βΊThe quantities in the symbol are called angular momenta.
…They therefore satisfy the triangle conditions
…where is any permutation of .
The corresponding projective quantum numbers
are given by
…
Blanch and Clemm (1965) includes values of ,
for , ; ,
. Also ,
for , ; ,
. In all cases
. Precision is generally 7D.
Approximate formulas and graphs are also included.
Ince (1932) includes eigenvalues , , and Fourier coefficients
for or , ; 7D. Also
, for ,
, corresponding to the eigenvalues in the tables; 5D. Notation:
, .
National Bureau of Standards (1967) includes the eigenvalues , for
with , and with ; Fourier
coefficients for and for
, , respectively, and various values of in the
interval ; joining factors ,
for
with (but in a different notation). Also,
eigenvalues for large values of . Precision is generally 8D.
Zhang and Jin (1996, pp. 521–532) includes the eigenvalues
, for ,
; (’s) or 19 (’s), .
Fourier coefficients for , ,
. Mathieu functions ,
, and their first -derivatives for ,
. Modified Mathieu functions
, , and
their first -derivatives for , , . Precision is
mostly 9S.
…
βΊ2 in Abramowitz and Stegun (1964) gives values of , , and to 7 or 8D in the rectangular and rhombic cases, normalized so that and (rectangular case), or and (rhombic case), for = 1.
…05, and in the case of the user may deduce values for complex by application of the addition theorem (23.10.1).
βΊAbramowitz and Stegun (1964) also includes other tables to assist the computation of the Weierstrass functions, for example, the generators as functions of the lattice invariants and .
…
…
βΊIn this case the lattice roots , , and are real and distinct.
…
βΊ
and have the same sign unless when both are zero: the pseudo-lemniscaticcase.
As a function of the root is increasing.
For the case
see §23.5(v).
…
βΊNote also that in this case
.
…
…
βΊThe rational solutions when the parameters satisfy (32.8.22) are special cases of §32.10(iv).
…
βΊCases (a) and (b) are special cases of §32.10(v).
…
βΊFor the case
see Airault (1979) and LukaΕ‘eviΔ (1968).
…
βΊIn the general case, has rational solutions if
…These are special cases of §32.10(vi).
…
…
βΊEuclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
…
βΊIt is the special case
of the function that counts the number of ways of expressing as the product of factors, with the order of factors taken into account.
…
βΊ
…
βΊAll terms are taken to be case-insensitive, except those taken to represent math expressions (see Case Sensitivity).
…
βΊNote that the first form may match other functions than the Bessel function, so if you are sure you want Bessel , you might as well enter one of the other 3 forms.
βΊ
Case Sensitivity
βΊDLMF search is generally case-insensitive except when it is important to be case-sensitive, as when two different special functions have the same standard names but one name has a lower-case initial and the other is has an upper-case initial, such as si and Si, gamma and Gamma.
In the following situations, DLMF search is case-sensitive:
…