…
►Addendum: For a companion
equation see (
20.7.34).
…
►In the following
equations
, and all square roots assume their principal values.
…
►
20.7.33
►These are specific examples of
modular transformations as discussed in §
23.15; the corresponding results for the general case are given by
Rademacher (1973, pp. 181–183).
…
►
20.7.34
…
…
►These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§
1.10(vi),
4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§
1.17); numerically satisfactory solutions of differential and difference
equations (§§
2.7(iv),
2.9(i)); and numerical analysis for complex variables (Chapter
3).
…
►This is because
is akin to the notation used for Bessel functions (§
10.2(ii)), inasmuch as
is an entire function of each of its parameters
,
, and
: this results in fewer restrictions and simpler
equations.
…
►
…
►For all
equations and other technical information this Handbook and the DLMF either provide references to the literature for proof or describe steps that can be followed to construct a proof.
…
►For
equations or other technical information that appeared previously in AMS 55, the DLMF usually includes the corresponding AMS 55
equation number, or other form of reference, together with corrections, if needed.
…
…
►The
modular functions
,
, and
are also obtainable in a similar manner from their definitions in §
23.15(ii).
…
►Suppose that the invariants
,
, are given, for example in the differential
equation (
23.3.10) or via coefficients of an elliptic curve (§
23.20(ii)).
…
►
(a)
In the general case, given by , we compute the roots ,
, , say, of the cubic equation
; see
§1.11(iii). These roots are necessarily distinct and represent ,
, in some order.
If and are real, and the discriminant is positive, that is ,
then , , can be identified via
(23.5.1), and , obtained from (23.6.16).
If , or and are not both real, then we label , , so
that the triangle with vertices , , is positively
oriented and is its longest side (chosen arbitrarily if
there is more than one). In particular, if , , are
collinear, then we label them so that is on the line segment
. In consequence, ,
satisfy
(with strict inequality unless
, , are collinear); also , .
Finally, on taking the principal square roots of and we obtain
values for and that lie in the 1st and 4th quadrants, respectively,
and , are given by
23.22.1
where denotes the arithmetic-geometric mean (see §§19.8(i) and
22.20(ii)). This process yields 2 possible pairs
(, ), corresponding to the 2 possible choices of the
square root.
…
…
►Equations (
24.5.3) and (
24.5.4) enable
and
to be computed by recurrence.
…
►For number-theoretic applications it is important to compute
for
; in particular to find the
irregular pairs
for which
.
…