§19.21 Connection Formulas
►
§19.21(i) Complete Integrals
…
►The complete case of
can be expressed in terms of
and
:
…
►
§19.21(ii) Incomplete Integrals
…
►
§19.21(iii) Change of Parameter of
…
…
►
§7.22(i) Main Functions
►The methods available for computing the main functions in this chapter are analogous to those described in §§
6.18(i)–
6.18(iv) for the exponential
integral and sine and cosine
integrals, and similar comments apply.
…
►
§7.22(ii) Goodwin–Staton Integral
…
►
§7.22(iii) Repeated Integrals of the Complementary Error Function
►The recursion scheme given by (
7.18.1) and (
7.18.7) can be used for computing
.
…
§19.15 Advantages of Symmetry
…
►Symmetry in
of
,
, and
replaces the five transformations (
19.7.2), (
19.7.4)–(
19.7.7) of Legendre’s
integrals; compare (
19.25.17).
…
…
►For the many properties of ellipses and triaxial ellipsoids that can be represented by elliptic
integrals, any symmetry in the semiaxes remains obvious when symmetric
integrals are used (see (
19.30.5) and §
19.33).
…
…
►All derivatives are denoted by differentials, not by primes.
►The first set of main functions treated in this chapter are Legendre’s complete
integrals
…of the first, second, and third kinds, respectively, and Legendre’s incomplete
integrals
…
►However, it should be noted that in Chapter 8 of
Abramowitz and Stegun (1964) the notation used for elliptic
integrals differs from Chapter 17 and is consistent with that used in the present chapter and the rest of the NIST Handbook and DLMF.
…
►The first three functions are incomplete
integrals of the first, second, and third kinds, and the
function includes complete
integrals of all three kinds.
§36.9 Integral Identities
►
36.9.1
…
►
36.9.8
…
►For these results and also
integrals over doubly-infinite intervals see
Berry and Wright (1980).
…
§6.4 Analytic Continuation
►Analytic continuation of the principal value of
yields a multi-valued function with branch points at
and
.
The general value of
is given by
…
►Unless indicated otherwise, in the rest of this chapter and elsewhere in the DLMF the functions
,
,
,
, and
assume their principal values, that is, the branches that are real on the positive real axis and two-valued on the negative real axis.
§36.10 Differential Equations
►
§36.10(i) Equations for
…
►
, cusp:
…
►
, swallowtail:
…
►In terms of the normal forms (
36.2.2) and (
36.2.3), the
satisfy the following operator equations
…