# in terms of Whittaker functions

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## 21—30 of 38 matching pages

##### 21: 2.11 Remainder Terms; Stokes Phenomenon

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►Taking $m=10$
in (2.11.2), the first three terms give us the approximation
…The error term is, in fact, approximately 700 times the last term obtained in (2.11.4).
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►These answers are linked to the terms involving the complementary error function in the more powerful expansions typified by the combination of (2.11.10) and (2.11.15).
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►However, to enjoy the resurgence property (§2.7(ii)) we often seek instead expansions in terms of the $F$-functions introduced in §2.11(iii), leaving the connection of the error-function type behavior as an implicit consequence of this property of the $F$-functions.
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►Subtraction of this result from the sum of the first 5 terms in (2.11.25) yields 0.
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##### 22: 3.10 Continued Fractions

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►if the expansion of its $n$th convergent ${C}_{n}$
in ascending powers of $z$ agrees with (3.10.7) up to and including the term in
${z}^{n-1}$, $n=1,2,3,\mathrm{\dots}$.
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►For applications to Bessel functions and Whittaker functions (Chapters 10 and 13), see Gargantini and Henrici (1967).
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►We say that it is

*associated*with the formal power series $f(z)$ in (3.10.7) if the expansion of its $n$th convergent ${C}_{n}$ in ascending powers of $z$, agrees with (3.10.7) up to and including the term in ${z}^{2n-1}$, $n=1,2,3,\mathrm{\dots}$. … ►For special functions see §5.10 (gamma function), §7.9 (error function), §8.9 (incomplete gamma functions), §8.17(v) (incomplete beta function), §8.19(vii) (generalized exponential integral), §§10.10 and 10.33 (quotients of Bessel functions), §13.6 (quotients of confluent hypergeometric functions), §13.19 (quotients of Whittaker functions), and §15.7 (quotients of hypergeometric functions). … ►The ${A}_{n}$ and ${B}_{n}$ of (3.10.2) can be computed by means of three-term recurrence relations (1.12.5). …##### 23: 28.31 Equations of Whittaker–Hill and Ince

###### §28.31 Equations of Whittaker–Hill and Ince

►###### §28.31(i) Whittaker–Hill Equation

►Hill’s equation with three terms …and constant values of $A,B,k$, and $c$, is called the*Equation of Whittaker–Hill*. … ►in (28.31.1). …

##### 24: Bibliography L

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Microwave specific attenuation by oblate spheroidal raindrops: An exact analysis of TCS’s in terms of spheroidal wave functions.
J. Electromagn. Waves Appl. 12 (6), pp. 709–711.
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Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions.
Stud. Appl. Math. 103 (3), pp. 241–258.
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Asymptotic expansions of the Whittaker functions for large order parameter.
Methods Appl. Anal. 6 (2), pp. 249–256.
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Monotonicity in terms of order of the zeros of the derivatives of Bessel functions.
Proc. Amer. Math. Soc. 108 (2), pp. 387–389.
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Integral representation of the Hankel function in terms of parabolic cylinder functions.
Quart. J. Mech. Appl. Math. 23 (3), pp. 315–327.
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##### 25: 33.22 Particle Scattering and Atomic and Molecular Spectra

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►The negative-energy functions are widely used in the description of atomic and molecular spectra; see Bethe and Salpeter (1977), Seaton (1983), and Aymar et al. (1996).
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►For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions, ${F}_{\mathrm{\ell}}(\eta ,\rho )$ and ${G}_{\mathrm{\ell}}(\eta ,\rho )$, or $f(\u03f5,\mathrm{\ell};r)$ and $h(\u03f5,\mathrm{\ell};r)$, to determine the scattering $S$-matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951).
►For bound-state problems only the exponentially decaying solution is required, usually taken to be the Whittaker function
${W}_{-\eta ,\mathrm{\ell}+\frac{1}{2}}\left(2\rho \right)$.
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►The penetrability of repulsive Coulomb potential barriers is normally expressed in terms of the quantity $\rho /({F}_{\mathrm{\ell}}^{2}(\eta ,\rho )+{G}_{\mathrm{\ell}}^{2}(\eta ,\rho ))$ (Mott and Massey (1956, pp. 63–65)).
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►The Coulomb functions given in this chapter are most commonly evaluated for real values of $\rho $, $r$, $\eta $, $\u03f5$ and nonnegative integer values of $\mathrm{\ell}$, but they may be continued analytically to complex arguments and order $\mathrm{\ell}$ as indicated in §33.13.
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##### 26: 22.2 Definitions

###### §22.2 Definitions

►The*nome*$q$ is given in terms of the

*modulus*$k$ by … ►The Jacobian functions are related in the following way. … ►The six functions containing the letter $\mathrm{s}$ in their two-letter name are odd in $z$; the other six are even in $z$. ►In terms of Neville’s theta functions (§20.1) …

##### 27: 13.24 Series

###### §13.24 Series

►###### §13.24(i) Expansions in Series of Whittaker Functions

►For expansions of arbitrary functions in series of ${M}_{\kappa ,\mu}\left(z\right)$ functions see Schäfke (1961b). ►###### §13.24(ii) Expansions in Series of Bessel Functions

… ►Additional expansions in terms of Bessel functions are given in Luke (1959). …##### 28: 23.2 Definitions and Periodic Properties

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►

###### §23.2(i) Lattices

… ► … ►###### §23.2(ii) Weierstrass Elliptic Functions

… ►Hence the order of the terms or factors is immaterial. … ► …##### 29: 22.18 Mathematical Applications

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►For these and other examples see Lawden (1989, Chapter 4), Whittaker and Watson (1927, §22.8), and Siegel (1988, pp. 1–7).
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►Bowman (1953, Chapters V–VI) gives an overview of the use of Jacobian elliptic functions in conformal maps for engineering applications.
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►in which $a,b,c,d,e,f$ are real constants, can be achieved in terms of single-valued functions.
…Discussion of parametrization of the angles of spherical trigonometry in terms of Jacobian elliptic functions is given in Greenhill (1959, p. 131) and Lawden (1989, §4.4).
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►a construction due to Abel; see Whittaker and Watson (1927, pp. 442, 496–497).
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##### 30: 5.17 Barnes’ $G$-Function (Double Gamma Function)

###### §5.17 Barnes’ $G$-Function (Double Gamma Function)

… ►In this equation (and in (5.17.5) below), the $\mathrm{Ln}$’s have their principal values on the positive real axis and are continued via continuity, as in §4.2(i). ►When $z\to \mathrm{\infty}$ in $$, ►
5.17.5
$$\mathrm{Ln}G\left(z+1\right)\sim \frac{1}{4}{z}^{2}+z\mathrm{Ln}\mathrm{\Gamma}\left(z+1\right)-\left(\frac{1}{2}z(z+1)+\frac{1}{12}\right)\mathrm{Ln}z-\mathrm{ln}A+\sum _{k=1}^{\mathrm{\infty}}\frac{{B}_{2k+2}}{2k(2k+1)(2k+2){z}^{2k}}.$$

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►and ${\zeta}^{\prime}$ is the derivative of the zeta function (Chapter 25).
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