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basic elliptic integrals

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1: 19.1 Special Notation
l , m , n nonnegative integers.
2: 19.29 Reduction of General Elliptic Integrals
§19.29(ii) Reduction to Basic Integrals
All other cases are integrals of the second kind. …
3: 29.3 Definitions and Basic Properties
§29.3 Definitions and Basic Properties
For each pair of values of ν and k there are four infinite unbounded sets of real eigenvalues h for which equation (29.2.1) has even or odd solutions with periods 2 K or 4 K . … In this table the nonnegative integer m corresponds to the number of zeros of each Lamé function in ( 0 , K ) , whereas the superscripts 2 m , 2 m + 1 , or 2 m + 2 correspond to the number of zeros in [ 0 , 2 K ) . … For dn ( z , k ) see §22.2. To complete the definitions, 𝐸𝑐 ν m ( K , k 2 ) is positive and d 𝐸𝑠 ν m ( z , k 2 ) / d z | z = K is negative. …
4: Bibliography G
  • F. Gao and V. J. W. Guo (2013) Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.
  • M. L. Glasser (1976) Definite integrals of the complete elliptic integral K . J. Res. Nat. Bur. Standards Sect. B 80B (2), pp. 313–323.
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  • R. A. Gustafson (1987) Multilateral summation theorems for ordinary and basic hypergeometric series in U ( n ) . SIAM J. Math. Anal. 18 (6), pp. 1576–1596.
  • A. J. Guttmann and T. Prellberg (1993) Staircase polygons, elliptic integrals, Heun functions, and lattice Green functions. Phys. Rev. E 47 (4), pp. R2233–R2236.
  • 5: Bibliography
  • H. Alzer and S. Qiu (2004) Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172 (2), pp. 289–312.
  • G. D. Anderson, S.-L. Qiu, M. K. Vamanamurthy, and M. Vuorinen (2000) Generalized elliptic integrals and modular equations. Pacific J. Math. 192 (1), pp. 1–37.
  • G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen (1990) Functional inequalities for complete elliptic integrals and their ratios. SIAM J. Math. Anal. 21 (2), pp. 536–549.
  • G. D. Anderson and M. K. Vamanamurthy (1985) Inequalities for elliptic integrals. Publ. Inst. Math. (Beograd) (N.S.) 37(51), pp. 61–63.
  • R. Askey (1980) Some basic hypergeometric extensions of integrals of Selberg and Andrews. SIAM J. Math. Anal. 11 (6), pp. 938–951.
  • 6: Bibliography F
  • H. E. Fettis (1965) Calculation of elliptic integrals of the third kind by means of Gauss’ transformation. Math. Comp. 19 (89), pp. 97–104.
  • H. E. Fettis (1970) On the reciprocal modulus relation for elliptic integrals. SIAM J. Math. Anal. 1 (4), pp. 524–526.
  • C. H. Franke (1965) Numerical evaluation of the elliptic integral of the third kind. Math. Comp. 19 (91), pp. 494–496.
  • T. Fukushima (2010) Fast computation of incomplete elliptic integral of first kind by half argument transformation. Numer. Math. 116 (4), pp. 687–719.
  • T. Fukushima (2012) Series expansions of symmetric elliptic integrals. Math. Comp. 81 (278), pp. 957–990.
  • 7: Bibliography P
  • F. A. Paxton and J. E. Rollin (1959) Tables of the Incomplete Elliptic Integrals of the First and Third Kind. Technical report Curtiss-Wright Corp., Research Division, Quehanna, PA.
  • H. N. Phien (1990) A note on the computation of the incomplete beta function. Adv. Eng. Software 12 (1), pp. 39–44.
  • B. Pichon (1989) Numerical calculation of the generalized Fermi-Dirac integrals. Comput. Phys. Comm. 55 (2), pp. 127–136.
  • R. Piessens (1982) Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.
  • W. H. Press and S. A. Teukolsky (1990) Elliptic integrals. Computers in Physics 4 (1), pp. 92–96.
  • 8: Bibliography M
  • J. N. Merner (1962) Algorithm 149: Complete elliptic integral. Comm. ACM 5 (12), pp. 605.
  • P. Midy (1975) An improved calculation of the general elliptic integral of the second kind in the neighbourhood of x = 0 . Numer. Math. 25 (1), pp. 99–101.
  • S. C. Milne (1985c) A new symmetry related to 𝑆𝑈 ( n ) for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.
  • S. C. Milne (1997) Balanced Θ 2 3 summation theorems for U ( n ) basic hypergeometric series. Adv. Math. 131 (1), pp. 93–187.
  • T. Morita (1978) Calculation of the complete elliptic integrals with complex modulus. Numer. Math. 29 (2), pp. 233–236.
  • 9: Bibliography C
  • B. C. Carlson (1965) On computing elliptic integrals and functions. J. Math. and Phys. 44, pp. 36–51.
  • B. C. Carlson (1970) Inequalities for a symmetric elliptic integral. Proc. Amer. Math. Soc. 25 (3), pp. 698–703.
  • B. C. Carlson (1977a) Elliptic integrals of the first kind. SIAM J. Math. Anal. 8 (2), pp. 231–242.
  • B. C. Carlson (1978) Short proofs of three theorems on elliptic integrals. SIAM J. Math. Anal. 9 (3), pp. 524–528.
  • D. Cvijović and J. Klinowski (1999) Integrals involving complete elliptic integrals. J. Comput. Appl. Math. 106 (1), pp. 169–175.
  • 10: Bibliography S
  • J. L. Schonfelder (1980) Very high accuracy Chebyshev expansions for the basic trigonometric functions. Math. Comp. 34 (149), pp. 237–244.
  • L. L. Schumaker (1981) Spline Functions: Basic Theory. John Wiley & Sons Inc., New York.
  • R. G. Selfridge and J. E. Maxfield (1958) A Table of the Incomplete Elliptic Integral of the Third Kind. Dover Publications Inc., New York.
  • L. Shen (1981) The elliptical microstrip antenna with circular polarization. IEEE Trans. Antennas and Propagation 29 (1), pp. 90–94.
  • S. K. Suslov (2003) An Introduction to Basic Fourier Series. Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.