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1: 19.36 Methods of Computation
Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18). …
19.36.2 1 3 14 E 2 + 1 6 E 3 + 9 88 E 2 2 3 22 E 4 9 52 E 2 E 3 + 3 26 E 5 1 16 E 2 3 + 3 40 E 3 2 + 3 20 E 2 E 4 + 45 272 E 2 2 E 3 9 68 ( E 3 E 4 + E 2 E 5 ) .
For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). … When the values of complete integrals are known, addition theorems with ψ = π / 2 19.11(ii)) ease the computation of functions such as F ( ϕ , k ) when 1 2 π ϕ is small and positive. …These special theorems are also useful for checking computer codes. …
2: Tom M. Apostol
Apostol was born on August 20, 1923. … In 1998, the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled “What Is the Most Surprising Result in Mathematics?” (Answer: the prime number theorem). …In addition, he was the co-author of New Horizons in Geometry, published by the MAA, which received the CHOICE “Outstanding Academic Title” award in 2013. …He additionally served as a visiting lecturer for the MAA, and as a member of the MAA Board of Governors. …
3: Bibliography K
  • R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.
  • B. J. King and A. L. Van Buren (1973) A general addition theorem for spheroidal wave functions. SIAM J. Math. Anal. 4 (1), pp. 149–160.
  • T. H. Koornwinder (1975a) A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. 13, pp. 145–159.
  • T. H. Koornwinder (1975b) Jacobi polynomials. III. An analytic proof of the addition formula. SIAM. J. Math. Anal. 6, pp. 533–543.
  • T. H. Koornwinder (1977) The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.
  • 4: Bibliography C
  • L. Carlitz (1961b) The Staudt-Clausen theorem. Math. Mag. 34, pp. 131–146.
  • B. C. Carlson (1971) New proof of the addition theorem for Gegenbauer polynomials. SIAM J. Math. Anal. 2, pp. 347–351.
  • B. C. Carlson (1978) Short proofs of three theorems on elliptic integrals. SIAM J. Math. Anal. 9 (3), pp. 524–528.
  • F. Clarke (1989) The universal von Staudt theorems. Trans. Amer. Math. Soc. 315 (2), pp. 591–603.
  • H. S. Cohl (2013a) Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental solution of the polyharmonic equation and polyspherical addition theorems. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 042, 26.
  • 5: Bibliography D
  • S. C. Dhar (1940) Note on the addition theorem of parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 29–30.
  • H. Ding, K. I. Gross, and D. St. P. Richards (1996) Ramanujan’s master theorem for symmetric cones. Pacific J. Math. 175 (2), pp. 447–490.
  • B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.
  • J. Dougall (1907) On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.
  • T. M. Dunster (1989) Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.