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21: 30.14 Wave Equation in Oblate Spheroidal Coordinates

… ►If $b_1=b_2=0$, then the function (30.13.8) is a twice-continuously differentiable solution of (30.13.7) in the entire $(x,y,z)$-space. …

22: 30.13 Wave Equation in Prolate Spheroidal Coordinates

… ►If $b_1=b_2=0$, then the function (30.13.8) is a twice-continuously differentiable solution of (30.13.7) in the entire $(x,y,z)$-space. …

23: 18.32 OP’s with Respect to Freud Weights

§18.32 OP’s with Respect to Freud Weights

►A Freud weight is a weight function of the form …where $Q(x)$ is real, even, nonnegative, and continuously differentiable, where $x{Q}^{\prime }(x)$ increases for $x>0$, and $Q^{\prime }(x)\to \mathrm{\infty }$ as $x\to \mathrm{\infty }$, see Freud (1969). …However, for asymptotic approximations in terms of elementary functions for the OP’s, and also for their largest zeros, see Levin and Lubinsky (2001) and Nevai (1986). For a uniform asymptotic expansion in terms of Airy functions (§9.2) for the OP’s in the case $Q(x)=x^4$ see Bo and Wong (1999). …

24: Bibliography

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Computation of the regular confluent hypergeometric function. The Mathematica Journal 5 (4), pp. 74–76.

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External Links:

ISSN 1047-5974, FullText

Referenced by:

§13.32(iii).

Encodings:

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Regular and irregular Coulomb wave functions expressed in terms of Bessel-Clifford functions. J. Math. Physics 33, pp. 111–116.

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External Links:

MathReview (E. Weber), ZentralBlatt

Referenced by:

§33.9(ii).

Encodings:

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Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions. J. Heat Transfer 118 (3), pp. 789–792.

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External Links:

Document

Referenced by:

§10.73(iv), §8.24(iii).

Encodings:

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Functional inequalities for hypergeometric functions and complete elliptic integrals. SIAM J. Math. Anal. 23 (2), pp. 512–524.

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External Links:

ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§19.9(i).

Encodings:

BibTeX

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Singularities of Differentiable Maps. Vol. II. Birkhäuser, Boston-Berlin.

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Notes:

Monodromy and asymptotics of integrals, Translated from the Russian by Hugh Porteous, Translation revised by the authors and James Montaldi

External Links:

ISBN 0-8176-3185-2, MathReview, ZentralBlatt

Referenced by:

§36.12(iii).

Encodings:

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25: 10.20 Uniform Asymptotic Expansions for Large Order

§10.20 Uniform Asymptotic Expansions for Large Order

… ►that is infinitely differentiable on the interval , including $z=1$. … ►In this way there is less usage of many-valued functions. … ►Each of the coefficients $A_k(\zeta )$, $B_k(\zeta )$, $C_k(\zeta )$, and $D_k(\zeta )$, $k=0,1,2,\mathrm{\dots }$, is real and infinitely differentiable on the interval . … ►

§10.20(iii) Double Asymptotic Properties

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26: 2.4 Contour Integrals

… ►On the interval let $q(t)$ be differentiable and ${\mathrm{e}}^{-ct}q(t)$ be absolutely integrable, where $c$ is a real constant. … ►Now assume that $c>0$ and we are given a function $Q(z)$ that is both analytic and has the expansion …Assume also (2.4.4) is differentiable. … ►For large $t$, the asymptotic expansion of $q(t)$ may be obtained from (2.4.3) by Haar’s method. This depends on the availability of a comparison function $F(z)$ for $Q(z)$ that has an inverse transform … ►and assigning an appropriate value to $c$ to modify the contour, the approximating integral is reducible to an Airy function or a Scorer function (§§9.2, 9.12). …

27: 3.3 Interpolation

… ►If $f$ and the $z_k$ ($=x_k$) are real, and $f$ is $n$ times continuously differentiable on a closed interval containing the $x_k$, then … ►

Example

… ►For interpolation of a bounded function $f$ on $ℝ$ the cardinal function of $f$ is defined by …where …is called the Sinc function. …

28: 10.43 Integrals

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§10.43(i) Indefinite Integrals

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§10.43(iii) Fractional Integrals

… ► … ►The Kontorovich–Lebedev transform of a function $g(x)$ is defined as … ►

On the interval , $x^{-1}g(x)$ is continuously differentiable and each of $xg(x)$ and $xd(x^{-1}g(x))/dx$ is absolutely integrable.

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29: 18.40 Methods of Computation

… ►Given the power moments, ${\mu }_n={\int }_a^b{x}^{n}d\mu (x)$, $n=0,1,2,\mathrm{\dots }$, can these be used to find a unique $\mu (x)$, a non-decreasing, real, function of $x$, in the case that the moment problem is determined? Should a unique solution not exist the moment problem is then indeterminant. … ►In what follows we consider only the simple, illustrative, case that $\mu (x)$ is continuously differentiable so that $d\mu (x)=w(x)dx$, with $w(x)$ real, positive, and continuous on a real interval $[a,b].$ The strategy will be to: 1) use the moments to determine the recursion coefficients ${\alpha }_n,{\beta }_n$ of equations (18.2.11_5) and (18.2.11_8); then, 2) to construct the quadrature abscissas $x_i$ and weights (or Christoffel numbers) $w_i$ from the J-matrix of §3.5(vi), equations (3.5.31) and(3.5.32). … ►The question is then: how is this possible given only $F_N(z)$, rather than $F(z)$ itself? $F_N(z)$ often converges to smooth results for $z$ off the real axis for $\mathrm{\Im }z$ at a distance greater than the pole spacing of the $x_n$, this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to $F_N(z)$ and evaluating these on the real axis in regions of higher pole density that those of the approximating function. … ► $H\left(x\right)$ being the Heaviside step-function, see (1.16.13). … ►The example chosen is inversion from the ${\alpha }_n,{\beta }_n$ for the weight function for the repulsive Coulomb–Pollaczek, RCP, polynomials of (18.39.50). …

30: 2.10 Sums and Sequences

… ►Then … ►

§2.10(iii) Asymptotic Expansions of Entire Functions

… ►Hence … ►We need a “comparison function” $g(z)$ with the properties: … ►Secondly, when $f(z)-g(z)$ is $m$ times continuously differentiable on $|z|=r$ the result (2.10.29) can be strengthened. …