,N e t GL k,N f pt GL k,N q . ( 13
)
Sometimes, this is explained by the difficulty of managing the infinite upper integration limit, in opposition with the Stieljes result (see [START_REF] Stieljes | Quelques recherches sur la théorie des quadratures dites mécaniques[END_REF]) saying that for all continuous function f : r´1; 1s ÝÑ C the N th -point Gauss-Legendre quadrature scheme converge, when N goes to infinity, to the value of the integral of f over r´1; 1s.

Actually, this is due to the growth rate of r f : t Þ ÝÑ f ptqe t , or saying it differently, r f can not be well-approximate by polynomials over r0; `8r. More precisely, in 1928, Uspensky (see [START_REF] Uspenksy | On the convergence of quadrature formulas related to an infinite interval[END_REF]) has shown convergence properties for the Gauss-Laguerre quadrature:

Dρ ą 0 , Dx 0 ą 0 , DC ą 0 , @x ě x 0 , |f pxq| ď Ce x x 1`ρ , (14)
then e GL N pf q ÝÑ ´C? N , but there is still a long way to go.

Uspensky's theorem is a theoretical theorem which is nowadays a bit frustrating for the numerical scientist working with high-power computation tools. Our main goal is to provide an explicit formula for the error e GL n pf q for large class of functions f , as well as find out a way to describe the rate of convergence of `eGL N pf q ˘NPN or `EN pf q ˘NPN to 0.

In this direction, Mastroianni & Monegato's result (see [START_REF] Mastroianni | Convergence of product integration rules over p0, 8q for functions with weak singularities at the origin[END_REF]) is important:

Theorem 2.2. (Mastroianni & Monegato, 1995)
Let p and q be two non-negative integers such that 0 ď p ď q. Let us denote C q p r0; 8q the subset of C p pr0; `8rq X C q ps0; `8rq defined by: C q p r0; 8q " tf P C p pr0; `8rq X C q ps0; `8rq ;

x Þ ÝÑ x i f pp`iq pxq P C 0 pr0; `8rq, i " 1, ¨¨¨, q ´pu . (17)

With f P C q p r0; 8q, we associate the auxiliary function Φ P C q pr0; `8rq defined by Φpxq " x q´p f pxq, so that e GL N pf q satisfies:

|e GL N pf q| " $ ' & ' % OpN ´q 2 q E N ´p´1 `Φpqq , e ´x 2 ˘, if q ď 2p `1 , OpN ´pp`1q ln nq E N ´p´1 `Φpqq , e ´x 2 ˘, if q ď 2p `2 , OpN ´pp`1q q E N ´p´1 `Φpqq , e ´x 2 ˘, if q ě 2p `3 , (18)
where E n pf, ωq " inf

PnPCn´1rXs

||ωpf ´Pn q|| 8,r0;`8r is the error of the best polynomial approximation of f .

Moreover, following the notations introduced in [START_REF] Mastroianni | Polynomial approximation on the real semiaxis with generalized Laguerre weights[END_REF], (at the beginning of Section 3), Mastroianni and Szabados have proved estimation of the quantity E n pf, e x q for a special class of functions: Lemma 2.3. [START_REF] Mastroianni | Polynomial approximation on the real semiaxis with generalized Laguerre weights[END_REF] For all functions f P W 8 1 pe ´xq, there exists C ą 0 and a sequence of numbers pa n q nPN satisfying a n " nÝÑ`8

n such that we have: The function a is an element of C q p r0; `8q for all integers p and q such that 0 ď p ď q. In particular, when p ě 0 and q " 2p, we have:

E n `f,
e GL N paq " N ÝÑ`8 OpN ´pq ¨EN´p´1 `Φp2pq , e ´x 2 ˘, (20)
where Φpxq " x p f pxq.

According to Φ ppq P W 8 1 pe ´xq and Lemma 2.3, we have:

E N ´p´1 `Φpqq , e ´x 2 ˘" N ÝÑ`8 OpN ´1 2 q . ( 21
)
Therefore, we deduce that:

Lemma 2.4. For all non-negative integers p, we have:

e GL N paq " N ÝÑ`8 OpN ´pq . ( 22
)
Example 5. Following the same scheme of proof, as in Example 4, we prove the following result:

Lemma 2.5. For all non-negative integers p, we have:

E N p1q " N ÝÑ`8
OpN ´pq .

Even if it will not be possible to apply it in our context of holomorphic functions over C ´Spf q, where Spf q is the closed subset of C of the singularities of f , let us remind, for completeness, the Lubinsky's result (see [START_REF] Lubinsky | Geometric convergence of Lagrangian interpolation and numerical integration rules over unbounded contours and intervals[END_REF]) Theorem 2.6. [START_REF] Lubinsky | Geometric convergence of Lagrangian interpolation and numerical integration rules over unbounded contours and intervals[END_REF] Let f be an entire function defined by its series expansion f pzq "

ÿ ně0 c n z n for all z P C.

Let us consider the real number

A " lim sup nÝÑ`8 ˜n|c n | 1 n 2 ¸.
If A ă 1, we have for sufficiently large n:

ˇˇe GL n pf q ˇˇď A 2n . ( 24
)
3 Error formula of the quadrature, by contour integration

Without forgetting methods based on Peano error estimates (see [START_REF] Stroud | Peano Error Estimates for Gauss-Laguerre Quadrature Formulas[END_REF] for example), it is well known that, when applied to holomorphic functions, there exist three main methods to obtain estimations of the remainder of a quadrature (see [START_REF] Gautschi | A Survey of Gauss-Christoffel Quadrature Formulae[END_REF], §4). Among the oldest are the estimates based on contour integration ; there are also these based on the Hilbert space norm estimates, as well as the estimates obtained via approximation theory (see [START_REF] Xiang | Asymptotics on Laguerre or Hermite polynomial expansions and their applications in Gauss quadrature[END_REF] for a recent result).

Here, we will be interested in the contour integration method, applied to meromorphic functions over C.

Proof of Theorem 1.2 concerning the error term of

Gauss-Laguerre quadrature

Let us fix a positive integer n. Let also f : C ÝÑ C be an holomorphic function on C ´Spf q such that:

• Spf q is a closed subset of C called the set of the isolated singular points of f ;

• f has only isolated singularities on C and no singularities in s0; `8r;

• the integral

ż `8 0
f ptq e ´t dt is well-defined.

According to the general method described in [START_REF] David | Methods of Numerical Integration[END_REF] (see §4.6, p. 303) or in [START_REF] Mcnamee | Error-bounds for the Evaluation of Integrals by the Euler-Maclaurin Formula and by Gauss-type Formulae[END_REF] (see §3.2), let us consider a closed circle Cp0, Rq, centered in 0, with radius R, enclosing all the zeros x GL 1,n , ¨¨¨, x GL n,n of the n-th Laguerre ploynomial L n . such that no pole of f are on Cp0, Rq.

Thus, according to Cauchy's residue theorem, we have for all positive real numbers t such that t ď R and t ‰ x GL k,n , k P rr1; nss:

1 2πi ż C R f pzq dz pz ´tqL n pzq " f ptq L n ptq `n ÿ k"1 f px GL k,n q px GL k,n ´tqL 1 n px GL k,n q `ÿ sPSpf q |s|ăR Res ˆz Þ ÝÑ f pzq pz ´tqL n pzq , s ˙. ( 25
)
Let us moreover assume that there exists an increasing sequence pR n q nPN of positive real numbers growing to `8 such that ' no singular point of f is located on the circles Cp0, R k q, k " 0, 1, ¨¨¨; ' no zero of L n is located on the circles Cp0, R k q, k " 0, 1, ¨¨¨; ' f does not grow faster than a polynomial on all the circles Cp0, R k q, k " 0, 1, ¨¨¨:

Dn 0 P N , DC ą 0 , @k P N , @z P C , |z| " R k ùñ |f pzq| ď C|z| n0 . (26)
Therefore, we have the following: Lemma 3.1. If such a sequence pR k q kPN is available for the function f : C ÝÑ C, holomorphic on C ´Spf q, for all n ą n 0 and all t P R `, we have:

1 2πi ż Cp0,R k q f pzq dz pz ´tqL n pzq ÝÑ kÝÑ`8 0 . ( 27
)
Proof. The dominant term of the polynomial L n pXq is p´Xq n n! . Therefore, there exist an integer k 1 ě 0 such that for all integers k ě k 1 and all z P Cp0, R k q, we have:

|z| n 2 ¨n! ď |L n pzq| ď 2|z| n n! (28)
Let us now fix t ě 0. So, there exists an integer k 2 ě k 1 such that for all k ě k 2 , we have for all z P Cp0, R k q:

$ ' ' & ' ' % R k ą t . ˇˇˇf pzq pz ´tqL n pzq ˇˇˇď CR n0 k pR k ´tq ¨Rn k 2n! . ( 29
)
Finally, this gives us:

ˇˇˇˇ1 2πi ż Cp0,R k q f pzq dz pz ´tqL n pzq ˇˇˇˇď 2Cn!R n0`1 k pR k ´tqR n k (30)
and proves for all non-negative real numbers t the convergence to 0 when k ÝÑ `8 of the sequences of integrals since n ą n 0 .

As a corollary, from Equation (25), we deduce that

ÿ sPSpf q Res ˆz Þ ÝÑ f pzq pz ´tqL n pzq , s is well-defined if n ą n 0 and t P R `´! x GL k,n ; 1 ď k ď n ) .
Now, it is possible to send R ÝÑ `8 in Equation (25) and then reorganising it. This gives the following Equation, valid a priori for all t P R

`´! x GL k,n ; 1 ď k ď n
) , but which can be extended to all t P R `using the continuity of each term: Res ˆz Þ ÝÑ f pzq pt ´zqL n pzq , s ˙¨L n ptq e ´t dt is also well-defined.

f ptq e ´t " n ÿ k"1 f px GL k,n q L 1 n px GL k,n q L n ptq e ´t t ´xGL k,n `ÿ sPSpf q Res ˆz Þ ÝÑ f pzq pt ´zqL n pzq , s ˙¨L n ptq e ´t . (31
Therefore, we have:

ż `8 0 f ptq e ´t dt " n ÿ k"1 f px GL k,n q L 1 n px GL k,n q ż `8 0 L n ptq e ´t t ´xGL k,n dt `ż `8 0 ÿ sPSpf q Res ˆz Þ ÝÑ f pzq pt ´zqL n pzq , s ˙¨L n ptq e ´t dt . (32)
According to Equations ( 5) and ( 6), this finally gives us a nice expression of the error term of the Gauss-Laguerre quadrature for such functions f :

e GL n pf q " ż `8 0 ÿ sPSpf q Res ˆz Þ ÝÑ f pzq pt ´zqL n pzq , s ˙¨L n ptq e ´t dt . (33)
Therefore, using Rodrigue's formula L n pxq " e x n! `xn e ´x˘p nq and successive integrations by parts, we have:

e GL N pf q " 1 n! ż `8 0 ÿ sPSpf q Res ˆz Þ ÝÑ f pzq pt ´zqL n pzq , s ˙¨`t n e ´t˘p nq dt " p´1q n n! ż `8 0 ¨ÿ sPSpf q Res ˆz Þ ÝÑ f pzq pt ´zqL n pzq , s ˙' pnq ¨tn e ´tdt ( 34
)
Using Laurent's expansion, it is easy to see that, if z Þ ÝÑ apzq is a meromorphic function near s P C, then we have for all integers n:

d n dt n Res ˆz Þ ÝÑ apzq t ´z , s ˙" p´1q n n! Res ˆz Þ ÝÑ apzq pt ´zq n`1 , s ˙. (35)
Therefore, plugging Equation (35) to Equation (34) concludes the proof of Theorem 1.2.

Remark 1. Let us remark that if the function f is entire and satisfies the hypothesis of Theorem 1.2, then, necessarily, f is a polynomial with degree d satisfying d ď n 0 . Then, by definition, the N -point Gauss-Laguerre quadrature applied to f is exact if n 0 ď 2N `1. This is, of course, stronger than the result of Theorem 1.2 in this particular case.

Consequently, this theorem is useful only for functions which have singularities and explain why Lubinski's result on geometric convergence to 0 of the error can not be applied in our context. Let us also fix a positive integer N and consider an increasing sequence pR k q kPN growing to `8 such that R 0 ě 2x GL N `1,N `1.

The function a has two simple poles, i and ´i. Therefore, Theorem 1.2 applied to a, with n 0 " 0, gives us:

e GL N paq " ż `8 0 ˆ´i 2pt ´iq N `1L N piq `i 2pt `iq N `1L N p´iq ˙tN e ´t dt " e ˆi L N p´iq ż `8 0 t N e ´t pt `iq N `1 dt ˙. ( 36
)
As explicit examples, we have:

e GL 1 paq " 1 2 ż `8 0 t 3 `2t 2 ´t p1 `t2 q 2 e ´t dt (37)
e GL 2 paq "

1 17 ż `8 0 8t 5 `6t 4 ´24t 3 ´2t 2 p1 `t2 q 3 e ´t dt (38)
e GL 3 paq "

3 148 ż `8 0 17t 7 ´12t 6 ´102t 5 `12t 4 `17t 3 p1 `t2 q 4 e ´t dt (39)
and we effectively have, using successive integration by parts:

e GL 1 paq " ż `8 0 e ´t 1 `t2 dt ´1 2 . ( 40
)
e GL 2 paq "

ż `8 0 e ´t 1 `t2 dt ´11 17 . (41)

The error of the Gauss-Laguerre-like quadrature

From now on, we want to change the weight function x Þ ÝÑ e ´x to x Þ ÝÑ 1 1 `ex to prove an analogue of Theorem 1.2 for the Gauss-Laguerre-like quadrature.

So, we will look for precise estimations of g 0 : z Þ ÝÑ 1 1 `e´z on circles centered in 0. Then, we will easily be able to state a derivation of Theorem 1.2 for the Gauss-Laguerre-like quadrature.

Estimations on circles centered in 0 of the weight function

If z " Re iθ P CpR, 0q, where the radius R satisfies R P R `´p2Z `1qπ, we have:

|g 0 pzq| 2 " ˇˇˇ1 1 `e´z ˇˇˇ2 " 1 1 `2e ´R cos θ cospR sin θq `e´2R cos θ . ( 42
)
The denominator of the right-hand side of Equation ( 42) is not so simple to handle as a function of θ Ps´π; πs. Nevertheless, the following Proposition gives us a uniform (in the variable R) upper bound of the function g 0 . This result will be sufficient to prove that the hypothesis (3) are satisfied with n 0 " 0 for a large range of values of R: Proposition 3.2. Let δ Ps0; πs. There exist a constant Cpδq ą 0, depending only on δ, such that for all positive real numbers R satisfying distpR, p2Z `1qπq ě δ, we have:

@z P CpR, 0q , ˇˇˇ1 1 `e´z ˇˇˇď Cpδq . ( 43
)
More precisely, Cpδq could be chosen to be defined by:

Cpδq " 1 min ˜0.93; sin δ; 1 ´e´δ ; c 1 ´cos 4 ´δ 2 ¯¸. (44)
The result would have been easy to prove if an integer n would have been fixed and R P rp2n ´1qπ `δ; p2n `1qπ ´δs, according to the continuity of z Þ ÝÑ p1`e ´z q ´1 of the compact annulus tz P C; p2n ´1qπ `δ ď |z| ď p2n `1qπ ´δu. Nevertheless, the theoretical upper bound found would have been a function of n...

Consequently, the proof of this Proposition is long, delicate, technical and non really informative, according to the subtle behaviour of the denominator of the right-hand side of Equation (42) as a function of the parameter θ. Therefore, it is merely postponed to Annex 6.

Application to the Gauss-Laguerre-like quadrature

Applying Theoreme 1.2 to the fonction r f : z Þ ÝÑ f pzq 1 `e´z , where f : C ÝÑ C satisfies the hypothesis of Theorem 1.2 gives us: Theorem 3.3. Let f : C ÝÑ C be an holomorphic function on C ´Spf q such that:

• Spf q is a closed subset of C, called the set of the isolated singular points of f ;

• f has only isolated singularities on C and no singularities in s0; `8r;

• the integral

ż `8 0 f ptq 1 `et dt is well-defined.
Let us also consider δ Ps0; πs and an increasing sequence pR n q nPN of positive real numbers growing to `8 such that:

' for all integers n, dist `Rn ; p2Z `1qπ ˘ě δ .

' no pole of f and no zero of L n are located on the circles Cp0, R k q, k " 0, 1, ¨¨¨; ' f does not grow faster than a polynomial on all the circles Cp0, R k q, k " 0, 1, ¨¨¨:

Dn 0 P N , DC ą 0 , @k P N , @z P C , |z| " R k ùñ |f pzq| ď C|z| n0 . ( 45
)
If N ą n 0 , the error E N pf q of the N -point Gauss-Laguerre-like quadrature is then given by:

E N pf q " ż `8 0 ÿ sPS Res ˆz Þ ÝÑ 1 pt ´zq N `1L n pzq f pzq 1 `e´z , s ˙¨t n e ´t dt ,( 46
)
where S denotes the set of isolated singularities of the function z Þ ÝÑ f pzq 1 `e´z .

Proof. From Equation ( 10), we know that the error of a Gauss-Laguerre-like quadrature is related to the error of a Gauss-Laguerre quadrature by:

E N pf q " e GL N ˆt Þ ÝÑ f ptq 1 `e´t ˙. (47)
We will compute this last value using Theorem (1.2).

Let us also define r f " f ¨g0 , i.e.:

r f pzq " f pzq 1 `e´z , for all z P C ´Sp r f q , (48)
where Sp r f q " Spf q Y p2Z `1qiπ. Firstly, as a product of functions defined over C, holomorphic respectively over C´Spf q and C´p2Z`1qiπ, with isolated singularities, r f is an holomorphic functions over C ´Sp r f q, with isolated singularities. Moreover, no pole of r f is

located on a circle CpR k , 0q, k " 0, 1, ¨¨¨and ż `8 0 f ptq 1 `et dt is assumed to be well-defined.
Finally, according to Proposition 3.2, we know that there exists a constant Cpδq ą 0 such that for all z P C, we have: dist `|z|, p2Z `1qπ ˘ě δ ùñ |g 0 pzq| ď Cpδq .

(49)

According to the assumption on the function f (see Equation ( 45)), we are now able to claim that for all z P CpR k , 0q, we have:

| r f | " |f pzq ¨|g 0 pzq| ď C|z| n0 ¨Cpδq . ( 50
)
This proves that r f satisfies Hypothesis (3).

Consequently, Theorem 1.2 can be applied to the function r f :

E N pf q " ż `8 0 ÿ sPSp r f q Res ˆz Þ ÝÑ 1 pt ´zq N `1L n pzq f pzq 1 `e´z , s ˙¨t n e ´t dt . (51)
Let us emphasize that when f has only a finite number of poles, the residues in Equation ( 46) should be easy to compute explicitly. Using an estimation of Laguerre polynomials evaluated on iX (see Subsection 5.1), we can show that the series and the integral can be permuted.

This is exactly what we can now do, coming back to Example 2:

Example 7. Using R n " 2nπ for all n P N and δ " π, Theorem 1.2 applied to the constant function 1 (with n 0 " 0) gives us:

E N p1q " ż `8 0 ˜ÿ kPZ 1 L N `p2k `1qiπ ˘tN e ´t `t ´p2k `1qiπ ˘N`1 ¸dt , (52)
for all N ą 0.

Let us emphasize that even if this is an explicit result of the error of the quadrature, there is unfortunately, still a long way to go to quantify numerically this error. . . This will be done in Section 5, especially in Subsection 5.4.

4 Asymptotic of the integral U N pzq for z P C ´RẀ e shall now study precisely the fundamental integral ż `8 0

t N e ´t pt ´zq N `1 dt, z P C ´R`a nd N P N, which naturarly appears in the explicit expression of the error E n pf q when f has a finite number of poles.

In particular, we want to be able to quantify each term of the summation E N pf q. Consequently, the main goal of this section is to prove the following Proposition 4.1. For all z P C ´R`, we have:

ż `8 0 t n pt ´zq n`1 e ´t dt " nÝÑ`8 ? πe ´z 2 p´nzq 1 4 e ´2? ´nz . ( 53
)
using the Laplace method (for example, see [START_REF] Flajolet | Analytic Combinatorics[END_REF], chapter 8 for an introduction to this asymptotic method).

To prove this result, let us denote the left-hand side of the equivalent sign in Equation (53) by U n pzq:

U n pzq " ż `8 0 t n pt ´zq n`1 e ´t dt . (54)
We will also denote respectively the principal branch of the logarithm and the square root by log and ? . During this Section, we will consider a fixed complex number z P C´R `, so that s " ? ´z is a well-defined complex number satisfying e s ą 0.

Finally, let us fix an integer n.

4.1 Modification of the integration path.

χ n pχ ´zq n`1 e ´χ dχ ˇˇˇď ż θ0 0 R n`1 |Re iθ ´z| n`1 e ´R cos θ dθ ď ż θ0 0 R n`1 ˇˇR ´|z| ˇˇn`1 e ´R cos θ dθ ď θ 0 ˇˇ1 ´|z| R ˇˇn `1 e ´R cos θ0 . ( 55
) R ? ´nz z θ0
According to the residue theorem applied to the path described on Figure 1, we conclude that:

U n pzq " ż ? ´nz¨8 0 t n pt ´zq n`1 e ´t dt . (57)
It turns out that the integral on the right-hand side of Equation (57) can also be written as:

ż `8 0 t n pt `sq n`1 e ´st dt , with s " ? ´z . ( 58
)
From now on, for all z P C ´R`, we denote s " ? ´z. Therefore, for all s P C such that e s ą 0, we define by I n psq the integral (58):

I n psq " ż `8 0
t n pt `sq n`1 e ´st dt , for all s P C , e s ą 0 , (59) so that we have: U n pzq " I n psq, with s " ? ´z .

(60)

A first elementary estimation

When s is a positive real number, we have:

0 ď I n psq ď 1 s ż `8 0 ´1 `s x ¯´n e ´sx dx ď 1 s ż `8 0 exp ˆ´s ´x `n s `x ¯˙dx , (61)
according to lnp1`xq ě x 1 `x for all x ě 0. The integral on the right-hand side of Inequation (61) can be generalised slightly, in order to find out an explicit upper bound of I n psq: Lemma 4.2. Let u ą 0, v ě 0 and n a positive integer. Therefore, the integral r I n pu, vq defined by

r I n pu, vq " ż `8 0 exp ˆ´u ´x `n x `v ¯˙dx (62)
satisfies:

r I n pu, vq ď $ ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' % e ´u? n ˆ1 u `4 ? n ¨?π ? u ˙, if 0 ď v ď ? n . e ´u ? n ? n`π u ˆ3 `2n π ˙, if v P ‰? n; ? n `π‰ . 2e ´un v u , if v ą ? n `π . (63)
Proof. Let us remark that when v " 0, the integral r I n pu, 0q is still a convergent one for all u ą 0.

' For all u ą 0, the function v Þ ÝÑ
To conclude the proof, we just have to check whether the lower bound of the Gaussian integral is null or not.

Case 1: 0 ď v ď ? n
Using the explicit values of Gaussian integral, Equation (66) becomes:

r I n pu, vq ď r I n pu, ? nq ď e ´u? n ˆ1 u `4 ? n ¨?π ? u ˙, (67)
for all u ą 0.

Case 2: v ą ? n

Yet, an easy integration by parts shows that for all α ą 0, we have: 69) that for all u ą 0:

ż `8 α e ´t2 dt ď e ´α2 2α . (68
r I n pu, vq ď e ´un v u ˜1 `4 ? n ? v ´c n v ḑ e ´un v u ˜1 `4 ? n ? n `π ´?n ? n `π ḑ e ´un v u ˆ1 `4 ? n ? n `π ´1 ˙ď 2e ´un v u . (71)
From that result, we can deduce an elementary estimation of U n pzq, z P C ´R`. It will be useful in Section 5 to estimate explicitly and simply some terms in error's expression of Examples 2 and 6.

We can in particular emphasize that this elementary upper bound gives us an exponentially decreassing character in the variable s. Nevertheless, it is not an optimal one, according to Proposition 4.1 which will be proved in the following Subsections.

Notations and Laplace's method implementation

The integrand function inside the integral of Equation ( 59) has a paek near ? n and is nearly null outside of it. Consequently, only a small interval around ? n contributes to the whole integral I n psq.

Let us fix

θ P  1 4 ; 1 3
" , so that our small interval around ? n will be "? n ´nθ ; ? n `nθ ‰ .

Then, let us now decompose I n psq into three parts:

I n psq " J n psq `Kn psq `Ln psq , (75)
where:

J n psq " ż ? n´n θ 0 t n pt `sq n`1 e ´st dt , (76)
K n psq "

ż ? n`n θ ? n´n θ t n pt `sq n`1 e ´st dt , (77)
L n psq "

ż `8 ? n`n θ t n pt `sq n`1 e ´st dt . (78
ζ n " a p1 `s2 q 2 `4s 2 n ´p1 `2s ? n `s2 q 2s . ( 80
)
4.4 Tail pruning.

Using similar substitutions to those used in Lemma 4.2, the main goal of this Subsection is to show the following:

Lemma 4.4. For all s P C such that e s ą 0, we have:

J n psq "
Proof. ' Firstly, let us focus on the case of the integral J n psq. Since n goes to `8, we can assume that n θ ě e psq.

We successively have:

|J n psq| ď ż ? n´n θ 0 t n pt ` e sq n`
according to e psq ď n θ ď ? n .

Consequently, we finally have: ' Let us now focus on the integral L n psq, using one more time a similar substitution to this used in Lemma 4.2.

We successively have:

|L n psq| ď ż `8 ? n`n θ t n pt ` e sq n`
where f n is defined by Equation (79).

We will prove the wanted equivalent using two steps, first a uniform central approximation, and then showing that K n psq is essentially a Gauss integral.

Uniform central approximation.

In this Paragraph, we will show the following Lemma 4.5. Let θ be a positive real number. There exists a positive constant Cpsq and a positive integer N 0 such that for all integers n ě N 0 , we have:

sup tPr´n θ ;n θ s ˇˇˇf n ptq ´fn pζ n q ´1 2 pt ´ζn q 2 f 2 n pζ n q ˇˇˇď Cpsq n 1´3θ . ( 92
)
Proof. ' The sequence pζ n q nPN of saddle nodes is a converging sequence to ζ 8 " ´1 `s2 2s . Consequently, there exist two positive integers N 1 and N 2 such that:

# n ě N 1 ùñ |ζ n ´ζ8 | ď 1 . n ě N 2 ùñ |ζ 8 | ď n θ . (93)
Consequently, if n ě maxpN 1 , N 2 q, t P r´n θ ; n θ s and u P rt, ζ n s, we successively have:

e u " e `κt `p1 ´κqζ n ˘, for a κ P r0; 1s

" κt `p1 ´κq e pζ n ´ζ8 q `p1 ´κq e pζ 8 q ě κt ´p1 ´κq|ζ n ´ζ8 | ´p1 ´κq|ζ 8 | ě κt ´p1 ´κqp1 `nθ q ě ´`κn θ `p1 ´κqp1 `nθ q ě

´`1 ´κ `nθ ˘ě ´p1 `nθ q .

(94) with a unique zero located in x 0 « 1.46557. From 10 ě x 6 0 , we therefore have for all n ě maxp10, N 1 , N 2 q:

´nθ n θ t ζn ζ8 1 u
?

n ´nθ ´1 ą ? n ´3 ? n ´1 ě ? 10 ´3 ? 10 ´1 ą 0 . (95)
' Let us now fix an integer n ě maxp10, N 1 , N 2 q. Then, the function f n is C 8 over C´s ´8; ´?ns and we have for all u P C´s ´8; ´?ns:

f 3 n puq "
we can conclude that if n P N satisfies n ě N 0 , we have:

@t P r´n θ ; n θ s , @u P rt, ζ n s , |f 3 n puq| ď cpsq n . (101)
' Finally, let us fix an integer n ě N 0 and a real number t P r´n θ , n θ s.

We remind that f n ptq ´fn pζ n q ´1 2 pt ´ζn q 2 f 2 n pζ n q can be expressed as an iterated integral, according to f 1 n pζ n q " 0:

f n ptq ´fn pζ n q ´1 2 pt ´ζn q 2 f 2 n pζ n q " ż ζn t ˜ż ζn u2 ´ż ζn u1 f 3 n pu 0 q du 0 ¯du 1 ¸du 2 " pζ n ´tq 3 ż 1 0 ˜ż 1 0 " ż 1 0 f 3 n ´t `"t 2 ``t 1 `t0 p1 ´t1 q ˘p1 ´t2 q ‰ pζ n ´tq ¯dt 0 ı ¨p1 ´t1 q dt 1 ¸p1 ´t2 q 2 dt 2 . ( 102
)
For all real numbers t 0 , t 1 and t 2 between 0 and 1, it is clear that we have:

0 ď t 2 ``t 1 `t0 p1 ´t1 q ˘p1 ´t2 q ď 1 . (103)
Therefore, according to Equations ( 101) and (102), we deduce that:

ˇˇˇf n ptq ´fn pζ n q ´1 2 pt ´ζn q 2 f 2 n pζ n q ˇˇˇď cpsq 6n |ζ n ´t| 3 . ( 104
)
To conclude, it is enough to remark that:

|t ´ζn | ď |t| `|ζ n ´ζ8 | `|ζ 8 | ď n θ `1 `nθ ď 3n θ , (105)
i.e. ˇˇˇf n ptq ´fn pζ n q ´1 2 pt ´ζn q 2 f 2 n pζ n q ˇˇˇď 9 cpsq 2n 1´3θ .

Therefore, the constant Cpsq can be defined by:

Cpsq " 9cpsq 2 . ( 107
)
Tail completion. In this Paragraph, we are shoving the following Lemma 4.6. For all θ ą 1 4 , M n psq "

ż `nθ ´nθ exp ˆ1 2 pχ ´ζn q 2 f 2 n pζ n q ˙dχ sat- isfies M n psq " nÝÑ`8 n 1 4 c π s . (108)
Proof. ' According to Equation 7.4.32 of [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF], for all complex numbers a, b and c such that a ‰ 0, we know that:

ż e ´pax 2 `2bx`cq dx " 1 2 c π a exp ˆb2 ´ac a ˙erf ˆx? a `b ? a ˙`Cst , (109)
where the function erf is defined for all complex numbers z by: erfpzq " 2 ? π

ż z 0 e ´t2 dt . (110)
Therefore, we have:

ż R ´R e ´apx´bq 2 dx " 1 2 c π a
ˆerf ´?apR `bq ¯`erf ´?ap´R `bq ¯˙. (111)

Consequently, we deduce an explicit expression of M n psq from Equation (111)

with R " n θ , a " ´1 2 f 2 n pζ n q and b " ζ n . From now on, we will denote R n , a n and b n instead of R, a and b.

' According to Equation 7.1.16 of [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF], we know that lim zÝÑ`8 |argpzq|ă π 4

erfpzq " 1 . We will use that result to conclude the proof.

First, we remind that the sequence pζ n q nPN of saddle nodes is a converging sequence to ζ 8 " ´1 `s2 2s . Moreover, we also have: Let us come back to the integral K n psq. The following lemma will be useful:

f 2 n pζ n q " n `
Lemma 4.7. Let ps n q nPN and pζ n q nPN be two converging sequences of complex numbers. Let also θ ď 1 2 be a real number.

Then, there exists a sequence pα n q nPN of positive real numbers converging to 0 such that for all t P r´n θ ; n θ s, we have: Let us denote by Ă f n the function defined by Ă f n ptq " f n ptq ´fn pζ n q ´1 2 pt ´ζn q 2 f 2 n pζ n q (116) for all t P r´n θ , n θ s, as well as s n "

e
´?n 2 f 2 n pζ n q. Therefore, we have: In this section, we will first find a lower bound of Laguerre's polynomials on iR, losing the asymptotic exponential character of Equation ( 125). Then, in the next subsection, we will develop an explicit upper bound of U n pzq for z P C ´R`. Finally, we will review our two key examples (Examples 1 and 2) to find out upper bounds of the quadrature error involved in each example.

e

A lower bound of the modulus of Laguerre polynomials on iR

The n-th Laguerre polynomials can be explicitly defined by the following sum:

L n pXq " n ÿ k"0 ˆn k ˙p´Xq k k! . ( 127
)
Therefore, we have the following explicit expression for L n piXq| 2 : Proposition 5.2. Let n be a non-negative integer. Then:

|L n piXq| 2 " n ÿ d"0 ˆn d ˙ˆn `d d ˙X2d p2dq! . ( 128
)
Proof. Let us remind that the modified Bessel function I 0 is defined for all complex number z by:

I 0 pzq " ÿ kě0 1 k! 2 ´z 2 ¯2k . ( 129
)
According to the Hardy-Hille formula (see [START_REF] Szegö | Orthogonal Polynomials[END_REF], Theorem 5.1, p. 102), we successively have:

ÿ ně0 |L n piXq| 2 Y n " 1 1 ´Y I 0 ˜2X ? Y 1 ´Y ¸" ÿ dě0 1 d! 2 X 2d Y d p1 ´Y q 2d`1 " ÿ dě0 ÿ ně0 1 d! 2 ˆn `2d n ˙X2d Y n`d " ÿ dě0 ÿ něd 1 d! 2 ˆn `d n ´d˙X 2d Y n " ÿ ně0 ˜n ÿ d"0 1 d! 2 ˆn `d n ´d˙X 2d ¸Y n , (130)
which directly leads to the announced expression of | n piXq| 2 , n P N.

As a direct consequence, we deduce:

Corollary 5.3. Let n and d be non-negative integers.

Let us denote by rX d sP the d-th coefficient of the polynomial P . Then:

1. rX 2d`1 s|L n piXq| 2 " 0.

2. rX 2d s|L n piXq| 2 " 1 p2dq! ˆn d ˙ˆn `d d ˙ě 0, if 0 ď d ď n.
3. rX 2d s|L n piXq| 2 " 0 if d ą n.

Corollary 5.4. Let n and d be two non-negative integers such that 0 ď d ď n.

Let us define the constant C N,d by:

C N,d " d p2dq! `N d ˘`N`d N ˘. (131)
Then, for all real numbers t, we have:

|L n pitq| ě |t| d C N,d . ( 132
)
As explicit examples, we therefore have: Then, `rX 2d s|L n piXq| 2 ˘0ďdďn is a unimodal sequence (i.e. is first increasing and then decreasing). So, Corollary 5.4 can be used with the value d " d 0 pnq for which this sequence is maximum.

@n ě 0 , |L n pitq| ě 1 . ( 133
) @n ě 1 , |L n pitq| ě c npn `1q 2
The following Proposition gives us some characterization to compute easily and rapidly these values d 0 pnq for any power of 2. From the increassing character, this can be done for any non-negative integer n by exploration. For example, Table 3 gives some values d 0 p2 k q for k P rr0; .13ss. Proposition 5.5. Let us denote by d 0 pnq for all non-negative integer n, the maximum value of the unimodal sequence `bn,d ˘0ďdďn , with

b n,d " 1 p2dq! ˆn d ˙ˆn `d n ˙. ( 138
)
Let also k be a non-negative integer. Then, 1. d 0 p2 2k`1 q " 2 k .

2. d 0 p2 2k q ě t2 n lnp2qu.

3. d 0 ppq ď d 0 pqq if p ď q.

Proof. We have for all non-negative integer n and d such that 0 ď d ď n:

b n,d`1 b n,d ´1 " n 2 `n ´pd `1qp4d 3 `10d 2 `9d `2q 2pd `1q 3 p2d `1q . ( 139
)
Let us denote by P the polynomial P pn, dq " n 2 `n´pd`1qp4d Therefore, we deduce from Equation (140) that d 0 p2 2k`1 q ě 2 k . If d ě 2 k , we have 2d 2 ě n, so that P pn, dq ď ´p14d 3 `17d 2 `11d `2q ď 0. Therefore, we deduce from Equation (141) that d 0 p2 2k`1 q ď 2 k , which concludes the proof of the first point.

' Using Equation ( 140), we deduce that d 0 p2 2k q ě t2 n lnp2qu if, and only if P `22k , e k ´1˘ě 0, where e k " t2 k lnp2qu. But, from P pn, d ´1q " n 2 `n ´4d 4 `2d 3 ´d2 `d ě pn 2 ´4d 4 q `pn ´d2 q, 2 4k ´4e 4 k ě 2 4k `1 ´4 lnp2q 4 ˘ě 0 and 2 2k ´e2 k ě 2 2k `1 ´lnp2q 2 ˘ě 0, we deduce that P `22k , e k ´1˘ě 0 for all non-negative integer k.

' We have d 0 pn `1q ě d 0 pnq if, and only if, for all integer d such that d ă d 0 pnq, P pn `1, dq ě 0, which is clear according to P pn `1, dq " P pn, dq `2n 2 and Equation ( 140). This proves the increasing character of the sequence `d0 pnq ˘nPN .

A computational upper bound of the integral U n pzq

In this Section, we will adapt from Section 4 and sum up all the necessary Equations to have a computational and explicit upper bound of U n pzq for alls z P C ´R`.

Let us emphasize that, during the proof of Lemma 4.4, we have proven that, if n θ ě e psq (see Equations ( 86) and ( 89

" .

Therefore, the integral U n pzq satisfies

|U n pzq| ď exp ´´2 e psq ¨p? n ´nθ q ¯¨M n ps, θq (151)
if n θ ě e psq, where: Op1q.

s " ? ´z . (152)
Proof. The proof is straightforward, using Equations ( 60) and (75), as well as the upper bounds given by Equations ( 144) -(146).

Let us emphasize that Proposition 4.3 and Corollary 5.7 pursue the same goal: having a upper bound of U n pzq for z P C ´R`a nd n P N ˚. Proposition 4.3 is not a sharp upper bound, while Corollary 5.7 is, but it has an advantage on Corollary 5.7: the upper bound is a decreassing function of s " e p ? ´zq. We will use this fact in Proposition 5.9 of Subsection 5.4.

Back to the Example 1

We have proven in Example 6, i.e. the continuation of Example 1, a simple and explicit error formula:

e GL N paq " e ˆi L N p´iq ż `8 0 t N e ´t pt `iq N `1 dt ˙" e ˆi U N p´iq L N p´iq ˙, (158)
where the function a is defined by apzq " 1 1 `z2 for all z P C ´ti; ´iu. According to Proposition 5.1, we conjecture that: Conjecture 1.

1. e GL N paq "

N ÝÑ`8 4πe ´2? 2N .
2. |e GL N paq| ď 4πe ´2? 2N for all non-negative integers N .

If there is still a long way to go to reach it, we have created a path in its direction in Section 4 and Subsections 5.1 and 5.2.

The error e GL N paq containing a unique term U N pzq, we therefore prefer use Corollary 5.7 to Proposition 4.3. This gives directly the following upper bound:

|e GL N paq| ď ˇˇˇU N p´iq L N p´iq ˇˇď 1 |L N p´iq| e ´?2p ? N ´N θ q M N p ? ´i, θq . (159)
We can now compute the N -th Laguerre polynomial explicitly, using its explicit definition given by Equation (127), or using Corollary 5.4 to obtain a weaker (but easier and more rapid to compute) upper bound of the error:

|e GL N paq| ď c N e ´?2p ? N ´N θ q M N p ? ´i, θq , (160)
where 159) and (160) give explicit upper bounds of the error, so that we can now compute the integral ż `8 0 e ´t 1 `t2 dt by the Gauss-Laguerre quadrature to find out a predefined numbers of exact digits ; 2. the error e N paq nearly converges to 0 as rapidly as e ´2? 2N π .

c N " 1 C N,d0pN q " g f f f f e ´2d 0 pN q ¯! ˆN d 0 pN q ˙ˆN `d0 pN q N ˙. (161)
3. the upper bound in Equation ( 160) is quite sharp, only a few digits are lost compared to the exact values of the error e GL N paq.

Back to the Example 2

Let us now focus on our second main example, i.e. Example 2 which is pursued in Example 7: the example of application of the Gauss-Laguerre-like quadrature to the constant function 1.

A new expression of E N p1q

We have proven in Example 7 an explicit error formula given by Equation (52):

E N p1q " e ˜ż `8 0 ´ÿ kě0 2 L N `p2k `1qiπ ˘tN e ´t `t ´p2k `1qiπ ˘N`1 ¯dt ¸. (162)
We are now able to prove the permutation of the symbols ř and ş in Equation (52) using term-by-term integration. Let us denote by f k the function defined over R `by

f k ptq " 1 L N `p2k `1qiπ ˘tN e ´t `t ´p2k `1qiπ ˘N`1 . (163)
According to Equation (134), we have for all non-negative integers k and all non-negative real numbers t:

|f k ptq| ď ? 2e ´t N p2k `1q 2 π 2 ď ? 2 N p2k `1q 2 π 2 . ( 164
)
Therefore, the Weierstrass' M test shows that the series

ÿ k f k is normally con- vergent on R `and its sum is a continuous function on R `.
Moreover, we have:

ż `8 0 |f k ptq| dt ď ? 2 N p2k `1q 2 π 2 ż `8 0 e ´t dt " ? 2 N p2k `1q 2 π 2 , (165) so that ÿ
k ż `8 0 |f k ptq| dt is a convergent series.
Consequently, it is possible to permute the symbol ř and ş in Equation (52), which gives the following: Proposition 5.8. For all positive integer N , we have:

E N p1q " ÿ kě0 e ˜2 L N `p2k `1qiπ ˘ż `8 0 t N e ´t `t ´p2k `1qiπ ˘N`1 dt ¸(166) " ÿ kě0 e ˜UN `p2k `1qiπ LN `p2k `1qiπ ˘¸. (167)

Upper bounds of E N p1q

According to Propositions 5.1 and 5.8, we conjecture the following upper bound of the error:

Conjecture 2. 1. E N p1q " N ÝÑ`8 e ˆ2 |U N piπq| |L N piπq| ˙.

|E N p1q| ď 4πe

´2? 2N π for all non-negative integers N .

In this direction, we can prove the following explicit upper bound of E N p1q:

˜4 c 2 π `4 ? N ? π ¸ˆ1 `1 ? N π ˙exp ´´b N π `K `1 2 ˘c N p2N q d0pN q ˜ˆ1 π `3 2N ˙exp ´´N ? 2N ? N `π ¯`2 πd 0 pN q e ´N ¸, if K ď f l N . c N p2N q d0pN q ˜ˆ1 π `3 2N ˙exp ´´N ? 2N ? N `π ¯`2 πd 0 pN q e ´N ¸, if K " f l N `1 . 2 πd 0 pN q c N e ´N `p2K `1qπ ˘d0pNq , if K ą f l N `1 . (169)
Proof. To prove the Proposition, let us use the elementary upper bound of Proposition 4.3 instead of this given by Corollary 5.7, according to its decreassing caracter in s " e p ? ´zq. Consequently, the remainder of the series defining E N p1q will be estimated using an upper bound on the U N part, the convergence of the remainder being guaranteed using Corollary 5.4.

Consequently, we cut E N p1q in three parts, relatively to the integer

f l N " Z N π ´1 2 ^: E p1q N p1q " f l N ÿ k"0 ˇˇˇˇUN `p2k `1qiπ LN `p2k `1qiπ ˘ˇˇˇˇ( 170) E p2q N p1q " ˇˇˇˇUN `p2f l N `3qiπ LN `p2f l N `3qiπ ˘ˇˇˇˇ( 171) E p3q N p1q " ÿ kěf l N `2 ˇˇˇˇUN `p2k `1qiπ LN `p2k `1qiπ ˘ˇˇˇˇ( 172)
Let us also assume that N ě 2, so that d 0 pN q ą 0. We will prove the Proposition only when K " 0, the extension to the other values being left to the reader.

The E p1q N p1q part. The first case of Proposition 4.3 gives us:

E p1q N p1q ď f l N ÿ k"0 1 |L N ´p2k `1qiπ ¯¨exp ´´b N π `k `1 2 ˘k `1 2 ¯π¯3 4 ˜1 ´k `1 2 ¯1 4 4 ? π `4 ? N ? π ḑ ˜4 c 2 π `4 ? N ? π ¸cN 2 d0pN q f l N ÿ k"0 exp ´´b N π `k `1 2 ˘k `1 2 ¯π¯d 0pN q`3 4 , (173)
according to Corollary 5.4 used with d " d 0 pN q .

Therefore, we successively have, according to d 0 pN q ą 0:

E p1q N p1q ď c N 2 d0pN q ˜4 c 2 π `4 ? N ? π ¸¨e xp ´´b N π 2 π 2 ¯d0pNq`3 4 `ż `8 1 2
e ´?N πt

pπtq d0pN q`3 4 dt ‹ ' ď c N 2 d0pN q ˜4 c 2 π `4 ? N ? π ¸¨e xp ´´b N π 2 π 2 ¯d0pNq`3 4 `exp ´´b N π 2 π 2 ¯d0pNq`3 4 c 1 N π ‹ ' ď 2 3 4 c N π d0pN q`3 4 ˜4 c 2 π `4 ? N ? π ¸ˆ1 `1 ? N π ˙exp ˜´c N π 2 ¸. ( 174
)
The E p2q N p1q part. The second case of Proposition 4.3 gives us:

E p2q N p1q ď exp ´´b `f l N `3 2 ˘π ¨N ? N `π f l N `3 2 ˘π 3 `2N π ˇˇLN ´p2f l N `3qiπ ¯ˇˇ. (175)
Corollary 5.4, used with d " d 0 pN q, gives now:

E p2q N p1q ď c N ˆ3 `2N π ˙2 exp ´´b `f l N `3 2 ˘π ¨N ? N `π p2f l N `3qπ ¯d0pNq`1 ď c N ˆ3 `2N π ˙exp ´´N ? 2N ? N `π p2N q d0pN q`1 , (176)
according to p2f l N `3qπ ě 2π ˆN π ´1 2 ˙`π " 2N .

The E p3q N p1q part. The third case of Proposition 4.3 and Corollary 5.4, used with d " d 0 pN q ą 0 gives:

E p3q N p1q ď 2e ´N ÿ kěf l N `2 1 ˇˇLN ´p2k `1qiπ ¯ˇˇ¨1 `k `1 2 ˘π ď 2 π c N e ´N p2πq d0pN q ÿ kěf l N `2 1 `k `1 2 ˘d0pNq`1 ď 2 π c N e ´N d 0 pN q 1 p2πq d0pN q 1 `f l N `3 2 ˘d0pNq ď 2 π
c N e ´N d 0 pN q 1 p2N q d0pN q .

(177)

Numerical computations

Now, we can estimate the error E N p1q and then compute the integral ż `8 0 dt 1 `et up to d digits, where the integer d is defined in advance: using Proposition 5.9 with K " 0, we just have to find out the lowest value of N which gives the required precision, according to the inequation |E N p1q| ď 10 ´d and then perform the N -points Gauss-Laguerre-like quadrature associated with the constant function 1.

Once this is done, we obtain Table 5 which shows the difference of values between the exact value of |E N p1q| and the estimation given by Proposition 5.9. It also gives the calculation time of the quadrature of degree N .

Consequently, we see in Table 5 that:

1. the dominant part of an equivalent of the error E N p1q seems to be in e ´c? N .

2. the upper bound of |E 2N p1q| given by Proposition 5.9 is not really sharp, but is in the same order of magnitude as the exact value of |E N p1q|.

3. the calculation time of the quadrature of degree 2N seems to be two time longer than the calculation time of the quadrature of degree N , which is quite predictable according to the double numbers of nodes and weights for the first quadrature in comparison to the second one.

Another program to estimate the error E N p1q more precisely could be used, since each term of the summation is relatively small in comparison to the previous term. Consequently, E N p1q can be very well estimated by computing a few of its first terms. Then, the integral ż `8 0 dt 1 `et is computed up to d digits by determining two integers, N and K: N names the order of the Gauss-Laguerre-like quadrature we will have to perform to estimate the integral, while K names the number of terms we will use to estimate E N p1q.

On the first hand, according to Proposition 4.3, the integer K is chosen in such a way that the pK `1qth-remainder of the series E N p1q be small relatively to the order of magnitude of the first term of the sum. On the other hand, N will be the first positive integer we find out satisfying |E N p1q| ď 10 ´d. Now, we have to compute the first K terms of the sum E N p1q. This is done by computing the terms U N `p2k `1qiπ ˘using a Gauss-Laguerre quadrature, similar to this used in Example 1! Even if the upper bound of the remainder of |E N p1q| given in Equation ( 169) is not so precise, this program reduces the order of the Gauss-Laguerrelike quadrature to perform, to reach the required precision. This has a drastic effect on the computation time.

A quasi equivalent of |E N p1q|

Using Proposition 4.3, we are now able to prove the following Proposition 5.10. We have:

E N p1q " N ÝÑ`8 e ˆUN piπq L N piπq ˙`o ´e´2 ? 2N π ¯. (178)
This is a significative advance in the direction of the first point of Conjecture 2.

Proof. According to Proposition 5.9, we know that the Kth-remainder of E N p1q, denoted here by R N,K satisfies:

|R N,K | " N ÝÑ`8
Ope ´N q , (179)

when K ą Z N π ´1 2 ^.
Therefore, we have:

|R N,K | " N ÝÑ`8 pe ´2? 2N π q . ( 180
)
From Proposition 5.1, we also easily see that for all integers k, we have:

e 2 ? 2N π ¨ˇˇˇˇU N `p2k `1qiπ LN `p2k `1qiπ ˘ˇˇˇˇ" N ÝÑ`8 2π exp ´´2 ? 2N π cdotp ? 2k `1 ´1q ¯, (181)
so that we have for all positive integers k:

ˇˇˇˇUN `p2k `1qiπ LN `p2k `1qiπ ˘ˇˇˇˇ" N ÝÑ`8 o ´e´2 ? 2N π ¯. (182)
Consequently, we have proven that |R N,1 | "

N ÝÑ`8 pe ´2? 2N π q, which concludes the proof of the Proposition.

6 Annex A: proof of Proposition 3.2

In this Annex, we will prove the delicate Proposition 3.2, where great caution will be given to the possible values of R.

If R P R, let us denote by f R and g R the 2π-periodic and even functions defined for all θ P r0; 2πs by: g R pθq " 1 `2e ´R cos θ cospR sin θq `e´2R cos θ (183)

" `cospR sin θq `e´R cos θ ˘2 `sin 2 pR sin θq .

f R pθq " a g R pθq .

(184)

We shall study g R on r0; πs to find out a uniform (in the variable R) lower bound, valid for all R P R ˚´ď nPN " p2n `1qπ ´δ; p2n `1qπ `δ‰ .

In this direction, let us fix once and for all δ Ps0; πr, n P N ˚and R P sp2n ´1qπ `δ; p2n `1qπ ´δrXR ˚.

General behaviour of the functions f R . Figure 3 shows a few graphs of functions f R for a few values of R P t1; 5; 2π; 9.3; 10; 50u. We can see here that f R seems to have an approximative flat section, centered in 0, larger and larger with increasing values of R. It also seems to reach its maximum in π as an extremely intense peak with huge values of R.

In reality, the "flat section" is not so simple... On " 0; π 2 ı , the flat section turns out to be an oscillatory part, with very small amplitudes, but increasing in amplitudes with θ (see Fig. 4a to Fig. 4g). However, the easy part is on " π 2 ; π ı : f R is an increasing function on this interval (see Fig. 4h to Fig. 4i). Let us emphasize that the graphs of Figures 3 and4 have been obtained using the Python programming language (see [START_REF] Van Rossum | Python tutorial[END_REF]) as well as the Matplotlib library (see [START_REF] Hunter | Matplotlib: A 2D graphics environment[END_REF]). The quantity 1e ´n `u on top of the y-axis means that the scale has to be multiplied by 10 ´n, and then, we have to add u to the graphed quantity (for example, 1e ´13 `1 in Fig. 4a means 1 `10 ´13 ). Therefore, according to Figure 4, we see that f 30 pθq « 1 for all θ P r0; 1.45s up to two digits.

The increasing part of f R on " π 2 ; π ı . On this subinterval, it is sufficient to prove the following: Lemma 6.1. The function g R , as well as the function f R , are increasing functions on " π 2 ; π ı .

Proof. The fonction g R is C 8 on r0; 2πs and for all θ, we have: g 1 R pθq " 2R sin θe ´R cos θ cospR sin θq ´2R cos θe ´R cos θ sinpR sin θq `2R sin θe ´2R cos θ .

Let us define C " R cos θ and S " R sin θ, so that C ď 0 and S ě 0. Therefore, g 1 R pθq can be expressed as:

g 1 R pθq " 2Se ´C `cospSq `e´C ˘´2Ce ´C sin S " 2Se ´C ˆ`1 `cospSq ˘´C ¨´1 `sin S S ¯``e ´C ´1 `C˘˙. (186)
Each quantity inside the parenthesis of the right-hand side of Equation ( 186) is actually a positive one. So, g 1 R pθq is positive and g R is an increasing function in " π 2 ; π ı , as well as the function f R .

In contrast with Lemma 6.1, let us emphasize that the increasing part of f R can begin before π 2 (see for example Fig. 4f). Now, we can easily derive a uniform bound on the sub-interval " π 2 ; π ı .

Therefore, if z P CpR, 0q is such that e pzq ď 0, then, we have:

ˇˇˇ1 1 `e´z ˇˇˇď 1 2 sin ˆδ 2 ˙. (188)
Proof. We remind that, if z " Re iθ , θ P

" ´π; ´π 2  Y " π 2 ; π ı , we have: ˇˇˇ1 1 `e´z ˇˇˇ" 1 a g R pθq ď 1 c g R ´π 2 ¯" 1 a 2p1 `cos Rq ď 1 a 2p1 `cospπ ´δqq " 1 2 sin ˆδ 2 ˙, (189)
according to the 2π-periodicity of g R , its parity and its increasing caracter on " π 2 ; π ı .

Subdivision of the interval

k max pRq " max " k P N ; ˆk `1 2 ˙π R ď 1 * `1 " Z R π ´1 2 ^`1 . (190)
Now, we can split

" 0; π 2 ı : I 0 " " 0; arcsin ´π 2R ¯ı . ( 191
) I k " " arcsin ˆ´k ´1 2 ¯π R ˙; arcsin ˆ´k `1 2 ¯π R ˙ , k P rr1, k max pRq ´1ss . ( 192
) I kmax " " arcsin ˆ´k max pRq ´1 2 ¯π R ˙; π 2  . (193)
A lower bound of f R in easy subintervals I k . With these definitions, θ Þ ÝÑ cos `R sinpθq ˘is positive on each sub-interval I k with even indices k, and negative with odd indices k. Consequently, half part of the oscillatory part is easy: Lemma 6.3. Let n P N and R P sp2n ´1qπ; p2n `1qr X R ˚.

Then, f R pθq ě 1 for all θ P ď kPN 0ď2kďkmaxpRq I 2k .

Proof. With the previous notations, it is sufficient to remark that the function cospRsinq is positive on I 2k for all integers k such that 0 ď 2k ď k max .

Let us now focus when θ P I 1 , I 3 , ¨¨¨, i.e. when the function cospR sinq is negative. Lemma 6.4. Let n P N and R P sp2n ´1qπ; p2n `1qπr X R ˚.

Then, for all non-negative integers k such that 2k `1 ă k max pRq and all θ P I 2k`1 , we have:

f R pθq ě 0.93 . (194)
Proof. Let us fix n P N, R P sp2n ´1qπ; p2n `1qr X R ˚, an integer k satisfying 2k `1 ă k max pRq and θ P I 2k`1 .

Therefore, cos `R sinpθq ˘ď 0 and cospθq ě 0, so that:

f R pθq ě b p1 ´e´R cos θ q 2 " 1 ´e´R cos θ ě 1 ´e´R cos ˜arcsin ˜´2k`3 2 ¯π R ¸" 1 ´e´dR 2 ´´2k`3 2 ¯2π 2 ě 1 ´e´dR 2 ´´kmaxpRq´1 2 ¯2π 2 . ( 195
)
We can now remark that we can assume that n P N ˚. Otherwise, we would have n " 0 and R Ps0; πr. So that k max P t0; 1u. Consequently, there would be no non-negative integer k such that 2k `1 ă k max .

We therefore have, according to n P N ˚:

' if R P  p2n ´1qπ; ˆ2n ´1 2 ˙π" , k max " 2n ´2 and R 2 ´ˆk max ´1 2 ˙2 π 2 ě 3π 2 2 ˆ4n ´7 2 ˙ě 3π 2 4 . ( 196
) ' if R P "ˆ2 n ´1 2
which directly implies that cos R ď cospR sin θq ď 0 .

Consequently, we successively have:

g R pθq " 2e ´R cos θ `coshpR cos θq `cospR sin θq ě 2e ´R cos θ `coshpR cos θq `cospRq ě 1 ´cos 2 R, according to Lemma 6.5, since R cos θ ě 0 and ´1 ă cos R ă 0 .

Therefore, f R pθq ě sin R ě sin δ for all θ P I 2n`1 .

2. Then, let us consider n P N ˚and R P " p2n ´1qπ `δ; ˆ2n ´1 2 ˙π" .

The method, here, is to use the simple idea developed in the previous part of the proof. Nevertheless, we will have to split in three part the sub-interval I 2n´1 "

" arcsin ˆ´2n ´3 2 ¯π R ˙; π 2  .
On the interval J " " arcsin ´p2n ´1q π R ¯; π 2 ı , cospR sinq is an increasing function from ´1 to cos R ă 0. Let θ 0 P J Ă I 2n´1 be the unique solution on J of the equation cos `R sinpθq ˘" ´1 `cos R 2 " ´sin 2 ˆR 2 ˙:

θ 0 " arcsin ˆ1 R ´p2n ´1qπ `arccos `sin 2 R 2 ˘¯˙, (210)
which is a well defined quantity, according to Lemma 6.6.

'
The inequalities proven in Corollary 6.8 are more and more sharper when R approaches π modulo 2π. We refer to Figure 5 to see how these inequalities are accurate for values of R " 3π ´δ (first array) and R " 3π `δ (second array) and conclude that Inequality (207) is more precise than Inequality (206).

The oscillatory part of f R . Nevertheless, Inequality (206) will be sufficient to derive a uniform (in the parameter R) upper bound of the function f R on the sub-interval

" 0; π 2

Theorem 1 . 1 .ptq dt ´i ż ` 8 0f

 118 [START_REF] Lindelöf | Le calcul des résidus et ses applications à la théorie des fonctions[END_REF] Let m 0 ą 1 2 be a real number, m a positive integer such that m ě m 0 and f : Ω ÝÑ C be an holomorphic function over Ω " z P C , Re z ě m 0 ´1 2 ( satisfying:' the 1D-Lindelöf hypothesis ; ' ÿ νěm f pνq is a convergent series.Then, f P L 1 prm 0 ´1 2 ; `8rq and ÿ νěm pm ´1 2 `itq ´f pm ´1 2 ´itq e 2πt `1 dt .[START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF]

Theorem 2 . 1 .

 21 (Uspensky, 1928) If the function f satisfies the following property:

Example 6 .

 6 Let us come back to Example 1 defining the meromorphic function a over C by apzq " 1 1 `z2 for all z P C ´ti; ´iu.

Figure 1 :; π 2 "

 12 Figure 1: Paths used to modify the path of integration of U n pzq.

)

  Let us also define the function f n by f n pζq " n logpζ `?nq ´pn `1q logpζ `s `?nq ´s ¨ζ (79) for all ζ P C such that e ζ ą 0. Finally, let us denote by ζ n the saddle-node points of f n (i.e. a point ζ where f 1 n pζq " 0) defined by:

?n J n psq ˇˇď n 5 4 p e sq 2 5 4 p e sq 2 exp ¨´ e s n 2pθ´1 4 q 1 8 o

 425418 exp ˆ e s ´?n `nθ ´n ? n ´nθ ` e s ¯ď n show that J n psq " nÝÑ`´n´1 4 e ´2s ? n ¯, according to θ ą 1 4 .

Figure 2 :

 2 Figure 2: Illustration of the used variables in the proof Let us remark that x Þ ÝÑ x 3 ´x2 ´1 is an increasing function on " 2 3 ; `8"

  (a) Function f1 (b) Function f5 (c) Function f2π (d) Function f9.3 (e) Function f10 (f) Function f50

Figure 3 :

 3 Figure 3: Graphs of function f R for R P t1; 5; 2π; 9.2; 10; 50u. 47

Figure 4 :Corollary 6 . 2 .

 462 Figure 4: The oscillatory part of f 30 on " 0; π 2 ı and the increasing part on " π 2 ; π ı

2 ı; π 2 ı' If R ą π 2 ,

 222 in sub-intervals such that θ Þ ÝÑ cos `R sinpθq ˘has a constant sign on each sub-interval.' If R P " 0, we just have to define I 0 " we first define the positive integer k max pRq by

Table 1 :

 1 The 16-points Gauss-Laguerre quadrature.

		N odes		W eights
	0.p1q 87649 41047 89278 40360	0.	20615 17149 57800
	0.	46269 63289 15080 83188	0.	33105 78549 50884
	1.	14105 77748 31226 85688	0.	26579 57776 44214
	2.	12928 36450 98380 61633	0.	13629 69342 96377
	3.	43708 66338 93206 64523	0.p1q 47328 92869 41252
	5.	07801 86145 49767 91292	0.p1q 11299 90008 03394
	7.	07033 85350 48234 13040	0.p2q 18490 70943 52631
	9.	43831 43363 91938 78395	0.p3q 20427 19153 08278
	12.	21422 33688 66158 73694	0.p4q 14844 58687 39812
	15.	44152 73687 81617 07676	0.p6q 68283 19330 87119
	19.	18015 68567 53134 85466	0.p7q 18810 24841 07967
	23.	51590 56939 91908 53182	0.p9q 28623 50242 97388
	28.	57872 97428 82140 36752	0.p11q 21270 79033 22410
	34.	58339 87022 86625 81453	0.p14q 62979 67002 51786
	41.	94045 26476 88332 63547	0.p17q 50504 73700 03551
	51.	70116 03395 43318 36434	0.p21q 41614 62370 37285

Table 2 :

 2

The 16-points Gauss-Laguerre-like quadrature.

  Here again, a library for arbitrary-precision floating-point arithmetic is needed to obtain more precise results for greater powers of two. This table suggests that the logarithm of the error ln |E n p1q| has a simple equivalent: there exists C P R, ln |E n p1q| "

				nÝÑ`8	C	?	n. More precisely, we
	conjecture that C "	´2? 2π (see Conjecture 2).
	1 0. 73105 85786	30005	0.p1q	379113980700596
	2 0. 69010 24700	33026	´0.p2q	304471052691971
	4 0. 69312 37952	35122	´0.p4q	233853248232775
	8 0. 69314 27730	86691	´0.p5q	440747325425636
	16 0. 69314 71785	75662	´0.p8q	198428359047497
	32 0. 69314 71805	54994	´0.p11q 495161506186579
	64 0. 69314 71805	59946	´0.p16q 387982796897492

  Proposition 4.3. For all z P C ´R`a nd all positive integer n, let us denote

	According to
					z P C ´R`ù ñ Argp	?	´zq P	ı ´π 2	;	π 2	"	ùñ s ą 0 ,	(74)
	Lemma 4.2 now concludes the proof.
	s " e p	?	´zq. Therefore, we have:
						$ ' ' ' ' '	e s ´s? n ? s	ˆ1 ? s	`4 ?	n	?	π ˙, if 0 ď s ď	?	n .
						'
	ˇˇˇż	`8 0	x n px ´zq n`1 e ´x dx ˇˇˇď	' ' & ' ' '	e	´s ? n ? n`π s 2	ˆ3	`2n π	˙, if s Ps ?	n;	?	n `πs .
						'
						' ' ' ' %	2e s 2 ´n				, if s ą	?	n `π .	(72)
	Proof. Equations (58) and (60) as well as Inequation (61) can be used together
	and rewritten as:
					|U (73)

n pzq| ď I n psq ď 1 s ¨r I n ps, sq .

  1 e ´t e s dt

	deduce that	n e psq	ě	?	n ´nθ `n ? n´n θ ` e psq . Therefore, we successively have:
	|J n psq| ď	1 2 e psq	ż n e psq n´n θ `n ? n´n θ ` e psq ?	e ´u e psq	b	`u ` e psq ˘2 ´4n u ` e psq	du
				ˆ´
		ď	exp	ď ď ˆ´	ż ? 0 e psq n´n θ ¨`? n ´nθ `n ? ˆ1 ´ e s t ` e s ˙n e ´t e s t ` e s n ´nθ ` e psq e psq dt ˘2 "b 1 e s ż ? n´n θ 0 exp ˆ´ e s ¨´t ¯˙dt . (82)  n e psq `u ` e psq ˘2 ´4n ? n´n θ `n ? n´n θ ` e psq `n t ` e s
	The functions ϕ : r2 ? s ´ e psq; ? n ´ e psqs ÝÑ r2 n ´ e psq; `8rÝÑs ´ e psq; ? n ´ e psq; `8r defined by ? n ´ e psqs and ψ : ď exp e psq ˘2 ¨`? n ´nθ `n ? n ´nθ ` e psq e psq ˇˇˇn e psq ´ e psq ˇˇď
			ϕpuq " n 2 e psq 2 ¨exp 1 2 ˆu ´ e psq ˆ´ e psq ¨`? n ´nθ `n ? ´b`u ` e psq ˘2 ´4n ˙(83) n ´nθ ` e psq ˘˙,
								ψptq " t	`n t ` e psq	(84)
	are inverse functions.
		Therefore, using the substitution t " ϕpuq, we have:
	ż ?	n´n θ	ˆ´					¯˙dt
	0	exp	e s ¨´t	`n t ` e s
					ż ψp ?	n´n θ q
				"				e ´u e s ϕ 1 puq du
					ψp0q
				"	1 2	n e psq ż ? n´n θ `n ? n´n θ ` e psq	e ´u e s ¨1	`u ` e psq ˘2 ´4n ´u ` e psq b	' du (85)
		According to the decreassing caracter of ψ on s ´ e psq;	?	n ´ e psqs, we

  1 e ´t e s dt On the integral K n psq. main goal of this Subsection is to show that K n psq "

	Denoting N pn, s, θq " t " 1 2 ˆu ´ e psq `b`p u ` e psq ˘2 ´4n ˙, v " u `s ? n `nθ ` e psq and using successively the substitutions ´2? n and then w " v ? s, we have: 4.5 The nÝÑ`8 ? πe ´z 2 4 p´nzq 1 e ´2s ? n .
	Using the substitution u " t ´?n, we have:
	|L n psq| ď	1 2N pn, s, θq K n psq " e ż `8 N pn,s,θq´s`n N pn,s,θq ´s? n ż `nθ ´nθ pu `?nq n e ´u e psq ¨1 pu `s `?nq n`1 e ´su du `u b `u ` e psq ˘2 ´4n `s	' du .
	"	e e psq¨` e psq´2 ? pn, s, θq " e ´s? n n 2N ż `8 `?Npn,s,θq´? n N pn,s,θq ´nθ ż `nθ e fnpuq du ,	˘2 e ´v e psq ˜1	`v a v 2 `4v `2? n ?	n	¸dv
	ď	e e psq¨` e psq´2 ? pn, s, θq	n	N ż `8 `?Npn,s,θq´? n N pn,s,θq	˘2 e ´v e psq ˆ1	`4 ? ? n v	˙dv
	"	e e psq¨` e psq´2 ? pn, s, θq	n	N ˜1 e psq	exp	ˆ´	N pn,s,θq e psq ´aN pn, s, θq ´b n	¯2˙2
									a	4 ? e psq n	? ż `8 e psq `?Npn,s,θq´? n N pn,s,θq	˘e´w 2	dw	¸,
									according to N pn, s, θq ě n
	ď	e e psq¨` e psq´2 ? pn, s, θq	n	N ¨exp ˆ´	e psq ´aN pn, s, θq	´c	N pn, s, θq n	¯21
		e psq	¨˜1	`4 ?	n	N pn, s, θq ´?n N pn, s, θq a	ḑ
									ḑ
		1 e psqN pn, s, θq	¨exp	ˆ´	e psq ´?n `nθ `n N pn, s, θq	¯˙¨˜1	`N pn, s, θq N pn, s, θq ´?n
		1 e psq	¨exp	ˆ´		e psq ´?n `nθ `n N pn, s, θq	¯˙¨ˆ1 ? n `1 n θ ˙.	(89)
	We finally have:
	ď n L n psq ˇˇď ż `8 ? n`n θ ď ? n `nθ ` e psq ˆ1 ´ t ` e psq e s ˆ´ 1 e psq ¨exp 1 2 e psq ¨exp ˆ´ ż `8 ? n`n θ e psq ´nθ ´?n ˙n e ´t e s t ` e s dt ˆ´ e psq n 2θ ´ e psqp `n N pn, s, θq ? n ´nθ q ¯˙¨ˆ1 4 ? n `1 n θ´1 4 ˙. ? (90) ď n `nθ ` e psq exp e psq ´t `n t ` e psq ¯˙dt . (88) This concludes the proof of L n psq " ˇˇn 1 4 e 2s ? nÝÑ`8 o ´n´1 4 e ´2s ? n ¯, according to θ ą 1 4 .

  1 pζ n `?n `sq 2 ´n pζ n `?nq 2 "

	i.e. a n " nÝÑ`8 Thus, ? a n pR n `bn q " s ? . n nÝÑ`8 From z P C ´R`, we know that argp n θ´1 4 Consequently, we have, since θ ą	? ? sq P s and ı ´π 4 ? a n p´R n `bn q " nÝÑ`8 ; " π . 4 π 4 :	´nθ´1 4	?	s.
	erf ´?a n pR n `bn q ¯`erf ´?a n p´R n `bn q ¯ÝÑ nÝÑ`8	2 ,	(113)
	which finally gives us
	M n psq " nÝÑ`8	c	π a n	" nÝÑ`8	n	1 4	c	π s	.	(114)
	4.6 End of the proof of Proposition 4.1.
								nÝÑ`8	´2s ? n	,	(112)

  As a final remark, let us mention that the sequence `rX 2d s|L n piXq| 2 ˘0ďdďn is log-concave for all integers n: `rX 2d s|L n piXq| 2 ˘2 rX 2d s|L n´1 piXq| 2 ¨rX 2d s|L n`1 piXq| 2

	"	pd `1qp2d `1q p2d ´1qpd ´1q	¨pd `1q 2 d 2	¨n ´d n ´d `1	`d n `d ¨n `1
	"	p2d `1qpd `1q 3 p2d ´1qpd ´1qd 2	¨n2 ´d2 `n n 2 ´d2 `n	`d ´d ě 1	(137)
							|t| ě	n|t| ? 2	.	(134)
	@n ě 2 , |L n pitq| ě	c	pn ´1qnpn `1qpn `2q 96	t 2 ě	pn ´1q 2 t 2 4 ? 6	. (135)
	@n ě 0 , |L n pitq| ě	|t| n n!	.		(136)

Table 3 :

 3 Values of d 0 pnq when n is the first power of 2.

	3 `10d 2 `9d`2q.

' Let us now fix n " 2 2k`1 for a non-negative integer k. If d ď 2 k ´1, we have 2pd `1q 2 ď n, so that P pn, dq ě pd `1qp2d 2 `5d `4q ě 0.

  Moreover, using elementary techniques, we can obtain an upper bound of |K n psq|. Of course, this upper bound will not behave as the equivalent obtained in Equation (122), but the term expp´2 e psq ¨?nq will nevertheless naturally appear.Corollary 5.7. Let z P C ´R`a nd n be a positive integer and θ P

	Consequently, we have:
	ż n θ ´nθ	pu `?nq n pu `?n `sq n`1 e ´su du
	Lemma 5.6. For all s P C such that e psq ą 0, θ P ď ln ˆ1 `2n θ ? n ´nθ `s ˙¨exp ˆ´s n `2sn θ ´sn 2θ´1 2 `s2  1 4 ; 1 3 " and n ě 2, we 2 `s ¨cn ps, θq ?	ď
	have: |K n psq| ď ? 2n θ e e psq ¨cn ´nθ ` e psq ` e psq, θ 2n θ n ´nθ `s ¨exp ˆ´s ? ?n exp ˆ´ e psq ´2? n `2sn θ ´sn 2θ´1 2 `s2 2 n ´2n θ `n2θ´1 2 ´ `s ¨cn ps, θq ˙, (150) e psq ¯˙, 2 which concludes the proof of the Lemma.
						(146)
	where c n ps, θq " Let us finally summarize the situation in order to have a computational n 3θ´1 2 ? n `nθ `s n 1 2 ´θ ´1 `s 2pn 1 2 ´θ ´1q 2 " Op1q . nÝÑ`8 upper bound of the integral U n pzq " ż `8 0 t n pt ´zq n`1 e ´t dt:
						
				´s? n	ż `nθ ´nθ	pu `?nq n pu `?n `sq n`1 e ´su du ,	(147)
	so that we have: |K n psq| ď K n	`	e psq	˘.
	pu `?nq n pu `?n `sq n`1 e ´su
		" ˆ1	`s u `?n	˙´n	e ´su u `?n	`s
		ď exp	ˆ´ns u `?n	)): ˙e´su 2pu `?nq 2 `ns 2 u `?n	`s ,
	|J n psq| ď |L n psq| ď ď exp n 2 e psq 2 exp 1 e psq ¨exp ˆ´ ˆ´ns ˆ´ e psq e psq according to lnp1 `xq ě x ¨´? n ´nθ `n ? n ´nθ ` e psq ´x2 for all x ě 0 ¯˙, (142) 2 ¨´? n `nθ `n ? n `nθ ` e psq ¯˙¨ˆ1 ? ˙. ? n `nθ ˙esn θ `ns 2 2p ? n ´nθ q 2 u `?n (148) `s . n `1 n θ (143) Moreover, we have:
	This can also be rewritten, or weakened, into the simpler upper bounds: |J n psq| ď n 2 e psq 2 exp ˆ´ e psq ¨´2 ? n ´ e psq ˆ´ns ? n `nθ `ns 2 2p ? n ´nθ q 2 `sn θ ˙´ˆ´s " ¯˙, ? n `2sn θ ´sn 2θ´1 2 `s2 2 `pn θ ´ e psqq 2 ? n ´nθ ` e psq (144) |L n psq| ď 2 e psq ¨exp ˜´ e psq ¨´2 ? n `nθ ´nθ`1 2 `s? n ? n `nθ `s ¯¸. (145) sn 3θ´1 2 ? n `nθ `s2 n 1 2 ´θ ´1 `s2 2pn 1 2 ´θ ´1q 2 " s ¨cn ps, θq . (149)

Proof. ' First of all, let us remind that K n psq is defined as

K n psq " e

' From now on, let us assume that s P R ˚. For all u P r´n θ ; n θ s, we successively have:

  Remark 2. Let us emphasize that j n ps, θq ÝÑ

	M n ps, θq "	n ¨ejnps,θq 2 e psq 2	`2n θ ¨eknps,θq ? n ´nθ ` e psq	`2 e psq	e lnps,θq . (153)
	j n ps, θq "	e psq ¨˜ e psq	´`n θ ´ e psq ˘2 ? n ´nθ ` e psq	´2n θ ¸. (154)
	k n ps, θq " l n ps, θq "	e psq ¨´´n 2θ´1 2 ` e psq ¨˜´3n θ `nθ`1 2 ` e psq e psq `cn 2 ? ` ? n `nθ ` e psq e psq, θ ˘¯. ¸. n	(155) (156)
	c n ps, θq "	n 3θ´1 2 ? n `nθ `s n 1 2 ´θ	´1	`s 2pn 1 2 ´θ ´1q 2	.	(157)
						nÐÑ`8	´8, k n ps, θq ÝÑ nÐÑ`8	´8,
	and l n ps, θq ÝÑ nÐÑ`8	´8, so that M n ps, θq	" nÐÑ`8

Table 4

 4

	shows the exact values of e GL N paq, its explicit upper bound given by Equation (160) for θ " 7 24 , as well as its minimal value for θ P  1 4 ; " 1 3 for all
	N P t2 k ; rr0; 13ssu.

Table 4 :

 4 Explicit upper bounds of the error e GL N paq . .

	Consequently, we see here that:
	1. Equations (

Table 5 :

 5 Explicit upper bounds of the error E N p1q . .

		Upper bound			Calculation
	N	of |E N p1q| from	Exact value	time of the
		Prop. 5.9 and	of |E N p1q|	quadrature of
		Eq. (132).			degree N
	2	0.p1q	66667 0.p2q	30447	0. 738 ms
	4	0.p2q	75577 0.p4q	23385	1. 332 ms
	8	0.p3q	30836 0.p5q	44075	2. 659 ms
	16	0.p4q	17694 0.p8q	19843	5. 213 ms
	32	0.p7q	26138 0.p11q 49516	10. 257 ms
	64	0.p10q	20262 0.p16q 38798	17. 487 ms
	128 0.p15q	12205 0.p23q 26126	32. 623 ms
	256 0.p22q	18595 0.p34q 38050	55. 362 ms
	512 0.p32q	16620 0.p48q 60513	110. 334 ms
	1024 0.p46q	30823 0.p68q 25357	223. 837 ms
	2048 0.p66q	18811 0.p97q 27463	489. 749 ms
	4096 0.p94q	16064 0.p138q 51246	916. 177 ms
	8192 0.p134q 14522 0.p196q 20657 1853. 715 ms

  Then, the function x Þ ÝÑ e ´x pcosh x `αq is a decreasing function on r0; ´ln |α|s Let n be an integer. Let us also denote sgpxq the sign of the real number x.One can easily show that A : x Þ ÝÑ arccos ´sin 2 `x 2

	˙π; ˆ2n Lemma 6.7. For all non-negative integers n, the function `1 2 ˙π" , k max " 2n ´1 and On " ˙π" p2n `1qπ; ˆ2n `3 2 , A and Q are increasing function. Therefore,
	R 2 ´ˆk max x Þ ÝÑ x 2 ´ˆnπ `arccos ´1 2 ˙2 π 2 ě p4n ´2qπ 2 ě 2π 2 . `1 ´cos x ˘˙2 according to Equation (202), we have on this interval: 2 is an increasing function on " nπ; ˆn `1 2 ˙π . f 1 `2´p 2n `1qπ `A`p 2n `1qπ ˘¯¨lim 2n`1 pxq ě 2x tÝÑ `p2n`1qπ	(200) (197) ˘`Qptq
	' if R P Proof. ' ˘¯is an increasing func-"ˆ2 n `1 2 ˙π; p2n `1qπ " , k max " 2n and R 2 ´ˆk max ´1 2 " 2p2n `1qπ ´?2p2n `1qπ ě 0 . (205) Consequently, for all non-negative integers n, f 1 n is positive on " ˙π nπ; ˆn `1 2 ˙2 π 2 ě 4nπ 2 ě 4π 2 . (198) tion on rp2n ´1qπ; 2nπs and decreassing on r2nπ; p2n `1qπs. so that f n is finally increasing on this interval.	,
	Consequently, we have R 2 ´ˆk max f pθq ě 1 ´e´π ´1 2 ? 3 ˙2 π 2 ě 2 ě 0.93 . 3π 2 4 The remaining subintervals of " 0; π the remaining subintervals I k 's of I where we do not already have studied , i.e.: (199) 2 ı to study. If R Psp2n ´1qπ; p2n 1qπr, The opposite of the derivative of A, defined by Q : x Þ ÝÑ sg ´cos `x 2 ˘¯sin c1 Therefore, we have: ´x 2 `sin 2 ¯, Corollary 6.8. Let δ P ı 0; " π 2 . ´x 2 also turns out to be increasing on sp4n ´1qπ; p4n `1qπr and on sp4n `1qπ; p4n 3qπr, according to the increasing caracter of x Þ ÝÑ x ? 1. If n P N ˚, R P " ˙π" p2n ´1qπ `δ; ˆ2n and θ P I 2n´1 , we then have: ´1 2 on R. 1 `x2 c f R pθq ě min ˜1 ´e´δ ; 1 ´cos 4 ¯¸. ´δ 2 (206)
	a lower bound of g R are : I 2n´1 if R P  2. If n P N, R P ˆ2 n `1 2 p2n ´1qπ; ´2n ˙π; p2n `1qπ ¯π" ´1 2 and n P N ´δ and θ P I 2n`1 , we then have: (201) ˚. I 2n`1 if R P " ´2n `1 2 ¯π; p2n `1qπ " and n P N. This function is C 1 on " nπ; ˆn ˙ for all nonnegative integers n, and we f R pθq ě sin δ . (207) `1 2 have, if x R π 2 `πZ: Proof. 1. Let us consider first n P N and R P ˆ2 n ˙π; p2n `1qπ ´δ . `1 2 f 1 n pxq " 2x `2 pnπ `Apxqq Qpxq . (202) " 2 `nπ `Apxq ˘ˆx nπ `Apxq `Qpxq ˙. (203) For all θ P I 2n`1 " " arcsin ˆ´2n `1 2 ¯π R  ˙; π 2 , we therefore have:
	from 1 `α to approaching 2 one. Therefore, according to Equation (203), we have on this interval: 1 " near `8. 1 2 p1 ´α2 q, and then is an increasing function on r´ln |α|; `8r On " `1 2 , A is a decreasing function while Q is an increasing 2nπ; ˆ2n ˙π" 2nπ `π 2
	Lemma 6.6. The function x Þ ÝÑ x ´arccos f 1 2n pxq ě 2 `2nπ `Apxq ˘ˆ2nπ ˆ1 ´cos x nπ `Ap2nπq 2 tion on R. 2 `2nπ `Apxq ˘2nπ ě 8n 2 π 2 ˙is an increasing func-`Qp2nπq ě 0 , (204)
	We will also need the following	nπ	`π 2	nπ	`π 2
	for all x P	"	2nπ; ˆ2n	`1 2	˙π"	.

Some preliminaries lemma To cover these last two cases, let us state without proof the following two easy lemmas: Lemma 6.5. Let α Ps ´1; 0r.

' Let us now denote by f n the function defined on " nπ; ˆn `1 2 ˙ by:

f n pxq " x 2 ´´nπ `arccos ´sin 2 `x 2 ˘¯¯2 " x 2

´`nπ `Apxq ˘2 . ď R sin θ ď R ă p2n `1qπ ,

  For θ P R pθq " 2e ´R cos θ `coshpR cos θq `cospR sin θq ě 2e ´R cos θ `coshpR cos θq ´sin 2

	Therefore, we have:
					b
	f R pθq ě	p1 ´e´R cos θ q	2 " 1 ´e´R cos θ ě 1 ´e´δ .	(214)
	' As in the first part of the proof, for all θ P	" θ 0 ;	π 2	ı , we have:
					´sin 2	ˆR 2	˙ď cos `R sin θ ˘ď cos R ă 0 .	(215)
	Consequently, we successively have:
	g ˆR 2	˙ě
	" arcsin ˆ´2n 1 ´sin 4 ˆR 2 ´3 2 ¯π R ˙, according to Lemma 6.5, since ˙; arcsin ´´2n ´1¯π R ¯, we have: $ & R cos θ ě 0 % ´1 ă ´sin 2 ě 1 ´cos 4 ˆδ 2 ˙.	ˆR 2	˙ă 0
	R cos θ ě ě Therefore, we have: f R pθq ě a R 2 ´R2 sin 2 θ d a R 2 ´p2n ´1q 2 π 2 b 1 ´cos 4 ˆδ 2	˙for all θ P	" θ 0 ;	π 2	ı .
					ě		`p2n `1qπ	`δ˘2 ´p2n `1q 2 π 2 ě δ .	(211)
	Therefore, we have:
								b
					f R pθq ě	p1 ´e´R cos θ q	2 ě 1 ´e´δ .	(212)
	' For θ P	" arcsin ´´2n	´1¯π R	¯; θ 0	ı , we have:
							b
	R cos θ ě R cos θ 0 "	R 2 ´R2 sin 2 θ 0
			d
		"		R 2 ´ˆp2n ´1qπ `arccos ´sin 2 `R 2	˘¯˙2
			d
		ě		pp2n ´1qπ `δq	2 ´ˆp2n ´1qπ `arccos ´sin 2 `nπ	´π 2 `δ 2	˘¯˙2	,
								according to Lemma 6.7
		"	d pp2n ´1qπ `δq	2 ´ˆp2n ´1qπ	`δ 2	˙2 "	c p2n ´1qδπ	`3δ 2 4
		ě	?	δπ ě	?	δ 2 " δ .	(213)
								54

Finally, unifying all the results on the three sub-intervals of I 2n´1 , we have proven that f R pθq ě min ˜1 ´e´δ ; c 1 ´cos 4 ´δ 2 ¯¸for all θ P I 2n´1 .

  ı, i.e. on its oscillatory part.Corollary 6.9. Let R P R `´p2Z `1qπ, and δ P s0; πs.Let us also consider the integer n such that: Examples of tests of inequalities obtained in the cases 1 and 2 of Corollary 6.8.Therefore, if z P CpR, 0q is such that e pzq ě 0, then, we have:Proof. ' Let us assume that δ Ps0; πs. , I 1 , ¨¨¨, I 2n´1 . From Lemmas 6.3 and 6.4, as well as the first part of Corollary 6.8, we have for all θ P , I 1 , ¨¨¨, I 2n´1 , I 2n . From Lemmas 6.3 and 6.4, we have for all θ P , I 1 , ¨¨¨, I 2n`1 . From Lemmas 6.3 and 6.4, as well as the first part of Corollary 6.8, we have for all θ P As a conclusion, grouping the results of Equations (219) to (221) gives us for all δ Ps0; πs and all θ P As a conclusion of the proof, if z " Re iθ , e z ą 0, we have θ P Conclusion of the proof of Proposition 3.2. To conclude the proof of Proposition 3.2, we need now to put together Corollary 6.9 and 6.2. All we have to do is now to remark that, for x P r0; πs:

	p2n ´1qπ `δ ď R ď p2n `1qπ ´δ . 7.9 8.5 9 9.3 1.5248 0.9248 0.4248 0.1248 0.9992 0.8069 0.4126 0.1246 0.00477796 9.42 0.0048 0.9989 0.7985 0.4121 0.1245 0.00477794 relative error 0.03 % 1.04 % 0.12 % 0.08 % R δ inf r0; π 2 s f R sinpδq 0.0004 % R 10.9 10.3 9.8 9.5 δ 1.4752 0.8752 0.3752 0.07522 0.005222039 (217) 9.43 inf r0; π 2 s f R 0.9709 0.7611 0.3662 0.07520 0.005222033 min ˜1 ´e´δ ; c 1 ´cos 4 δ 2 ¸0.7713 0.5718 0.2615 0.0532 0.0037 relative error 20.55 % 24.88 % 28.59 % 29.26 % 28.85 % Figure 5: ˇˇˇ1 1 `e´z ˇˇˇď 1 min ˜0.93; sin δ; 1 ´e´δ ; c 1 ´cos 4 ´δ 2 ¯¸. (218) If p2n ´1qπ `δ ď R ă ˆ2n ´1 2 ˙π, " 0; π 2 ı is subdivided using the intervals I 0 " 0; π 2 ı : f R pθq ě min ˜0.93; 1 ´e´δ ; c 1 ´cos 4 ´δ 2 ¯¸. (219) If now ˆ2n ´1 2 ˙π ď R ă ˆ2n `1 2 ˙π, " 0; π 2 ı is subdivided using the intervals I 0 " 0; π 2 ı : f R pθq ě 0.93 . (220) Finally, if ˆ2n `1 2 ˙π ď R ă p2n `1qπ ´δ, " 0; π 2 ı is subdivided using the 0; π 2 ı : f R pθq ě min p0.93; sinpδqq . (221) ' " 0; π 2 ı : f R pθq ě min ˜0.93; sin δ; 1 ´e´δ ; c 1 ´cos 4 ´δ 2 ¯¸. (222) ı ´π 2 ; π 2 " . Therefore, we have: ˇˇˇ1 1 `e´z ˇˇˇ" 1 |f R pθq| " 1 ˇˇf R p|θ|q ˇď 1 min ˜0.93; sin δ; 1 ´e´δ ; c 1 ´cos 4 ´δ 2 ¯¸. (223) intervals I 0 " 2 sin ´x 2 ¯ě sin x . (224)

Proposition 5.9. Let N ě 2 be an integer, and let us denote f l N "

For all K ě 0, we have: