Location of the zeros of certain parametric families of functions of generalized Fresnel integral type

In this paper, two parametric families of functions, the so-called Complementary Fresnel Integral and the Lommel type, which are of generalized Fresnel integral type, are considered. We review the problems of existence and uniqueness of their zeros in certain determined intervals, called location intervals, which improve previous results of other authors. For the approximation error obtained, bounds, monotonicity as well as the asymptotic behavior are analyzed. The study uses results from the theory of xed point of real functions, introducing the concept of xed point sequential problem (FPSP) and the properties of certain special functions.


Introduction
It is considered in this work several families of functions of real variable depending on a real parameter: the family called Complementary Fresnel Integral (CFI), and a certain type of functions of the Lommel type whose precise denitions will be given later. If f α , α ∈ J, for J in an interval of R, is one of these families, the associated zeros are studied nding rst numerable families of disjoint intervals I n,α , n ∈ N (location intervals ), each interval containing one and only one zero of the function f α , and each zero of f α belonging to only one of them. Then, if a n,α , b n,α dene the lower end (respectively upper) of I n,α and z n,α is the only zero of f α in that interval we look for bounds of the successions of errors |a n,α − z n,α |, |b n,α − z n,α |, n ∈ N, α ∈ J been xed. We establish the monotonicity of these sequences of errors and nd asymptotic equivalents of them, whenever they converge to zero.
Our study constitutes a revision and at the same time an extension of the work done on the CFI family in [10] and those of the Lommel family carried out by S. Koumandos and M. Lamprecht in [7]. To carry out a joint study of the two mentioned families, we introduce in a preliminary section a basic theory of the so-called xed point sequential problem (FPSP), where we enunciate the existence and uniqueness of xed-point of the real variable. In other sections we study several families of auxiliary parametric functions dened by integrals, in several aspects: analytical properties with respect to the variable, basic identities and inequalities. All this provides a common basis for a unied approach in the study of the zeros of these families. * CIMPA, Universidad de Costa Rica, San Pedro de Montes de Oca,San José, Costa Rica. jaimelobosegura@hotmail.com same address, mario.villalobos@ucr.ac.cr
For each µ the zeros of this function coincide then with those of F µ .

Plan of the article
In the sections 2, 3 of preliminaries we expose the results of xed point problems and auxiliary functions respectively. The concept of FPSP exposed in the section 2 is original and allows to translate the concept of the zero of a function into a concept of xed point of a function, as it is explained in the respective sections. Some of the functions addressed in the section 3 appear more or less explicitly in the literature that deals with the problem of the asymptotic development of functions dened by integrals. However, the analytical study undertaken here seems original and has its own interest. The theory of probability is used (theory of gamma distribution for example) and in a certain way it extends the results of L. Gordon on the probabilistic approach of the gamma function (see [5]).
In sections 4, 5 we develop the theory about zeros properly. The property of zeroes of the CFI family is translated in xed point problem by a geometric argument that diers from the one used in [10] and that seems more natural. The theorem 4.1 also oers an error bound for T n,α whereby a problem raised in [10] about the asymptotic behavior of this error is solved (see section 6 in [10]).
The equivalence of zeros for the Lommel family is more direct by the decomposition lemma 1.4, and leads to two dierent FPSP that must be treated separately. We deduce from this translation not only the results of the quote 1.5 and 1.6, but also new location intervals (theorem 5.5) and their respective errors (theorem 5.6).

The xed point sequential problem
For the study of the zeros proposed in the Introduction, we develop here a topic related to xed point problems of real variable functions, whose interest will be revealed in later sections. We consider a general situation as the following: let I n = [a n , b n ] n∈N a sequence of intervals of R + which is increasing: b n < a n+1 for all n, and such that the sequence (a n ) n∈N diverges to innity.
Let U be an interval of +∞, J an open interval of R + and G α : U → R + , function that depends on the parameter α ∈ J.
It denes the sequence of functions: G n,α (t) ≡ a n + G α (t), t ∈ I n .
Sucient conditions are sought to guarantee the existence of a single xed point g n,α in each I n for each G n,α (·), as well as the most relevant properties of the sequence (g n,α ) n∈N under dierent conditions. We call this xed point sequential problem and, that we abbreviated by FPSP. We say that an FPSP has unique solution if for all n there is a single xed point of G n,α on I n .
We rst consider the particular problem of xed points for functions dened in a single bounded interval of the real line. Thereafter, I is a closed and bounded interval [a, b] of the real line.

Proof
The hypotheses about the domain and the codomain imply that the function Id − G changes sign at the ends of I and joint to the continuity property the existence of at least one xed point in I is guaranteed in all cases. In case 1) the uniqueness is derived from the fact that Id − G is strictly increasing. In case 2) the conditions ensure that Id − G is either strictly monotonic in I or unimodal in I with global extreme inside I; in both cases there can only be one zero of that function in I.

Proof
In case 1) g < h must be fullled because otherwise, using the strict decrease of G, we would have g = G(g) G(h) < H(h) = h, that is contradictory. The relation g < h implies that F (h) < F (g) < G(g) = g, the second relation in this case.

QED
We will consider the FPSP dened by (I n ) n∈N , an increasing sequence of intervals U, with I n = [a n , b n ], and by a continuous parametric function G α : U → R + .
In certain cases we can assure that this problem has a unique solution, for example: Corolary 2.3. Let l n the lengths of the intervals I n that satisfy L = inf n∈N {l n } > 0. We assume that the codomain of G α is included in (0, L). If G α is strictly decreasing or it is concave or convex in all its domain, the associated FPSP has a unique solution.

Proof
It is enough to observe that under the conditions of the proposition all the functions G n,α dened in each I n have codomain within int(I n ). On the other hand, the other conditions imply that each G n,α satises case 1) or case 2) of the proposition 2.1. The result is then derived from 2.1.

QED
It will be assumed in the rest of this subsection that the FPSP problem has a unique solution: for all n there exists a real only g n,α which is the only xed point of G n,α in I n . Since the intervals I n grow, meaning sup I n < inf I n+1 , the sequence (g n,α ) n∈N is strictly increasing and diverges to innity. We will suppose further that the intervals I n are of constant length c > 0 and that the sequence (a n ) n∈N is arithmetic of common dierence A > 0 (arithmetic conditions).
Also consider the condition that we call LIM : G α is strictly decreasing in all U and converges to 0 in ∞ for all α ∈ J.
We dene a special class of functions: Denition 2.4. A funtion f : R + → R satises condition @ if there are real K, β such that: "· ∼ · (t → ∞)" denotes the relation of asymptotic equivalence in innity.
The subindex α is omitted from now on to simplify the notation.
The following result follows from the denition of xed point itself: Proposition 2.5. (monotony of the distance to extremes): Let a FPSP of unique solution with arithmetic conditions, of constants A, c.
Then it holds that: If G satises condition LIM the sequence (g n − a n ) n∈N converges to 0 and is strictly decreasing and If c − G satises the condition LIM the sequence (g n − b n ) n∈N converges to 0 and is strictly increasing.
If in the rst case G meets the property @ then g n − a n ∼ G(a n ), n → ∞ and if in the second case it is met by G − c then g n − b n ∼ (G(a n ) − c), n → ∞.

Proof
Writing g n − a n = G(g n ) in the rst case and g n − b n = G(g n ) − c in the second and taking into account that lim n→∞ g n = ∞ and that (g n ) n∈N is strictly increasing, the conclusions follow directly from the conditions of the LIM condition.
The proof of the second part is almost immediate from the relation g n = a n + G(g n ) and in the rst case that property @ implies that G(a n ) ∼ G(g n ), n → ∞, given that a n ∼ g n , n → ∞ by the arithmetic conditions. Analogously the second case is dealt with.

QED 3 Auxiliary functions and gamma distribution
We study here the parametric functions f α , g α , introduced in Lemma 1.4 of introduction. To do this, we introduce certain special functions denoted J α , whose denition is given in terms of the gamma probability distribution of the parameter α, denoted by (γ α , α > 0) and dened by Remember also the bi-parametric gamma family (γ α,t , α, t > 0) that is dened by: The parametric function: is well dened for all positive t, E α denoting expectation operator with respect to the gamma law of parameter α.
Note that the functions f α , g α dened in the introduction can be redened as: This notation is the same one that is used in [13].
We dene for a α ∈ R + : We can nd the asymptotic development of order 2 of J α at 0: from the classical theory of asymptotic developments for parametric integrals (see Watson's lemma, for example in section 4.1 of [3]).
We study some properties of variable functions t , J α (t), f α (t), with t positive real, and α positive positive: For xed α, the function t → J α (t) is decreasing in R + and range (0, 1) .
This is veried, as the quantity J α (t) being the expectation with respect to γ α of the function h(x) = (1 + (tx ) 2 ) −1 , it suces to use the fact that h is bounded and decreasing with respect to t, convergent to the constant function 1 when t → 0, and to the constant 0 when t → ∞ and then we use the classic convergence theorems.
For all α > 0 the function This is proved in lemma 3.4 of [10].
We have: The rst inequality was demonstrated in Lemma 3.4 of [10], using Harris's inequality (see [9] Theorem 5.13, chap 4). For the second one we can use Jensen's inequality by writing using the convexity of h and then applying the formula for the second moment of the gamma distribution, that is (α) 2 .
The funtion f α is derivable in its domain, with f α (t) = −g α (t), g α (t) = f α (t) − t −α , (3.5) It is enough to use the Lebesgue theorem of derivation of parametric integrals, observing that integrand functions have derivatives with respect to t, which are integrable in R + .
We have: Just use the relation f α (t) = t −α J α (t −1 ) and the property cited in (3.2).
For all t ∈ R + , µ ∈ (0, 1) it is fullled: g 1−µ (t) < Γ(µ) cos(µπ/2), (3.8) Indeed, according to the lemma 1.4 the amounts f 1−µ (0), g 1−µ (0) exists and it is fullled: by applying the known identity: ∞ 0 t µ−1 e it dt = Γ(µ)e iµπ/2 (see [14], page 52). We now use the fact that the functions t → f 1−µ (t) and t → g 1−µ (t) are strictly decreasing in R + and we conclude (3.7) as a parameterized function of the angular measure of the curve t → c α (t) + is α (t). Therefore in the case of the function c α for example, since a real t is one of its zeros if and only the angular measure in the value t is an odd multiple of π/2, it follows from the above that t is a xed point of the function kπ/2 + arctan( gα fα ) for an even value of the integer k. The result is analogous for s α taking k as an odd integer. Which is the content of corollary 3.3 in [10], where a dierent reasoning was followed.
Then the zeros of c α and s α are exactly the xed points of the FPSP, in the sense of the section 2, dening G α (t) = arctan g α f α (t), the intervals I n = [nπ/2, (n + 1)π/2] n∈N and taking Then in this case a n = nπ/2 y b n = a n+1 . The existence and uniqueness of the xed points within each I n is guaranteed by the corollary 2.3 since case 1) is fullled (the functions G α are all decreasing as mentioned in the property 3.3).
In the rest of the section we denote z n,α the zero of the CFI belonging to the interval I n and dened by the value α of the parameter.
The result of the quote 1.3, except for the asymptotic result, it is obtained from the proposition 2.2, since the function z → nπ/2 + α z is bounded above by G n,α (see rst inequality of (3.4)) and it satises in each I n the conditions of that proposition with condition 1) as stated above.
The results enunciated in the quote 1.2 come from the proposition 2.5. In eect, according to what has been said above, the FPSP associated with this problem has a unique solution with arithmetic conditions (A = c = π/2 in this case). The function G satises in this case the LIM condition, according to what has been said above about the decrease of the G and its convergence to 0 at innity using for example the inequality in 3.4. The asymptotic behavior of (z n,α − nπ/2) n∈N is deduced from this same proposition since G fullls the property @ with K = α, β = −1, in virtue of the property (3.6).
It should be noted that a formulation of a FPSP like the one above appears in Macleod [12] for the case of c 1 for α = 1.

A theoretical lower bound and a bound of the error, CFI case
We now look for theoretical lower bounds of the zeros of the CFI family, which improve the theoretical levels a n , n ∈ N established in the previous subsection. The analysis of the error associated with these bounds allows to deduce a stronger version than that of the quote 1.3.

Proof
Thanks to the inequalities in (3.4) and the relationship gα fα (t) = αf α+1 (t) fα(t) we deduce the inequality: It is veried without diculty that F α satises the conditions of the corollary 2.3 because its codomain lies in (0, π/2) and it is strict decreasing for t > α + 2. Then case 1) of the proposition 2.2 is fullled in I n if α + 2 < a n , and therefore of the double inequality of this proposition we deduce that nπ/2 + F α (T n,α ) is a lower bound of zero z n,α .
Given that T n,α is a x point of z → nπ/2 + α z , it is obtained Expressing now F α (T n,α ) as arctan(x) and using inequality arctan(x) > x − x 3 /3 for |x| < 1, we nd that for α + 2 < a n : But in this last expression α(α + 1) 2 = (α) 3 on the one hand while on the other x < α T n,α from where we conclude. QED Note: A similar previous analysis applied to the xed point of t → nπ/2 + arctan(α/t) en I n , that we denote again T n,α , shows that this is an upper bound of z n,α and also an error bound is given as that of theorem 4.1 taking K α = (α) 3 . According to the theorem 4.1 the asymptotic relation |z n,α − T n,α | = o(1/n 2 ) is in particular fullled, responding in this way to the conjecture of [10] By the integral formula for c 1−µ (z) + is 1−µ (z) from lemma 1.4, and matching the imaginary parts of both members of the last expression we get (5.1).

The zeros as FPSP problems
We will denote from now on a n = nπ/2, and for J n the interval(a n , a n+1 ).
We translate the equation (5.2) in terms of xed point sequential problems as it was proposed in section 2. For this we need to dene the sequence of intervals (I n ) n∈N and the function associated with each of the FPSP. We need the following denitions of intervals and functions, for an integer n: a n+2 ) if n = 2 mod 4, (a n−1 , a n+1 ) if n = 0 mod 4, n = 0 (a 0 , a 1 ) if n = 0, (5.3) and the functions where in both cases U = R + , J = (0, 1), with notations of section 2.
Note: in this case the intervals (I 4n+k ) n∈N , k = 0, 2, correspond, in each case, to the intervals (I n ) n∈N from the section 2.
ii) The function G (2) µ is strictly increasing and concave.
Item a) is immediate from the condition J = (0, 1) and that the term arcsin is the arcsine evaluated in the real interval (0, 1).
where the factor(arcsin) is positive and decreasing since arcsine is increasing and convex and f 1−µ is decreasing; the factor f 1−µ (t) = −g 1−µ (t) (see property (3.5)) is increasing and negative strict; then the product of these two factors is a strict and negative increasing function, i.e. the convexity of G and its decrease. For part c): in the rst pair of limits we use the fact that arcsin(t) ∼ t at 0, while f 1−µ (t) ∼ t µ−1 at innity. For the second limit we use the expression

Proof
It is enough to invoke the corollary 2.3 from the section 2 that is fullled with L = π, since the length of the intervals is constant L and that the other conditions of this corollary (with the condition of convexity or concavity) are fullled thanks to the lemma 5.2.

QED
We denote by z n,µ the unique zero of F µ in the interval I n (µ). We complete the quote 1.6 by an asymptotic law for the convergent sequences mentioned there (they are denoted by a n,µ , b n,µ resp.) the lower end (upper end resp.) of I n (µ)): Proposition 5.4. We have the following asymptotic approximations

Proof
According to lemma 5.2, item c), the property @ is met in both reduced problems with G(t) ∼ t µ−1 /Γ(µ), t → ∞ in the reduced problem a) and G(t) − c ∼ −t µ−1 /Γ(µ) t → ∞, in the reduced problem b). It suces to apply then the proposition 2.5 and the interpretation of the zeros as xed points of each reduced FPSP.

New theoretical bounds and their error
The location intervals I n (µ), n ∈ N studied in the previous section have constant length µπ/2 and therefore the property (b n,µ − a n,µ ) −→ n→∞ 0 is not accomplished. We look for some intervals with this property resorting to proposition 2.2, for which it is required to nd a pair of functions dened in each interval I n (µ) which bounds function G.
We remember that z n,µ denotes the unique zero of F µ in the interval I n (µ).
We dene: 1. for n odd, let be H n,µ = a n,µ + H µ , F n,µ = a n,µ + F µ , 2. for n even, let be F µ (z) = arccos z µ−1 Γ(µ) + µπ/2 − π/2, F n,µ = a n,µ + F µ , Theorem 5.5. We assume that (a n,µ ) µ−1 < Γ(µ + 1). We have the following theoretical bounds of z n (µ): 1. if n is odd, the function H n,µ is well dened in I n (µ) and there is a unique xed point h n,µ of this function in this interval that fullls: F n,µ (h n,µ ) < z n,µ < h n,µ 2. if n is even( n ≥ 2 ) the function F n,µ is well dened in I n (µ) and there is a unique xed point f n,µ of this function in this interval that fullls: f n,µ < z n,µ < b n,µ Proof 1. For n odd the inequalities are satised for z ∈ I n (µ): F µ (z) < G (1) µ (z) − µπ/2 < H µ (z) as long as its members are dened. They result from the growth of the function arcsine on the one hand and on the other hand the inequalities of 3.4 for the function f µ .
The well denition of H n,µ in I n (µ) follows form a n,µ > 1 for n odd, so the function z → z µ−1 Γ(µ) , bounded above by (an,µ) µ−1 Γ(µ) in I n (µ), is less than 1 and then belongs to the arcsine domain. The existence and uniqueness of the xed point of function H n,µ in I n (µ) follows from the proposition 2.1. In eect, by inequality arcsin(x) < π/2x, x > 0 and the condition of the theorem, it is fullled that 0 < H µ < µπ/2 for all I n (µ). Then the codomain of H n,µ is in the interior of I n (µ) and as on the other hand H n,µ fullls case 1) of this proposition we conclude.
On the other hand, the function F n,µ is well dened in I n (µ) if H n,µ is. It satises the conditions of the proposition 2.2 and case 1) of the proposition 2.1. For the condition of the case 1) let's prove that for n odd the function z → z µ−1 (1 + (1 − µ)(2 − µ)z −2 ) −1 is decreasing in I n (µ).
It is easy to prove, using the logarithmic derivative, that this last condition is equivalent to z 2 > (1 + µ) (2) , z ∈ I n (µ) which is equivalent to a 2 n,µ > (1 + µ) (2) for n odd; since a n,µ is increasing for n odd, it is enough to prove it for n = 1, which is almost immediate. The result is then derived from the proposition 2.2.

Conclusions and future work
The results developed in this paper about the location of the zeros of the parametric functions considered above has been based on the theory that we called xed point sequential problems. In particular we have been able to nd bounds of approximation error that were not evident in the previous work [10] and that are absent in [7].
We hope to continue with this approach in a next work about on the problem of parametric dependence of the zeros of the same families of functions.