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11: 1.10 Functions of a Complex Variable
§1.10(i) Taylor’s Theorem for Complex Variables
Analytic continuation is a powerful aid in establishing transformations or functional equations for complex variables, because it enables the problem to be reduced to: (a) deriving the transformation (or functional equation) with real variables; followed by (b) finding the domain on which the transformed function is analytic. …
12: 19.16 Definitions
When one variable is 0 without destroying convergence, any one of (19.16.14)–(19.16.17) is said to be complete and can be written as an R -function with one less variable: …
13: Bibliography F
  • C. Ferreira, J. L. López, and E. Pérez Sinusía (2013a) The third Appell function for one large variable. J. Approx. Theory 165, pp. 60–69.
  • C. Ferreira, J. L. López, and E. P. Sinusía (2013b) The second Appell function for one large variable. Mediterr. J. Math. 10 (4), pp. 1853–1865.
  • 14: 10.73 Physical Applications
    and on separation of variables we obtain solutions of the form e ± i n ϕ e ± κ z J n ( κ r ) , from which a solution satisfying prescribed boundary conditions may be constructed. … On separation of variables into cylindrical coordinates, the Bessel functions J n ( x ) , and modified Bessel functions I n ( x ) and K n ( x ) , all appear. …
    15: Annie A. M. Cuyt
    A lot of her research has been devoted to rational approximations, in one as well as in many variables, and sparse interpolation. …
    16: Bibliography Y
  • Z. M. Yan (1992) Generalized Hypergeometric Functions and Laguerre Polynomials in Two Variables. In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemporary Mathematics, Vol. 138, pp. 239–259.
  • 17: 18.2 General Orthogonal Polynomials
    It is assumed throughout this chapter that for each polynomial p n ( x ) that is orthogonal on an open interval ( a , b ) the variable x is confined to the closure of ( a , b ) unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.) …
    18: Mathematical Introduction
    Special functions with one real variable are depicted graphically with conventional two-dimensional (2D) line graphs. …
    19: 36.12 Uniform Approximation of Integrals
    In the cuspoid case (one integration variable) …
    20: 9.14 Incomplete Airy Functions
    Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter. …