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Taylor theorem

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1: 12.13 Sums
§12.13(i) Addition Theorems
2: 1.4 Calculus of One Variable
§1.4(vi) Taylor’s Theorem for Real Variables
3: 1.10 Functions of a Complex Variable
§1.10(i) Taylor’s Theorem for Complex Variables
4: 1.5 Calculus of Two or More Variables
§1.5(iii) Taylor’s Theorem; Maxima and Minima
5: 25.11 Hurwitz Zeta Function
25.11.10 ζ ( s , a ) = n = 0 ( s ) n n ! ζ ( n + s ) ( 1 - a ) n , s 1 , | a - 1 | < 1 .
6: 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.
  • P. Dienes (1931) The Taylor Series. Oxford University Press, Oxford.
  • 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.
  • J. Dougall (1907) On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.
  • 7: Bibliography K
  • Y. S. Kim, A. K. Rathie, and R. B. Paris (2013) An extension of Saalschütz’s summation theorem for the series F r + 2 r + 3 . Integral Transforms Spec. Funct. 24 (11), pp. 916–921.
  • 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.
  • B. G. Korenev (2002) Bessel Functions and their Applications. Analytical Methods and Special Functions, Vol. 8, Taylor & Francis Ltd., London-New York.
  • 8: Bibliography B
  • H. F. Baker (1995) Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions. Cambridge University Press, Cambridge.
  • K. A. Berrington, P. G. Burke, J. J. Chang., A. T. Chivers, W. D. Robb, and K. T. Taylor (1974) A general program to calculate atomic continuum processes using the R-matrix method. Comput. Phys. Comm. 8 (3), pp. 149–198.
  • M. V. Berry (1976) Waves and Thom’s theorem. Advances in Physics 25 (1), pp. 1–26.
  • 9: 2.10 Sums and Sequences
    The asymptotic behavior of entire functions defined by Maclaurin series can be approached by converting the sum into a contour integral by use of the residue theorem and applying the methods of §§2.4 and 2.5. … and Cauchy’s theorem, we have …
    §2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method
    2.10.25 f ( z ) = n = - f n z n , 0 < | z | < r .
    What is the asymptotic behavior of f n as n or n - ? More specially, what is the behavior of the higher coefficients in a Taylor-series expansion? …