Revista de Matemática: Teoría y Aplicaciones 2016 23(2) : 321–338 cimpa – ucr issn: 1409-2433 (Print), 2215-3373 (Online)

zeroes of generalized fresnel complementary integral functions

ceros de funciones integrales complementarias de fresnel generalizadas

Jaime Lobo–Segura^{∗} Mario Alberto Villalobos–Arias^{†}

Received: 13 Feb 2015; Revised: 14 Apr 2016; Accepted: 6 May 2016

∗CIMPA, Universidad de Costa Rica, San José, Costa Rica. E-Mail: jaimelobosegura@hotmail.com

†Misma dirección que/Same address as: J. Lobo. E-Mail: mario.villalobos@ucr.ac.cr

Abstract

Theoretical upper and lower bounds are established for zeroes of a parametric family of functions which are defined by integrals of the same type as the Fresnel complementary integral. Asymptotic properties for these bounds are obtained as well as monotony properties of the localization intervals. Given the value of the parameter an analytical-numerical procedure is deduced to enclose all zeros of a given function with an a priori error.

Keywords: Fresnel complementary integrals; zeroes of functions; theoretical bounds for zeroes; gamma distribution.

Resumen

Se establecen cotas superiores e inferiores teï¿½ricas para los ceros de una familia paramï¿½trica de funciones que se definen por las integrales del tipo integral complementaria de Fresnel. Propiedades asintï¿½ticas de estas cotas se obtienen, asï¿½ como propiedades de monotonï¿½a de los intervalos de localizaciï¿½n. Dado el valor del parï¿½metro un procedimiento analï¿½tico-numï¿½rico se deduce para acotar todos los ceros de una funciï¿½n dada con un error a priori.

Palabras clave: complementarias de Fresnel; ceros de funciones; cotas teï¿½ricas para ceros; distribuciï¿½n gamma.

Mathematics Subject Classification: 65H99, 33E20, 26A99.

1 Introduction

Consider the family of functions s_{α}, c_{α} defined by a real parameter α, 0 < α, with domain ℝ_{+}, as follows:

s_{α}(t) = ∫_{t}^{∞} \frac{sin(u)}{u^{α}}du, c_{α}(t) = ∫_{t}^{∞} \frac{cos(u)}{u^{α}}du, t > 0, α > 0.

It is well known that these integrals exist as improper Riemann integrals although they could be also interpreted as proper integrals in the theory of generalized Riemann integral. Among them the best known cases correspond to α = 1, 1/2. For α = 1 the functions s_{1}, c_{1} are generally denoted as si, ci, the complementary of the integral sine and integral cosine respectively and are related to the integral sine and integral cosine functions, usually denoted by Si and Ci, the following way:

Si(t) = ∫_{0}^{t} \frac{sin(u)}{u}du, si(t) = π/2 − Si(t), ci(t) = −Ci(t).

zeroes of generalized Fresnel complementary integral ... 323

Concerning α = 1/2, s_{1/2}, c_{1/2} they correspond grosso modo to the complements of the well known Fresnel integrals

S_{1/2}(t) = ∫_{0}^{t} \frac{sin(u)}{u^{1/2}}du, C_{1/2}(t) = ∫_{0}^{t} \frac{cos(u)}{u^{1/2}}du, t > 0,

(although these denominations are also used for the integrals 1/\sqrt{2π} S_{1/2}(t^{2}π/2), 1/\sqrt{2π} C_{1/2}(t^{2}π/2)).

Many properties of functions s_{1}, c_{1}, s_{1/2}, c_{1/2} are mentioned in [1] in sections 5.2, 7.3. From now on, the parametric family s_{α}, c_{α} will be referred as generalized Fresnel complementary integral functions, and are denoted as FCI.

We study in this work the zeroes of the family FCI, that is the sets {s_{α} = 0}, {c_{α} = 0} for each value of the parameter α. Some numerical experiments with these functions using different values of α allow us to make some conjectures about this problem. Some plots of these functions with the aid of Mathematica are shown below (see figures).

Figure 1: Graphs of S_{1}(t), S_{1/2}(t).

It is noticed for example, that in each interval of monotonicity of the given FCI function there is a unique zero and that the set of zeroes is defined by an increasing sequence of reals which is very similar to an arithmetical one. In fact, some authors have found more accurate results of this kind for some particular cases of the parameter α. A. E. Livingstone and L. Lorch in [8] and A. E. Livingstone in [7] studied the issue of zeroes of functions of type t → ∫_{t}^{∞} f(x)g(x)dx

Figure 2: Graphs of C_{1}(t), C_{1/2}(t).

under conditions including the function s_{1} defined above. They proved that zeroes of a given function are actually an increasing sequence c(n) adjacent to the sequence nπ+π/2 with c(n)−(nπ+π/2) ↓ 0. However, their proof is not valid for c_{1} because of the conditions imposed on f and g. E. Kreyszig establishes in [5] some properties of functions s_{1/2}, c_{1/2}, the actual Fresnel complementary integrals, and approaches the issue of the approximation of zeros by establishing lower bounds and approximations based on asymptotical methods. However in all these works upper bounds for zeroes are unmentioned.

In [4] similar results for another family of function (a certain Lommel functions) is presented, where approximation intervals of fixed length are shown, and also a monotony property of the kind mentioned above.

The authors look for an extension of the results mentioned before to the whole family FCI for α > 0. It is shown that it is even possible to find theoretical upper bounds for the zeroes sought for and establish sharp asymptotic results for them as stated in the main theorem of the next section.

2 Main theorem

In the following theorem theoretical bounds are given in terms of approximation intervals for the zeros of the functions.

Theorem 2.1. For every integer n ≥ 0 and α > 0 let

T (α, n) = \frac{nπ/2 + \sqrt{(nπ/2)^{2} + 4α}}{2}.

The zeros of s_{α}, c_{α} all belong to the intervals of the form

J(n) = (nπ/2, min{(n + 1)π/2, T (α, n)})

(n even in case of c_{α}, n odd in case of s_{α}) and there is only one of them in each J(n). Let z(α, n) be the only zero on J(n). Then:

z(α, n) − nπ/2 ∼ \frac{2α}{nπ} , n → ∞, T (α, n) − z(α, n) = o (\frac{1}{n}) , n → ∞.

The sequences with general term z(α, n) − nπ/2 is strictly decreasing to zero.

Remark 2.2. The preceding result implies particularly that intervals of the form [nπ/2, (n + 1)π/2], n integer, enclose zeroes of s_{α}, c_{α}. This generalizes Kreyszig’s results ([5] lemma 1, pag. 503) where the case α = 1/2 is proven.

Remark 2.3. Since T (α, n) < \frac{2α}{nπ} + \frac{nπ}{2} , n ≥ 1 the right side of this inequality is also a theoretical upper bound of the zeros of index n ≥ 1. In Kreyszig ([5] section 5) this value is proposed as an approximation of zeroes in case α = 1/2, but without proving that these are actually upper bounds. In this same work, many other approximations of zeroes are proposed, some of them being larger than T (1/2, n) and so Theorem 2.1 is an improvement of the results in [5].

By means of the intervals J(n) all zeros of s_{α}, c_{α} can be enclosed with the peculiarity that their lengths decrease to zero as n approaches infinity. This property will be used in conjunction with numerical methods to construct a simple procedure to enclose simultaneously all zeroes of a given function in FCI (section 5). One immediate consequence of this theorem is the possibility to find, as T (α, n) ≤ (n+1)π/2 for every n, a sequence of enclosing intervals of the same length, taking as initial interval J(0) whose length is \sqrt{α}:

Corollary 2.4. For 0 < α ≤ π^{2}/4 the zeros of s_{α}, c_{α} are enclosed in intervals

(nπ/2, nπ/2 + \sqrt{α}),

n ≥ 0, with n odd in case s_{α}, n even in case c_{α}.

As an example of corollary 2.4 enclosing intervals of length 1/\sqrt{2} for each zero of s_{1/2}, c_{1/2} are obtained.

3 Proof of the main theorem

First, some preliminary results are established. The following is paramount and is related to the representation of the original functions by auxiliary ones.

Lemma 3.1. For α > 0, t > 0 we have: c_{α}(t) + is_{α}(t) = e^{it}(g_{α}(t) + if_{α}(t)), where

g_{α}(t) = \frac{1}{Γ(α)} ∫_{0}^{∞} \frac{exp(−ty)y^{α}}{1 + y^{2}} dy, f_{α}(t) = \frac{1}{Γ(α)} ∫_{0}^{∞} \frac{exp(−ty)y^{α−1}}{1 + y^{2}} dy.

(3.1)

This representation is mentioned in Abramowitz & Stegun [1] and in the NIST [10] for the case α = 1 ( formulae 5.2.8, 5.2.9, 5.2.12, 5.2.13 ) and α = 1/2 ( formulae 7.3.5, 7.3.6, 7.4.25, 7.4.26 ). The case 0 < α < 1 is mentioned in Erdelyi ([3] pag. 150) under the form of asymptotic relations but not the representation itself by g_{α}, f_{α}. The authors give here a proof of this result in the general case 0 < α that resorts to an argument similar to Loya’s ([9] theorem pag. 64), based on the Fubini theorem in the plane but simplified thanks to the Tonelli-Hobson theorem. Notice that integrals in the identity of lemma 3.1 are improper on the left hand and Lebesgue in the right hand. That allows to use the theory of Lebesgue Integral as we will see in the sequence.

As a first step, it is of primarily significance to interpreting the functions g, f in the lemma in probabilistic terms. Let

μ_{α,t}(dy) = \frac{t^{α}}{Γ(α)}e^{−ty}y^{α−1} 1_{\{y>0\}}dy

be the probability measure on ℝ whose density with respect the Lebesgue measure is the gamma density function of parameters α, t. It may be written:

f_{α}(t) = t^{−α} ∫ f̄(y)μ_{α,t}(dy), g_{α}(t) = t^{−α} ∫ ḡ(y)μ_{α,t}(dy),

where

ḡ(y) ≡ \frac{y}{1 + y^{2}} , f̄(y) ≡ \frac{1}{1 + y^{2}} .

Let us recall the identity:

\frac{Γ(α)}{t^{α}} = ∫_{0}^{∞} exp(−ty)y^{α−1}dy.

Proof of lemma 3.1: For each 0 < t < A we define F (t, A) as

F (t, A) = ∫_{t}^{A} \frac{exp(ix)}{x^{α}}dx.

For each x > 0 we have

x^{−α} = \frac{1}{Γ(α)} ∫_{0}^{∞} y^{α−1} exp(−xy)dy,

and then:

F (t, A) = \frac{1}{Γ(α)} ∫_{t}^{A}∫_{0}^{∞} y^{α−1} exp(−xy + ix)dydx.  (3.2)

The function |y^{α−1} exp(−xy+ix)| = y^{α−1} exp(−xy) is integrable in [t, A]× ℝ_{+},as the iterated integral

∫_{t}^{A}∫_{0}^{∞} y^{α−1} exp(−xy)dydx = Γ(α) ∫_{t}^{A} x^{−α}dx

exists. The function (x, y) → y^{α−1} exp(−xy + ix) is therefore integrable in [t, A] × ℝ_{+} ( Tonelli-Hobson theorem) and by virtue of the Fubini theorem the order of integration may be reverted into (3.2) to get:

F (t, A) = \frac{1}{Γ(α)} ∫_{0}^{∞} y^{α−1}∫_{t}^{A} exp(−xy + ix)dx dy.  (3.3)

By denoting G(x, y) = \frac{exp(x(i−y))}{i−y} = − exp(x(i − y))\frac{(y+i)}{y^{2}+1} the interior integral is expressed as G(A, y) − G(t, y) and replacing in (3.3):

F (t, A) = \frac{1}{Γ(α)} (∫_{0}^{∞} G(A, y)y^{α−1}dy − ∫_{0}^{∞} G(t, y)y^{α−1}dy) . (3.4)

On the other hand, for every A > t, the modulus of the function y → G(A, y)y^{α−1} is bounded by the function y → y^{α−1} exp(−ty)|\frac{y+i}{y^{2}+1}|, which is integrable in [0, ∞] . But in (0, ∞) the pointwise convergence G(A, y)y^{α−1} → 0 occurs when A → ∞ and so by the dominated convergence theorem it can be deduced that ∫_{0}^{∞}(G(A, y)y^{α−1})dy → 0, when A → ∞. Hence we obtain

lim_{A→∞} F (t, A) = 1/Γ(α) ∫_{0}^{∞}(−G(t, y)y^{α−1})dy.

The left member is c_{α}(t) + is_{α}(t), whereas the function under the integral sign in the right member is the sum of the functions y → exp(ti) exp(−ty)\frac{y^{α}}{y^{2}+1} and y → i exp(ti) exp(−ty)\frac{y^{α−1}}{y^{2}+1}, which are both integrable in [0, ∞) as their moduli are exponentially dominated in +∞. The integral may be separated as the sum of the integrals of each term, and factorizing exp(ti) the lemma is proven.

Remark 3.2. Formulae for f_{α}, g_{α} in terms of parametric integrals can also be found in NIST (see [10], 8.21.4, 8.21.5). However, in contrast with lemma 3.1, these integrals are not Lebesgue integrals for all values of parameter α.

For each integer n ≥ 0 we denote: I(n) = [nπ/2, (n + 1)π/2], where the functions s_{α}, c_{α} are both monotone. The function θ_{α}(t) is defined by

θ_{α}(t) = arctan (\frac{g_{α}(t)}{f_{α}(t)}) ,

with domain ℝ_{+}, where arctan(x) = ∫_{0}^{x} \frac{1}{1 + u^{2}} du. Actually, the range of this

function is included in (0, π/2) because θ_{α}(t) = π/2 − arg(g_{α}(t) + if_{α}(t)) ( arg is the principal argument function), where g_{α}(t) + if_{α}(t) lies in the first quadrant.

The following gives a first enclosing of zeroes.

Corollary 3.3. a) Zeroes of s_{α} (resp. those of c_{α}) lie in the interior of the intervals I(n) for n odd ( resp. in the interior of I(n), n even) and there is only one zero in each of them.

b) Zero in I(n) defined in a) (of s_{α} or c_{α} depending of the parity of n) is the only fixed point of the function θ_{α} + nπ/2 in this interval.

Proof. a) For the existence and uniqueness of one zero it is enough to prove that each function s_{α}, c_{α} changes sign exactly at the extrema of the intervals I(n) for the n having the same parity. Let us verify this for s_{α}: the relation of the preceding lemma applied at the points nπ/2 is written as:

s_{α}(nπ/2) = (−1)^{[[n/2]]}f_{α}(nπ/2), if n is even,
s_{α}(nπ/2) = (−1)^{[[n/2]]}g_{α}(nπ/2), if n es odd,

([[·]] designs the integer part of a given real number) and since f_{α}, g_{α} are positive the result is proved.

For the functions c_{α} the reasoning is the same.

b) According to lemma 3.1

c_{α}(t) = Re(e^{it}(g_{α}(t) + if_{α}(t)
s_{α}(t) = Im(e^{it}(g_{α}(t) + if_{α}(t).

By calculating the expressions in the right hand and taking into account that functions g_{α}, f_{α} never vanish, we get:

s_{α}(t) = 0 ⇔ cot(t) + \frac{g_{α}(t)}{f_{α}(t)} = 0, c_{α}(t) = 0 ⇔ tan(t) − \frac{g_{α}(t)}{f_{α}(t)} = 0

in the points where the cotangent and tangent functions are defined, which is true for every zero of s_{α}, c_{α} since neither their sine or cosine vanishes. Let us verify the property of s_{α}’s zero within I(2n + 1): the cotangent function being invertible in the interior of I = I(2n + 1) with inverse arccot+nπ, where arccot is the principal value of arcotangent function, and since the unique zero of s_{α} in I is an interior point of this interval (according to a) above), the first equivalence above implies that this zero is a fixed point of arccot(−g/f)+nπ in this interval. Besides arccot(−x) = π/2+arctan(x) and the assertion is proved in this case. The same procedure can be used for the zeroes of c_{α}.

The intervals I(n) constitute a first family of enclosing intervals for zeroes. In order to establish the intervals stated in the theorem we resort to the characterization of zeroes given in b) of corollary 3.3 as fixed points of the translations of the function θ_{α}. For this purpose, the authors denoted by z(α, n) the unique fixed point of θ_{α}+nπ/2 en I(n), that is the only zero of the equation θ_{α}(t)+nπ/2 = t in the interval I(n). Then, according to the preceding corollary, the set of zeroes we look for are:

{s_{α} = 0} = {z(α, n), n odd}, {c_{α} = 0} = {z(α, n), n even}.

The following properties are of key importance to achieve the author’s aim.

Lemma 3.4. For 0 < α ≤ 1 the following properties hold:

a) θ_{α} is decreasing in ℝ_{+},

b) θ_{α}(t) < arctan (\frac{α}{t}) , t > 0,

c) θ_{α}(t) ∼ \frac{α}{t}, t → ∞.

Proof. a) Since arctan is increasing in ℝ_{+}it is enough to prove that t → g_{α}(t)/f_{α}(t) is strictly decreasing in ℝ_{+}. The first derivative criterion is valid for this function since both f_{α} and g_{α} are derivable and f_{α} does not vanish. To determine the sign of the derivative, it was noticed that the rule for the derivation under the integral sign are applicable for f_{α}, g_{α}. Indeed to apply the classical rule of the Lebesgue theory [2], theorem 10.30, we only need to prove that the integrand function H_{α}(t, y) has partial derivative with respect t for all (t, y) and continuous for t > 0, y > 0 and bounded by an integrable function Ĥ_{α}(y) defined for t > a > 0; for that purpose take

Ĥ_{α}(y) = \frac{y^{α+1}e^{−ay}}{1 + y^{2}} for g_{α} and Ĥ_{α}(y) = \frac{y^{α}e^{−ay}}{1 + y^{2}} for f_{α}.

and thus obtaining:

\frac{d(g_{α}(t))}{dt} = \frac{−1}{Γ(α)} ∫_{0}^{∞} (\frac{e^{−ty}y^{α+1}}{1 + y^{2}} ) dy,
\frac{d(f_{α}(t))}{dt} = \frac{−1}{Γ(α)} ∫_{0}^{∞} (\frac{e^{−ty}y^{α}}{1 + y^{2}} ) dy.

It is observed that d(g_{α}(t))/dt = f_{α}(t) − t^{−α}, d(f_{α}(t))/dt = −g_{α}(t), and applying the rule for the derivative of the quotient to g_{α}(t)/f_{α}(t) the authors arrived at the following:

\frac{d}{dt} (\frac{g_{α}}{f_{α}} ) (t) = \frac{g_{α}^{2}(t) + f_{α}^{2}(t) − f_{α}(t)t^{−α}}{f_{α}^{2}(t)} .  (3.5)

Using the representation of f_{α}, g_{α} in terms of the gamma distribution μ_{α,t}(dy) ( see comments and notations for lemma 3.1), and resorting to the Cauchy-Schwartz inequality to the integrals defining f_{α}, g_{α}:

f_{α}^{2}(t) < t^{−2α} ∫ f̄^{2}(y)μ_{α,t}(dy), g_{α}^{2}(t) < t^{−2α} ∫ ḡ^{2}(y)μ_{α,t}(dy)

where the inequality is strict since f̄, ḡ are not constant, and then

f_{α}^{2}(t) + g_{α}^{2}(t) < t^{−2α} (∫ (f̄^{2}(y) + ḡ^{2}(y))μ_{α,t}(dy)) = t^{−α}f_{α}(t).

From this inequality and (3.5), it is deduced then that d(g_{α}(t)/f_{α}(t))/dt < 0 on ℝ_{+}, and thus the decreasing property is proved.

b) Resorting again to the representation of f_{α}, g_{α} mentioned in a) we get

\frac{g_{α}(t)}{f_{α}(t)} = \frac{∫ ḡ(y)μ_{α,t}(dy)}{∫ f̄(y)μ_{α,t}(dy)}.

On the other side ∫ ḡ(y)μ_{α,t}(dy) = ∫ yf̄(y)μ_{α,t}(dy), and since the identity function in ℝ is increasing while y → f̄(y) is decreasing in the support of the measure μ_{α,t}(dy), as a consequence of Harris inequality (see for example theorem 5-13, chap. IV in Lindvall) we get

∫ ḡ(y)μ_{α,t}(dy) ≤ ∫ yμ_{α,t}(dy) ∫ f̄(y)μ_{α,t}(dy)

and in conjunction with the above identity we get g_{α}(t)/f_{α}(t) ≤ ∫ yμ_{α,t}(dy). But the right member of this inequality is the mean of the distribution μ_{α,t}(dy) whose value is \frac{α}{t}. Therefore arctan(g_{α}(t)/f_{α}(t)) ≤ arctan(α/t), as we wished.

c) Since arctan is equivalent to the identity function in 0 it is enough to prove that t → g_{α}(t)/f_{α}(t) is equivalent to α/t in +∞. Now then, g_{α}, f_{α} are respectively the Laplace transforms of the functions G_{α} , F_{α}, with G_{α}(y) ∼ y^{α}/Γ(α), F_{α}(y) ∼ y^{α−1}/Γ(α) y → 0. According to Watson lemma g_{α}(t) ∼ Γ(α + 1)/Γ(α)t^{α+1} = α/t^{α+1}, f_{α}(t) ∼ 1/t^{α} when t → ∞ and therefore g_{α}(t)/f_{α}(t) ∼ α/t when t → ∞, thus this being the author s conclusion.

Summarizing the previous results we get:

End of the proof of theorem 2.1: From a) of corollary 3.3 to obtain the upper bound stated in the theorem it is sufficient to prove that T (α, n) is an upper bound of z(α, n). From the inequality θ_{α}(t) < \frac{α}{t} ( b) of lemma 3.4 and the fact that arctan(x) < x in ℝ_{+}) it is deduced that the fixed point of θ_{α}(t) + nπ/2 is lower than the correspondent of the function t → \frac{α}{t} + nπ/2 in ℝ. This last one can be calculated exactly by solving the equation \frac{α}{t} + nπ/2 = t, equivalent to a quadratic polynomial equation whose only root superior to nπ/2 corresponds to the value T (α, n) declared in the statement. The conclusion follows from b) of corollary 3.3

Let us prove the other properties.

Proof. a) Asymptotic properties:

To analyze the behavior of z(α, n) − nπ/2, n → ∞ let us observe first that since z(α, n) diverges to +∞, from c) of lemma 3.4 it is deduced that θ_{α}(z(α, n)) ∼ α/z(α, n), n → ∞ and by the fixed point property of corollary 3.3 it is obtained z(α, n) − nπ/2 ∼ α/z(α, n). Since 1/z(α, n) ∼ 2/nπ as n → ∞, it is finally deduced that z(α, n) − nπ/2 ∼ 2α/nπ, n → ∞.

For T (α, n) − z(α, n), n → ∞ we write

T (α, n) − z(α, n) = (T (α, n) − nπ/2) − (z(α, n) − nπ/2).  (3.6)

Writing now T (α, n) as nπ/4(1 + \sqrt{1 + 4α/(nπ/2)^{2}}) and applying a second order Taylor expansion around 0 of x → \sqrt{1 + x^{2}} we obtain T (α, n) = nπ/2+2α/nπ+o(1/n). Then (T (α, n)−nπ/2) = 2α/nπ+o(1/n), and since it was proved that z(α, n) − nπ/2 ∼ 2α/nπ, when replacing in equation (3.6) and substracting the asymptotic formulae the term 2α/nπ is eliminated remaining only the sum of terms of the form o(1/n), which is the sought relation. b) Growth and convergence properties:

The convergence to 0 is a consequence of the asymptotic properties. To establish these ones let us first consider the case (z(α, n) − nπ/2, n ∈ ℕ): given an integer n, from b) of corollary 3.3 we know that z(α, n) = θ_{α}(z(α, n)) + nπ/2, and on the other hand θ_{α}(z(α, n + 1)) < θ_{α}(z(α, n)) by virtue of the monotony property of θ_{α} ( a) from lemma 3.4). Then z(α, n) + π/2 > z(α, n + 1), which is equivalent to z(α, n) − nπ/2 > z(α, n + 1) − (n + 1)π/2.

4 An improved lower bound

The sequence of lower bounds stated in the theorem 2.1, nπ/2, can be improved in such a way that the the sequence of errors is asymptotically as good as that of the upper bounds. Let I(n) = [nπ/2, (n + 1)π/2], and let int(I(n)) be the interior of I(n).

Theorem 4.1. Let N be the integer such that for n ≥ N the inequality (n + 1)π/2 > T (α, n) holds. Let h be the function defined in ⋃_{n≥N} int(I(n)) by

h(t) = \{ \begin{matrix}c_{α}(t) & ift ∈ int(I(n)) and n even, \\ s_{α}(t) & ift ∈ int(I(n)) and n odd. \end{matrix}

Then for every n ≥ N the real real τ(α, n) defined by:

τ(α, n) = T (α, n) − (h′)^{−1}(T (α, n))h(T (α, n))

is well defined and is a lower bound of the zero z(α, n) in I(n). Besides the following relation holds:

τ(α, n) − z(α, n) = o (\frac{1}{n}) , n → ∞.

Proof. a) Bound property First it is proved that τ(α, .) are actually lower bounds. Let us consider an n ≥ N, that we suppose even . Let us prove that τ(α, n)

is a lower bound of the zero of c_{α} located in I(n). Since T (α, n) belongs to int(I(n)) their cosine is not 0, and then τ(α, n) is well defined . We observe that the real τ(α, n) is the abscissa of the point of intersection of the tangent line to y = c_{α}(x) and the abscissa axis, that is the value of the first iteration of the Newton method applied to the function c_{α} starting at T (α, n). To verify that τ(α, n) is actually a lower bound it is sufficient to prove that function c_{α} is convex in I(n) whenever is decreasing in I(n), and concave whenever is increasing in I(n). In order to reach this point the sign of the first and second derivatives of c_{α} are analyzed:

c′_{α}(t) = −\frac{cos(t)}{t^{α}}, c′′_{α}(t) = t^{−α−1}(tsin(t) + α cos(t)).

The derivative c′_{α} is negative in I(n) (c_{α} is decreasing in I(n)) only if the cosine function is positive in this interval. But the cosine and sine functions have the same sign in I(n) because n is even, and therefore from the above expression for c′′_{α} it is deduced that the second derivative is positive. Then c_{α} is convex in I(n). The case when c_{α} is increasing in I(n) is alike.

The case n ≥ N and n is odd is similar to the last one. In this case the problem is about a lower bound for the zero of s_{α} in I(n), that holds if we prove as before that s_{α} is convex in I(n) whenever is decreasing in I(n), concave whenever is increasing. Indeed, when the function is convex and decreasing in I(n), the tangent line of the function at T (α, n) is below the graph of the original function and with negative slope and then its intersection with the abscissa axis, τ(α, n), is at the left of the zero z(α, n). For the case concave and increasing the reasoning is similar.

In order to reach to this the calculation of derivatives is considered:

s′_{α}(t) = −\frac{sin(t)}{t^{α}}, s′′_{α}(t) = −t^{−α−1}(t cos(t) − α sin(t))

it is concluded by recalling that the sine and cosine functions are of different sign in I(n) when n is odd.

b) Asymptotic property:

Consider the case of an even integer n and n ≥ N and the reals τ(α, n), z(α, n) associated to it. It is obtained, recalling that cά(t) = − cos(t)/t^{α}:

τ(α, n) = T (α, n) + (cos(T (α, n))/T (α, n)^{α})^{−1} c_{α}(T (α, n)).

Using a first order Taylor expansion of c(α, n) at z(α, n) it is obtained:

c_{α}(T (α, n)) = − cos(w(α, n))/w(α, n)^{α}(T (α, n) − z(α, n))

where w(α, n) is a real number between z(α, n) and T (α, n). Hence

τ(α, n) − T (α, n)

= −(cos(w(α, n))/ cos(T (α, n)))(T (α, n)/w(α, n))^{α}(T (α, n) − z(α, n)).

But (w(α, n), n ≥ N), (T (α, n), n ≥ N) are sequences adjacent to the sequence (nπ/2, n ≥ N) with | cos(nπ/2)| = 1, and by virtue of the uniform continuity of the cosine function in ℝ the quotient cos(w(α, n))/ cos(T (α, n)) converges to 1 as n → ∞. On the other side the quotient w(α, n)/T (α, n) converges to 1 as n → ∞ and from the expression stated above it is concluded that |τ(α, n) − T (α, n)| ∼ |z(α, n) − T (α, n)| as n → ∞.

Thanks to theorem 2.1 τ(α, n) − T (α, n) = o(1/n) is met as n → ∞ and n is even, whence it is concluded that τ(α, n) − z(α, n) = o(1/n) as n → ∞ and n is even, since |τ(α, n) − T (α, n)| > |z(α, n) − τ(α, n)|.

The case of odd n is treated similarly as the previous one.

Remark 4.2. It is seen from the proof of the theorem above that if τ(α, n) is defined from any other real number in int(I(n)) instead of T (α, n) a lower bound of z(α, n) is obtained as well. In particular, this is true for nπ/2, but then an improvement of the asymptotic quality with respect to T (α, n) is not guaranteed. In fact, retracing the proof of theorem it can be asserted that |τ(α, n) − nπ/2| ∼ |z(α, n) − nπ/2|, and best to be concluded is that τ(α, n) − z(α, n) = O(1/n).

Remark 4.3. It cannot be ensured that τ(α, n) is larger than nπ/2 for all the integers n. However the asymptotic property of the sequence of the τ(α, n) together with the result of theorem 2.1proves that τ(α, n) > nπ/2 from a certain value n.

5 Numerical examples

As a numerical test of the bounds found in theorems 2.1 and 4.1, the case of parameter α = 1 is considered, and numerical approximations of si(t), ci(t) zeroes are compared with the bounds values. This issue being equivalent to that of finding zeroes of Si(x) − π/2 and Ci(x), the Newton method is applied to these functions in different intervals I(n), and the iterations are then of the form:

x_{k+1} = x_{k} (1 − \frac{Si(x_{k}) − π/2}{sin(x_{k})}) , x_{k+1} = x_{k} (1 − \frac{Ci(x_{k})}{cos(x_{k})}) . (5.1)

Here x_{0} = nπ/2 is taken as initial value in (5.1). The procedure (5.1) is programmed in Mathematica, taking advantage of the predefined functions “SinIntegral[t]” and “CosIntegral[t]” for Si(t), Ci(t) in this package. The calculations are stopped when |x_{k} − x_{k−1}| < 10^{−15} is reached, obtaining approximations z̄(1, n) for z(1, n). For some particular values of n the comparisons with the theoretical bounds are shown in Table 1. zeroes of generalized Fresnel complementary integral ... 335

Table 1: Approximations of zeroes and theoretical bounds for n.

\begin{matrix}n odd & z̄(1, n) & z̄(1, n) − nπ/2 & T (1, n) & z̄(1, n) − t(1, n) & τ(1, n) & z̄(1, n) − τ(1, n) \\ 1 & 1.9264476 & 0.3556513 & 2.0569524 & −0.1305047 & 1.9178190 & 0.0086285 \\ 11 & 17.3356640 & 0.0569044 & 17.3364415 & −0.0007774 & 17.3356640 & 3.48083 × 10^{−8} \\ 101 & 158.6567308 & 0.0063018 & 158.6567319 & −0.0000010 & 158.6567308 & 0. × 10^{−15} \\ 1001 & 1572.3677591 & 0.0006359 & 1572.3677591 & −0.000000 & 1572.3677591 & 0. × 10^{−14} \\ 10001 & 15709.534127 & 0.0000636 & 15709.534127 & −1.0 ∗ 10^{−12} & 15709.534127 & 0. × 10^{−13} \end{matrix} \begin{matrix}n even & z̄(1, n) & z̄(1, n) − nπ/2 & T (1, n) & z̄(1, n) − T (1, n) & τ(1, n) & z̄(1, n) − τ(1, n) \\ 2 & 3.3841804 & 0.2425877 & 3.4328922 & −0.04871179 & 3.3834919 & 0.0006884 \\ 10 & 15.7703496 & 0.0623863 & 15.771369 & −0.0010196 & 15.7703495 & 6.57931 × 10^{−8} \\ 100 & 157.0859975 & 0.0063648 & 157.085998 & −0.0000011 & 157.0859975 & 7.94 × 10^{−15} \\ 1000 & 1570.7969634 & 0.0006366 & 1570.796963 & −0.0000000 & 1570.7969634 & 0. × 10^{−17} \\ 10000 & 15707.963331 & 0.0000636 & 15707.963331 & −1.118 ∗ 10^{−12} & 15707.963331 & 0. × 10^{−16} \end{matrix}

Rev.Mate.Teor.Aplic. (ISSN print: 1409-2433; online: 2215-3373) Vol. 23(2): 321–338, July 2016

It is readily seen that in these cases, for large values of n, z(1, n) is closer to T (1, n) than nπ/2, which is consistent with theorem 2.1, and on the other hand that τ(1, n) is a better approximation than T (1, n), confirming theorem 4.1. Besides, the error associated to τ(1, n) seems to be the square of the error associated to T (1, n), confirming once again the quality of the bounds τ(1, n).

Now then, if enclosing intervals of predetermined length for the zeroes of s_{α}, c_{α} are looked for, first an priori error ϵ > 0 is chosen; it is easy verified that for n > \frac{2α}{πε} the length of J(n) = min(T (α, n), (n + 1)π/2) − nπ/2 is less than ε, the lengths of all J(m), m ≥ n being also less than ε according to theorem 2.1. For s_{α}, c_{α} s zeroes lying in J(m), m < n, a numerical procedure is applied in order to obtain enclosing intervals of the sought degree of approximation. In case α = 1 if ε = 10^{−1} then for n ≥ 7 the length of J(m) < 10^{−1}. For zeroes z(1, 0), z(1, 1), …, z(1, 6) numerical methods are used. It is useful to notice that for n = 2, 3, 4, 5, 6 the values τ(1, n), T (1, n) can be taken as extrema of localization intervals, whereas for n = 0, 1 the Newton method is applied. The localization intervals are shown in Table 2 (where a(1, n), b(1, n) denote the left and right extremum respectively ).

n  z̄(1, n)  a(1, n)  b(1, n)  b(1, n) − a(1, n)
0 0.6165054856 0.6100000000 0.630000000 0.0200000
1 1.9264476603 1.9178190733 1.940000000 0.022181
2 3.3841804226 3.3834919312 3.432892216 0.049400
3 4.8938359526 4.8937384827 4.915814086 0.02207560
4 6.4270477441 6.4270275142 6.438500963 0.01147345
5 7.9726826200 7.9726771577 7.979305819 0.006629
6 9.5255754576 9.5255736692 9.529712916 0.004139
> 6  J(n)  < 10^{−1}

Table 2: Approximation intervals of zeroes with error less than 10^{−1}.

Similar numerical experiments can be performed in the case α = 1/2, since Mathematica has programmed functions for the Fresnel integrals.

6 Conclusions and future work

In this work we have found theoretical bounds of the zeros of the parametric family of functions FCI presented in theorems 2.1 and 4.1. These theoretical bounds were proven by means of numerical calculations presented in the tables of the previous section.

We could improve the bounds found here for example by attempting to reduce the order of approximation error from o(1/n) to o(1/n^{p}), p > 1. On the other hand it would be important to know whether the property of monotony of theorem 2.1, is valid for the sequence (T (α, n) − τ(α, n); n ≥ 1), which would allow the improvement of the simultaneous approximation procedure for the zeroes described in section 5.

Acknowledgment

We wish to thank the anonymous referees for their valuable comments and suggestions.

References

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