Watson integrals of the cubic lattices

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Numbers

family

$n$

$W_L$

$\mathbb{Z}^n$

1.516386059151978018156012159681420779955387044452262676566980463658086320353521450401611741209688114

comment: The lattice is simple cubic.

$A_n$

1.344661183165144296349156350052455748708176605124362191475225098059839519670920404713506548841411718

comment: The lattice is face-centred cubic.

$A_n^{*}$

1.393203929685676859184246260325368242657481217515617878974281631880324012575036630678647329857809556

comment: The lattice is body-centred cubic.

Definition

For a three-dimensional cubic lattice $L$, let $S(L)$ be its set of minimal vectors, and set $\phi_L(u,v,w)=|S(L)|^{-1}\sum_{x\in S(L)}\cos(x\cdot(u,v,w))$ in the coordinates used for (1). Listed is the Watson integral $W_L=\pi^{-3}\int_0^\pi\int_0^\pi\int_0^\pi (1-\phi_L(u,v,w))^{-1}\,du\,dv\,dw$.

Parameters

family

— family of the lattice

$n$

— dimension ($n=3$)

Formulas

(1)

The three structure functions are $\phi_{\mathbb{Z}^3}(u,v,w)=(\cos u+\cos v+\cos w)/3$, $\phi_{A_3}(u,v,w)=(\cos u\cos v+\cos v\cos w+\cos w\cos u)/3$, and $\phi_{A_3^{*}}(u,v,w)=\cos u\cos v\cos w$.

(2)

The closed forms are $W_{\mathbb{Z}^3}=\dfrac{\sqrt6}{32\pi^3} \Gamma(\frac1{24})\Gamma(\frac5{24}) \Gamma(\frac7{24})\Gamma(\frac{11}{24})$, $W_{A_3}=\dfrac{9\Gamma(\frac13)^6}{2^{14/3}\pi^4}$, and $W_{A_3^{*}}=\dfrac{\Gamma(\frac14)^4}{4\pi^3}$ [1] [2].

Comments

(3)

The lattice names $\mathbb{Z}^3$, $A_3$ and $A_3^{*}$ are the ones used in Packing densities and Hermite numbers of the classical lattices. Thus $\mathbb{Z}^3$ is the simple cubic lattice, $A_3$ is the face-centred cubic lattice, and $A_3^{*}$ is the body-centred cubic lattice.

(4)

The Watson integral depends only on the similarity class of $L$: rescaling the lattice rescales the wave vector and leaves $\phi_L(u,v,w)$ unchanged after the coordinate change. No Gram-matrix normalisation is part of this table.

(5)

Watson's original triple integrals [1] are often written as $I_1$, $I_2$ and $I_3$ [3]. With the normalisation in this table, $W_{A_3^{*}}=I_1$, $W_{A_3}=3I_2$, and $W_{\mathbb{Z}^3}=3I_3$.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

R = RealBallField(numberdb.bits(100, losing=128))
pi = R.pi()
gamma = lambda a, b: R(QQ(a) / QQ(b)).gamma()
two_14_3 = (R(QQ(14) / QQ(3)) * R(2).log()).exp()

W_Z3 = (R(6).sqrt() * gamma(1, 24) * gamma(5, 24) *
        gamma(7, 24) * gamma(11, 24) / (32 * pi**3))
W_A3 = 9 * gamma(1, 3)**6 / (two_14_3 * pi**4)
W_A3dual = gamma(1, 4)**4 / (4 * pi**3)

References

[1]

G. N. Watson, Three triple integrals, Quarterly Journal of Mathematics os-10 (1939), no. 1, 266-276. (doi)

[2]

M. L. Glasser and I. J. Zucker, Extended Watson integrals for the cubic lattices, Proceedings of the National Academy of Sciences of the United States of America 74 (1977), no. 5, 1800-1801. (doi)

Links

[3]

MathWorld: Watson's Triple Integrals

Similar tables

Packing densities and Hermite numbers of the classical lattices — the same lattice family and dimension keys for classical lattices

Covering radii and covering densities of the classical lattices — another table using the same classical-lattice names and dimensions

Entropy constants of lattice models — lattice-model constants indexed by named regular lattices

Pólya's random walk constants — $p(3)=1-1/W_{\mathbb{Z}^3}$, the return probability of the simple random walk on $\mathbb{Z}^3$

Values of the Gamma function at rational numbers — contains the gamma values in the closed forms for the three Watson integrals

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: yes

How they were obtained:

Each value is a real ball at numberdb.bits(digits, losing=128) bits, computed from (2) by arb's gamma function. The generator writes 100 digits.

Before the draft was filled, each gamma-product value was compared with the corresponding complete-elliptic-integral form of Watson's triple integrals, using arb ball arithmetic.