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1: 3.8 Nonlinear Equations
2: 20 Theta Functions
Chapter 20 Theta Functions
…3: 22.17 Moduli Outside the Interval [0,1]
§22.17 Moduli Outside the Interval [0,1]
… ►Jacobian elliptic functions with real moduli in the intervals and , or with purely imaginary moduli are related to functions with moduli in the interval by the following formulas. … ►When is fixed each of the twelve Jacobian elliptic functions is a meromorphic function of . …In particular, the Landen transformations in §§22.7(i) and 22.7(ii) are valid for all complex values of , irrespective of which values of and are chosen—as long as they are used consistently. For proofs of these results and further information see Walker (2003).4: 6.19 Tables
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.
5: 8 Incomplete Gamma and Related
Functions
6: 8.26 Tables
Khamis (1965) tabulates for , to 10D.
Zhang and Jin (1996, Table 3.8) tabulates for , to 8D or 8S.
Abramowitz and Stegun (1964, pp. 245–248) tabulates for , to 7D; also for , to 6S.
Pagurova (1961) tabulates for , to 4-9S; for , to 7D; for , to 7S or 7D.
Zhang and Jin (1996, Table 19.1) tabulates for , to 7D or 8S.
7: 23 Weierstrass Elliptic and Modular
Functions
8: 7.23 Tables
Abramowitz and Stegun (1964, Chapter 7) includes , , , 10D; , , 8S; , , 7D; , , , 6S; , , 10D; , , 9D; , , , 7D; , , , , 15D.
Finn and Mugglestone (1965) includes the Voigt function , , , 6S.
Zhang and Jin (1996, pp. 637, 639) includes , , , 8D; , , , 8D.
Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of , , , 7D and 8D, respectively; the real and imaginary parts of , , , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.
9: 18.39 Applications in the Physical Sciences
10: 10.75 Tables
Achenbach (1986) tabulates , , , , , 20D or 18–20S.
Makinouchi (1966) tabulates all values of and in the interval , with at least 29S. These are for , 10, 20; , ; with and , except for .
Bickley et al. (1952) tabulates or , or , , (.01 or .1) 10(.1) 20, 8S; , , , or , 10S.
Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of and , for , 9S.