numerical inversion

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1: 8.25 Methods of Computation

… ►A numerical inversion procedure is also given for calculating the value of

x

(with 10S accuracy), when

a

and

P\left(a,x\right)

are specified, based on Newton’s rule (§3.8(ii)). …

2: Bibliography E

… ►

U. T. Ehrenmark (1995) The numerical inversion of two classes of Kontorovich-Lebedev transform by direct quadrature. J. Comput. Appl. Math. 61 (1), pp. 43–72.

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External Links:: ISSN 0377-0427, Document, MathReview (J. P. Singhal), ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

…

3: 3.11 Approximation Techniques

… ►

Laplace Transform Inversion

►Numerical inversion of the Laplace transform (§1.14(iii)) …

4: 4.45 Methods of Computation

… ►

4.45.8

2\operatorname{arctan}\frac{x}{1+(1+x^{2})^{1/2}}=\operatorname{arctan}x,

0<x<\infty

ⓘ

Symbols:: $\operatorname{arctan}\NVar{z}$ : arctangent function and $x$ : real variable
Referenced by:: Erratum (V1.0.7) for Equations (4.45.8), (4.45.9)
Permalink:: http://dlmf.nist.gov/4.45.E8
Encodings:: pMML, png, TeX
Substitution (effective with 1.0.7):: The equation that was given originally, $2\operatorname{arctan}\frac{(1+x^{2})^{1/2}-1}{x}=\operatorname{arctan}x$ , has been replaced with an equation that leads to a better recurrence for numerically computing $\operatorname{arctan}x$ ; see (4.45.9).
Suggested 2014-02-05 by Masataka Urago
See also:: Annotations for §4.45(i), §4.45(i), §4.45 and Ch.4

… ►

4.45.9

x_{n}=\frac{x_{n-1}}{1+(1+x^{2}_{n-1})^{1/2}},

n=1,2,3,\dots

ⓘ

Symbols:: $\operatorname{arctan}\NVar{z}$ : arctangent function, $n$ : integer and $x$ : real variable
Referenced by:: (4.45.8), Erratum (V1.0.7) for Equations (4.45.8), (4.45.9)
Permalink:: http://dlmf.nist.gov/4.45.E9
Encodings:: pMML, png, TeX
Substitution (effective with 1.0.7):: The original recurrence given in this equation, $x_{n}=\frac{(1+x^{2}_{n-1})^{1/2}-1}{x_{n-1}}$ is susceptible to cancellation error when $x_{n}$ is small. It has been replaced with another recurrence that is better for numerically computing $\operatorname{arctan}x$ .
Suggested 2014-02-05 by Masataka Urago
See also:: Annotations for §4.45(i), §4.45(i), §4.45 and Ch.4

…

5: 3.5 Quadrature

… ►

Example. Laplace Transform Inversion

… ►A special case is the rule for Hilbert transforms (§1.14(v)): …

6: Bibliography S

… ►

H. E. Salzer (1955) Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.

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External Links:: FullText, MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §3.5(viii).
Encodings:: BibTeX

…

7: Brian D. Sleeman

… ►Sleeman published numerous papers in applied analysis, multiparameter spectral theory, direct and inverse scattering theory, and mathematical medicine. …