Real zeros of the derivative of the Scorer function Gi

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Numbers

$n$

$g_n^{\prime}$

0.6090754170730509650643376980642754695116286129313403454823043673126924539100040656187817145753258088

comment: The unique positive real zero of $\operatorname{Gi}^{\prime}$. It is the location of a local maximum of $\operatorname{Gi}$.

-2.249954218638817535551920590312671028653669729574528428858487107180258062476786700331895112581008450

-4.083954088492768989665632454395987771479522256609746289167119975706211070153532625596148183314908682

-5.507430211106304220514638383070196129479449524757437419829772874287858424075816957229232747470072377

-6.784144057321159619329850111090593016915614472349302172061493107108987541316898071442713536069455884

-7.938313716297769428775096959257777073818941159536097584879974183295447087789374969168895851797710582

-9.020901668155199634667814626857141297522289208983980451562253824707148210362016279970147167298819551

-10.03671062970476229413280681448630660567262713253056495746825889036179886281504103891259043095545702

-11.00722880418930260632311024797807835922918239805857226612054326633138915892898746147989039990216413

-11.93364744097381438503254987063028781355468334675111942043460462001525909978911006699382343079478222

10:

-12.82776229042580250254021131766012900343306957750944586844072844230483769349479269567115544411547772

11:

-13.68973411504112805470257672048840721645976914151808301651806526739079004154675449363506381232515624

12:

-14.52700413794864318994249309685742072099973228231111848464816546225948798431461310214018542547369148

Definition

Let $\operatorname{Gi}(x)=\pi^{-1}\int_0^\infty\sin(xt+t^3/3)\,dt$ be the Scorer function, with its oscillatory improper integral convention [1]. This table gives the signed real zeros $g_n^{\prime}$ of its derivative with respect to $x$, with $g_0^{\prime}>0$ and $0>g_1^{\prime}>g_2^{\prime}>\cdots$.

Parameters

$n$

— zero index ($n\geq0$)

Formulas

(1)

$\operatorname{Gi}^{\prime\prime}(x)-x\operatorname{Gi}(x)=-1/\pi$ [1].

(2)

With $\Gamma$ denoting the Gamma function, $\operatorname{Gi}(0)=1/(3^{7/6}\Gamma(2/3))$ and $\operatorname{Gi}^{\prime}(0)=1/(3^{5/6}\Gamma(1/3))$ [3].

(3)

For the Airy functions $\operatorname{Ai}$ and $\operatorname{Bi}$, $\operatorname{Gi}^{\prime}(x)=\operatorname{Bi}^{\prime}(x)\left(\frac13-\int_0^x\operatorname{Ai}(t)\,dt\right)+\operatorname{Ai}^{\prime}(x)\int_0^x\operatorname{Bi}(t)\,dt$. This follows by differentiating the Airy integral representation of $\operatorname{Gi}$ [4] and using $\int_0^\infty\operatorname{Ai}(t)\,dt=1/3$ [5].

(4)

For the companion Scorer function $\operatorname{Hi}$, $\operatorname{Bi}^{\prime}(g_n^{\prime})=\operatorname{Hi}^{\prime}(g_n^{\prime})$, since $\operatorname{Gi}(x)+\operatorname{Hi}(x)=\operatorname{Bi}(x)$ [6].

Comments

(5)

The derivative of $\operatorname{Gi}$ has exactly one positive real zero and infinitely many negative real zeros [2]. Index $0$ selects the positive zero; the negative zeros are numbered from the origin towards $-\infty$.

(6)

The prime in $g_n^{\prime}$ distinguishes a zero of $\operatorname{Gi}^{\prime}$ from a zero of $\operatorname{Gi}$; it is not differentiation with respect to $n$. Each entry is a location on the real axis.

(7)

The normalization is that of DLMF §9.12 [1]. Derivatives are with respect to $x$, and real powers of $3$ are positive.

Programs

[7]

Python

from mpmath import mp
mp.dps = 120
f = lambda x: mp.diff(mp.scorergi, x)
roots = [mp.findroot(f, (0, 1))]
x = mp.mpf(0)
y = f(x)
while len(roots) < 13:
    t = x - mp.mpf(1)/16
    z = f(t)
    if y*z < 0:
        roots.append(mp.findroot(f, (t, x)))
    x, y = t, z
for n, root in enumerate(roots):
    print(n, mp.nstr(root, 100))

Links

[1]

DLMF §9.12: Scorer functions

[2]

DLMF §9.12(ix): Zeros

[3]

DLMF §9.12(iii): Initial values

[4]

DLMF 9.12.4: Airy integral representation of Gi

[5]

DLMF 9.10.11: Definite integral of Ai

[6]

DLMF 9.12.11: Sum of Gi and Hi

[7]

mpmath: Scorer function Gi

Similar tables

Values of the Scorer function Gi — values of the function whose stationary point locations are tabulated here

Real zeros of the Scorer function Gi — the negative real zeros of $\operatorname{Gi}$ interlace those of its derivative [2]

Local extrema of the Airy function of the second kind Bi — stationary points of the homogeneous Airy solution $\operatorname{Bi}$, which is related to $\operatorname{Gi}$ by (4)

Values of the Scorer function Hi — values of the companion Scorer function in (4)

Data properties

Entries are of type: real number

Table is complete: no (it holds the unique positive real zero, $n=0$, and the first twelve negative real zeros, $1\leq n\leq12$, a consecutive reference selection beginning nearest the origin)

How well the digits are known: proven

How they were obtained:

The values are written to 100 significant digits. Arb real balls at 700 bits enclose a 900-term Maclaurin expansion of the solution fixed by (1) and (2). With $\operatorname{Gi}(x)=\sum_{k\geq0}c_kx^k$, the coefficients satisfy $c_2=-1/(2\pi)$ and $c_{k+3}=c_k/((k+3)(k+2))$. Separate geometric tails for the three residue classes bound the function and its first two derivatives on $|x|\leq16$. Their bounds are respectively less than $7\cdot10^{-432}$, $4\cdot10^{-430}$ and $3\cdot10^{-428}$ and are added to the ball radii.

The generator covers $[-465/32,0]$ and $[0,1]$ by 994 adjacent rational intervals. Local enclosures follow from the differential equation and a Gronwall bound, with its derivation in the attached generator. On 976 intervals the enclosure of $\operatorname{Gi}^{\prime}$ excludes zero. On the other eighteen, $\operatorname{Gi}^{\prime\prime}$ has a fixed nonzero sign; five have equal endpoint signs and thirteen have opposite endpoint signs. This certifies twelve negative roots, one positive root and no omitted roots in the covered intervals. The uniqueness of the positive root on the entire nonnegative axis is established in [2].

Each root is further enclosed by exact rational endpoints separated by $2\cdot10^{-120}$, with opposite signs of $\operatorname{Gi}^{\prime}$ and a nonzero sign of $\operatorname{Gi}^{\prime\prime}$ throughout the bracket. The bracket is strictly inside the interval denoted by the stored decimal. Independent certified quadrature of (3) confirms the signs at both endpoints for all thirteen roots. Integrals of $t^2$ and $\operatorname{Ai}^{\prime}$ with known nonzero answers serve as controls, and each integral evaluation is finite with absolute radius less than $10^{-145}$.

Independent mpmath root computations at 150 and 200 decimal digits [7] agree to 140 significant digits and lie inside every certified bracket. Checks of the initial values, differential equation and differentiated sum identity use mpmath computations independent of the generator. These comparisons supplement the rigorous enclosures. The attached checks repeat the bracket and integral tests on all intervals read from the database.