inversion problem

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11: Brian D. Sleeman

… ► thesis was Some Boundary Value Problems Associated with the Heun Equation. ►Sleeman published numerous papers in applied analysis, multiparameter spectral theory, direct and inverse scattering theory, and mathematical medicine. …

12: 19.20 Special Cases

… ►

19.20.5

2R_{G}\left(x,y,y\right)=yR_{C}\left(x,y\right)+\sqrt{x}.

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind
Referenced by:: §19.20(ii)
Permalink:: http://dlmf.nist.gov/19.20.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.20(ii), §19.20 and Ch.19

… ►

19.20.13

2(p-x)R_{J}\left(x,y,z,p\right)=3R_{F}\left(x,y,z\right)-3\sqrt{x}R_{C}\left(% yz,p^{2}\right),

p=x\pm\sqrt{(y-x)(z-x)}

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind
Referenced by:: §19.20(iii), §19.22(iii)
Permalink:: http://dlmf.nist.gov/19.20.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §19.20(iii), §19.20 and Ch.19

… ►When the variables are real and distinct, the various cases of

R_{J}\left(x,y,z,p\right)

are called circular (hyperbolic) cases if

(p-x)(p-y)(p-z)

is positive (negative), because they typically occur in conjunction with inverse circular (hyperbolic) functions. Cases encountered in dynamical problems are usually circular; hyperbolic cases include Cauchy principal values. … ►

19.20.20

R_{D}\left(x,y,y\right)=\frac{3}{2(y-x)}\left(R_{C}\left(x,y\right)-\frac{% \sqrt{x}}{y}\right),

x\neq y

y\neq 0

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables
Referenced by:: §19.20(iv), §19.22(iii)
Permalink:: http://dlmf.nist.gov/19.20.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §19.20(iv), §19.20 and Ch.19

…

13: 12.16 Mathematical Applications

… ►In Brazel et al. (1992) exponential asymptotics are considered in connection with an eigenvalue problem involving PCFs. … ►PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …

14: 6.18 Methods of Computation

… ►However, this problem is less severe for the series of spherical Bessel functions because of their more rapid rate of convergence, and also (except in the case of (6.10.6)) absence of cancellation when

z=x

(

>0

). ►For large

x

and

\left|z\right|

, expansions in inverse factorial series (§6.10(i)) or asymptotic expansions (§6.12) are available. …

15: Bibliography P

… ►

M. D. Perlman and I. Olkin (1980) Unbiasedness of invariant tests for MANOVA and other multivariate problems. Ann. Statist. 8 (6), pp. 1326–1341.

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External Links:: ISSN 0090-5364, FullText, MathReview (Takeaki Kariya), ZentralBlatt
Referenced by:: §35.9.
Encodings:: BibTeX

… ►

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992b) Integrals and Series: Inverse Laplace Transforms, Vol. 5. Gordon and Breach Science Publishers, New York.

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External Links:: ISBN 2-88124-838-1, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §1.14(viii), §1.4(iv), §10.22(vi), §10.43(vi), §10.71(iii), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(ii), §13.10(vi), §13.23(i), §13.23(iv), §14.17(v), §15.14, §15.14, §16.15, §16.20, §16.5, §18.17(ix), §19.13(iii), §24.13(iii), §25.11(ix), §25.12(ii), §25.7, §5.13, §6.14(iii), §7.14(iii), §8.6(iii), §9.10(ix).
Encodings:: BibTeX

►

J. D. Pryce (1993) Numerical Solution of Sturm-Liouville Problems. Monographs on Numerical Analysis, The Clarendon Press, Oxford University Press, New York.

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External Links:: ISBN 0-19-853415-9, MathReview (L. Greenberg), ZentralBlatt
Referenced by:: §3.7(iv).
Encodings:: BibTeX

…

16: Bibliography H

… ►

E. Hairer, S. P. Nørsett, and G. Wanner (1993) Solving Ordinary Differential Equations. I. Nonstiff Problems. 2nd edition, Springer Series in Computational Mathematics, Vol. 8, Springer-Verlag, Berlin.

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Notes:: A reprinting with corrections appeared in 2000.
External Links:: ISBN 3-540-56670-8, MathReview, ZentralBlatt
Referenced by:: §3.7(v), §3.7(i).
Encodings:: BibTeX

►

E. Hairer, S. P. Nørsett, and G. Wanner (2000) Solving Ordinary Differential Equations. I. Nonstiff Problems. 2nd edition, Springer-Verlag, Berlin.

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External Links:: ISBN 3-540-56670-8, Document, MathReview
Referenced by:: §32.17.
Encodings:: BibTeX

►

E. Hairer and G. Wanner (1996) Solving Ordinary Differential Equations. II. Stiff and Differential-Algebraic Problems. 2nd edition, Springer Series in Computational Mathematics, Vol. 14, Springer-Verlag, Berlin.

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External Links:: ISBN 3-540-60452-9, MathReview, ZentralBlatt
Referenced by:: §3.7(v).
Encodings:: BibTeX

… ►

G. H. Hardy and J. E. Littlewood (1925) Some problems of “Partitio Numerorum” (VI): Further researches in Waring’s Problem. Math. Z. 23, pp. 1–37.

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Notes:: Reprinted in Collected Papers of G. H. Hardy (Including Joint papers with J. E. Littlewood and others), Vol. I, pp. 469–505, Clarendon Press, Oxford, 1966.
External Links:: ISSN 0025-5874, Document, MathReview Entry, ZentralBlatt
Referenced by:: §27.13(iii).
Encodings:: BibTeX

… ►

S. P. Hastings and J. B. McLeod (1980) A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation. Arch. Rational Mech. Anal. 73 (1), pp. 31–51.

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External Links:: ISSN 0003-9527, Document, MathReview (Richard Brown), ZentralBlatt
Referenced by:: §32.11(ii).
Encodings:: BibTeX

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17: 3.5 Quadrature

… ►

Example. Laplace Transform Inversion

… ►A frequent problem with contour integrals is heavy cancellation, which occurs especially when the value of the integral is exponentially small compared with the maximum absolute value of the integrand. … ►In fact from (7.14.4) and the inversion formula for the Laplace transform (§1.14(iii)) we have … ►A special case is the rule for Hilbert transforms (§1.14(v)): … ►Other contour integrals occur in standard integral transforms or their inverses, for example, Hankel transforms (§10.22(v)), Kontorovich–Lebedev transforms (§10.43(v)), and Mellin transforms (§1.14(iv)). …

18: Bibliography B

… ►

P. Baldwin (1991) Coefficient functions for an inhomogeneous turning-point problem. Mathematika 38 (2), pp. 217–238.

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External Links:: ISSN 0025-5793, MathReview (Archibald D. MacGillivray), ZentralBlatt
Referenced by:: §2.8(iii).
Encodings:: BibTeX

… ►

W. G. Bickley (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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External Links:: Document, ZentralBlatt
Referenced by:: §10.73(iv).
Encodings:: BibTeX

… ►

Å. Björck (1996) Numerical Methods for Least Squares Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:: ISBN 0-89871-360-9, MathReview (Hongyuan Zha), ZentralBlatt
Referenced by:: §3.11(v).
Encodings:: BibTeX

… ►

P. Bleher and A. Its (1999) Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. of Math. (2) 150 (1), pp. 185–266.

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External Links:: ISSN 0003-486X, Document, MathReview (Thomas Kriecherbauer), ZentralBlatt
Referenced by:: §18.32.
Encodings:: BibTeX

… ►

A. A. Bogush and V. S. Otchik (1997) Problem of two Coulomb centres at large intercentre separation: Asymptotic expansions from analytical solutions of the Heun equation. J. Phys. A 30 (2), pp. 559–571.

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External Links:: ISSN 0305-4470, Document, MathReview, ZentralBlatt
Referenced by:: §31.13.
Encodings:: BibTeX

…

19: 26.15 Permutations: Matrix Notation

… ►The inversion number of

\sigma

is a sum of products of pairs of entries in the matrix representation of

\sigma

: … ►For the problem of derangements,

r_{j}(B)=\genfrac{(}{)}{0.0pt}{}{n}{j}

. … ►

Example 1

►The problème des ménages asks for the number of ways of seating

n

married couples around a circular table with labeled seats so that no men are adjacent, no women are adjacent, and no husband and wife are adjacent. …

20: Bibliography T

… ►

J. G. Taylor (1978) Error bounds for the Liouville-Green approximation to initial-value problems. Z. Angew. Math. Mech. 58 (12), pp. 529–537.

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External Links:: ISSN 0044-2267, Document, MathReview, ZentralBlatt
Referenced by:: §2.7(iii).
Encodings:: BibTeX

… ►

N. M. Temme (1992a) Asymptotic inversion of incomplete gamma functions. Math. Comp. 58 (198), pp. 755–764.

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External Links:: ISSN 0025-5718, FullText, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §8.12, §8.12.
Encodings:: BibTeX

►

N. M. Temme (1992b) Asymptotic inversion of the incomplete beta function. J. Comput. Appl. Math. 41 (1-2), pp. 145–157.

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Notes:: Asymptotic methods in analysis and combinatorics
External Links:: ISSN 0377-0427, Document, MathReview (Dumitru Acu), ZentralBlatt
Referenced by:: §8.18(ii).
Encodings:: BibTeX

… ►

N. M. Temme (1995c) Uniform asymptotic expansions of integrals: A selection of problems. J. Comput. Appl. Math. 65 (1-3), pp. 395–417.

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External Links:: ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt
Referenced by:: Ch.2.
Encodings:: BibTeX

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