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Cayley identity for Schwarzian derivatives

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1: 1.13 Differential Equations
Then the following relation is known as Abel’s identity
Elimination of First Derivative by Change of Dependent Variable
Elimination of First Derivative by Change of Independent Variable
Here dots denote differentiations with respect to ζ , and { z , ζ } is the Schwarzian derivative: …
Cayley’s Identity
2: 19 Elliptic Integrals
3: Bibliography B
  • R. J. Baxter (1981) Rogers-Ramanujan identities in the hard hexagon model. J. Statist. Phys. 26 (3), pp. 427–452.
  • A. Berkovich and B. M. McCoy (1998) Rogers-Ramanujan Identities: A Century of Progress from Mathematics to Physics. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 163–172.
  • M. V. Berry and F. J. Wright (1980) Phase-space projection identities for diffraction catastrophes. J. Phys. A 13 (1), pp. 149–160.
  • J. M. Borwein and P. B. Borwein (1991) A cubic counterpart of Jacobi’s identity and the AGM. Trans. Amer. Math. Soc. 323 (2), pp. 691–701.
  • J. L. Burchnall and T. W. Chaundy (1948) The hypergeometric identities of Cayley, Orr, and Bailey. Proc. London Math. Soc. (2) 50, pp. 56–74.
  • 4: Bibliography C
  • A. Cayley (1895) An Elementary Treatise on Elliptic Functions. George Bell and Sons, London.
  • A. Cayley (1961) An Elementary Treatise on Elliptic Functions. Dover Publications, New York (English).
  • B. K. Choudhury (1995) The Riemann zeta-function and its derivatives. Proc. Roy. Soc. London Ser. A 450, pp. 477–499.
  • J. N. L. Connor and D. Farrelly (1981) Molecular collisions and cusp catastrophes: Three methods for the calculation of Pearcey’s integral and its derivatives. Chem. Phys. Lett. 81 (2), pp. 306–310.
  • A. Cruz and J. Sesma (1982) Zeros of the Hankel function of real order and of its derivative. Math. Comp. 39 (160), pp. 639–645.
  • 5: 15.17 Mathematical Applications
    The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. …
    §15.17(iv) Combinatorics
    In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. …
    6: Bibliography M
  • Magma (website) Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.
  • J. Miller and V. S. Adamchik (1998) Derivatives of the Hurwitz zeta function for rational arguments. J. Comput. Appl. Math. 100 (2), pp. 201–206.
  • S. C. Milne (1985b) An elementary proof of the Macdonald identities for A l ( 1 ) . Adv. in Math. 57 (1), pp. 34–70.
  • V. P. Modenov and A. V. Filonov (1986) Calculation of zeros of cylindrical functions and their derivatives. Vestnik Moskov. Univ. Ser. XV Vychisl. Mat. Kibernet. (2), pp. 63–64, 71 (Russian).
  • R. J. Moore (1982) Algorithm AS 187. Derivatives of the incomplete gamma integral. Appl. Statist. 31 (3), pp. 330–335.
  • 7: 21.7 Riemann Surfaces
    §21.7(ii) Fay’s Trisecant Identity
    where again all integration paths are identical for all components. Generalizations of this identity are given in Fay (1973, Chapter 2). …
    §21.7(iii) Frobenius’ Identity
    8: 20.11 Generalizations and Analogs
    This is the discrete analog of the Poisson identity1.8(iv)). … In the case z = 0 identities for theta functions become identities in the complex variable q , with | q | < 1 , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … Similar identities can be constructed for F 1 2 ( 1 3 , 2 3 ; 1 ; k 2 ) , F 1 2 ( 1 4 , 3 4 ; 1 ; k 2 ) , and F 1 2 ( 1 6 , 5 6 ; 1 ; k 2 ) . … Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas. …
    9: 9.18 Tables
  • Yakovleva (1969) tabulates Fock’s functions U ( x ) π Bi ( x ) , U ( x ) = π Bi ( x ) , V ( x ) π Ai ( x ) , V ( x ) = π Ai ( x ) for x = 9 ( .001 ) 9 . Precision is 7S.

  • Miller (1946) tabulates a k , Ai ( a k ) , a k , Ai ( a k ) , k = 1 ( 1 ) 50 ; b k , Bi ( b k ) , b k , Bi ( b k ) , k = 1 ( 1 ) 20 . Precision is 8D. Entries for k = 1 ( 1 ) 20 are reproduced in Abramowitz and Stegun (1964, Chapter 10).

  • Sherry (1959) tabulates a k , Ai ( a k ) , a k , Ai ( a k ) , k = 1 ( 1 ) 50 ; 20S.

  • National Bureau of Standards (1958) tabulates A 0 ( x ) π Hi ( x ) and A 0 ( x ) π Hi ( x ) for x = 0 ( .01 ) 1 ( .02 ) 5 ( .05 ) 11 and 1 / x = 0.01 ( .01 ) 0.1 ; 0 x A 0 ( t ) d t for x = 0.5 , 1 ( 1 ) 11 . Precision is 8D.

  • Nosova and Tumarkin (1965) tabulates e 0 ( x ) π Hi ( x ) , e 0 ( x ) = π Hi ( x ) , e ~ 0 ( x ) π Gi ( x ) , e ~ 0 ( x ) = π Gi ( x ) for x = 1 ( .01 ) 10 ; 7D. Also included are the real and imaginary parts of e 0 ( z ) and i e 0 ( z ) , where z = i y and y = 0 ( .01 ) 9 ; 6-7D.

  • 10: 36.10 Differential Equations
    §36.10(ii) Partial Derivatives with Respect to the x n
    K = 1 , fold: (36.10.6) is an identity. …
    36.10.10 3 n Ψ 3 y 3 n = i n 2 n Ψ 3 z 2 n .
    §36.10(iv) Partial z -Derivatives