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21: 19.21 Connection Formulas
§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
…22: 6.19 Tables
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§6.19(ii) Real Variables
►Abramowitz and Stegun (1964, Chapter 5) includes , , , , ; , , , , ; , , , , ; , , , , ; , , . Accuracy varies but is within the range 8S–11S.
Zhang and Jin (1996, pp. 652, 689) includes , , , 8D; , , , 8S.
Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of , , , 6D; , , , 6D; , , , 6D.
Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of , , , 8S.
23: 7.22 Methods of Computation
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§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 . …24: 19.15 Advantages of Symmetry
§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). …25: 19.1 Special Notation
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►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
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►The first three functions are incomplete integrals of the first, second, and third kinds, and the function includes complete integrals of all three kinds.
nonnegative integers. | |
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26: 36.9 Integral Identities
§36.9 Integral Identities
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36.9.1
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36.9.8
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►For these results and also integrals over doubly-infinite intervals see Berry and Wright (1980).
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