inversion formula

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21—30 of 56 matching pages

21: 6.14 Integrals

… ►

6.14.3

\int_{0}^{\infty}e^{-at}\operatorname{si}\left(t\right)\,\mathrm{d}t=-\frac{1}% {a}\operatorname{arctan}a,

\Re a>0

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\Re$ : real part and $\operatorname{si}\left(\NVar{z}\right)$ : sine integral
A&S Ref:: 5.2.29
Permalink:: http://dlmf.nist.gov/6.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §6.14(i), §6.14 and Ch.6

…

22: 6.18 Methods of Computation

… ►For large

x

and

\left|z\right|

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

\operatorname{ph}z

can be covered by use of the continuation formulas of §6.4. … ►For example, the Gauss–Laguerre formula (§3.5(v)) can be applied to (6.2.2); see Todd (1954) and Tseng and Lee (1998). For an application of the Gauss–Legendre formula (§3.5(v)) see Tooper and Mark (1968). …

23: 16.15 Integral Representations and Integrals

… ►For these and other formulas, including double Mellin–Barnes integrals, see Erdélyi et al. (1953a, §5.8). …For inverse Laplace transforms of Appell functions see Prudnikov et al. (1992b, §3.40).

24: Bibliography C

… ►

R. G. Campos (1995) A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.

ⓘ

External Links:: ISSN 1017-1398, Document, MathReview (V. I. Polovinkin), ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

… ►

L. Carlitz (1963) The inverse of the error function. Pacific J. Math. 13 (2), pp. 459–470.

ⓘ

External Links:: MathReview (D. P. Banerjee), ZentralBlatt
Referenced by:: §7.17(ii).
Encodings:: BibTeX

… ►

B. C. Carlson (2008) Power series for inverse Jacobian elliptic functions. Math. Comp. 77 (263), pp. 1615–1621.

ⓘ

External Links:: ISSN 0025-5718, Document, MathReview (Shaun Cooper), ZentralBlatt
Referenced by:: §19.25(v), §22.15(ii).
Encodings:: BibTeX

… ►

D. Colton and R. Kress (1998) Inverse Acoustic and Electromagnetic Scattering Theory. 2nd edition, Applied Mathematical Sciences, Vol. 93, Springer-Verlag, Berlin.

ⓘ

External Links:: ISBN 3-540-62838-X, MathReview, ZentralBlatt
Referenced by:: §10.73(i).
Encodings:: BibTeX

… ►

R. Cools (2003) An encyclopaedia of cubature formulas. J. Complexity 19 (3), pp. 445–453.

ⓘ

Notes:: Numerical integration and its complexity (Oberwolfach, 2001)
External Links:: ISSN 0885-064X, Document, Web Page, MathReview, ZentralBlatt
Referenced by:: §3.5(x).
Encodings:: BibTeX

…

25: 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.

ⓘ

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

… ►

R. Spira (1971) Calculation of the gamma function by Stirling’s formula. Math. Comp. 25 (114), pp. 317–322.

ⓘ

External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §5.11(i), §5.11(ii).
Encodings:: BibTeX

… ►

A. Strecok (1968) On the calculation of the inverse of the error function. Math. Comp. 22 (101), pp. 144–158.

ⓘ

External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §7.17(ii).
Encodings:: BibTeX

… ►

A. H. Stroud and D. Secrest (1966) Gaussian Quadrature Formulas. Prentice-Hall Inc., Englewood Cliffs, N.J..

ⓘ

External Links:: MathReview (P. C. Hammer), ZentralBlatt
Referenced by:: §3.5(v).
Encodings:: BibTeX

…

26: 19.26 Addition Theorems

… ►

§19.26(i) General Formulas

… ►

19.26.11

R_{C}\left(x+\lambda,y+\lambda\right)+R_{C}\left(x+\mu,y+\mu\right)=R_{C}\left% (x,y\right),

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\lambda$ : positive, $x$ : positive, $y$ : positive and $\mu>0$ : parameter
Referenced by:: §19.26(i)
Permalink:: http://dlmf.nist.gov/19.26.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §19.26(i), §19.26 and Ch.19

… ►

§19.26(iii) Duplication Formulas

… ►The equations inverse to

z+\lambda=(\sqrt{z}+\sqrt{x})(\sqrt{z}+\sqrt{y})

and the two other equations obtained by permuting

x, y, z

(see (19.26.19)) are … ►

19.26.25

R_{C}\left(x,y\right)=2R_{C}\left(x+\lambda,y+\lambda\right),

\lambda=y+2\sqrt{x}\sqrt{y}

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\lambda$ : positive, $x$ : positive and $y$ : positive
Referenced by:: §19.26(iii)
Permalink:: http://dlmf.nist.gov/19.26.E25
Encodings:: pMML, png, TeX
See also:: Annotations for §19.26(iii), §19.26 and Ch.19

…

27: Bibliography

… ►

M. J. Ablowitz and P. A. Clarkson (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society Lecture Note Series, Vol. 149, Cambridge University Press, Cambridge.

ⓘ

External Links:: ISBN 0-521-38730-2, Document, Link, MathReview (Walter Oevel), ZentralBlatt
Referenced by:: §22.19(ii), §22.19(iii), §32.11(ii), §32.5, §9.16, Profile Mark J. Ablowitz, Profile Peter A. Clarkson.
Encodings:: BibTeX

… ►

M. J. Ablowitz and H. Segur (1981) Solitons and the Inverse Scattering Transform. SIAM Studies in Applied Mathematics, Vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

ⓘ

External Links:: ISBN 0-89871-174-6, MathReview (Benno Fuchssteiner), ZentralBlatt
Referenced by:: §21.9, §21.9, §32.5, §9.16, Profile Mark J. Ablowitz.
Encodings:: BibTeX

… ►

K. Aomoto (1987) Special value of the hypergeometric function

{}_{3}F_{2}

and connection formulae among asymptotic expansions. J. Indian Math. Soc. (N.S.) 51, pp. 161–221.

ⓘ

External Links:: ISSN 0019-5839, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §16.4(iii).
Encodings:: BibTeX

… ►

T. M. Apostol (1985a) Formulas for higher derivatives of the Riemann zeta function. Math. Comp. 44 (169), pp. 223–232.

ⓘ

External Links:: ISSN 0025-5718, FullText, MathReview (Aleksandar Ivić), ZentralBlatt
Referenced by:: (25.11.19), (25.11.20), (25.11.6), §25.11(vi), §25.18(i), (25.4.5), (25.4.6), §25.4, (25.6.11), (25.6.12), (25.6.13), (25.6.14), §25.6(ii), §25.6.
Encodings:: BibTeX

… ►

T. M. Apostol (2006) Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.

ⓘ

Referenced by:: §24.4(iii).
Encodings:: BibTeX

…

28: 1.2 Elementary Algebra

… ►where

\det(\mathbf{A})

is defined by the Leibniz formula … ►

The Inverse

►If det(

\mathbf{A}

)

\neq

0

\mathbf{A}

has a unique inverse,

{\mathbf{A}}^{-1}

, such that … ►has a unique solution,

\mathbf{b}={\mathbf{A}}^{-1}\mathbf{c}

. … ►Formula (1.2.77) is more generally valid for all square matrices

\mathbf{A}

, not necessarily non-defective, see Hall (2015, Thm 2.12).

29: 22.20 Methods of Computation

… ►

22.20.4

\phi_{n-1}=\frac{1}{2}\left(\phi_{n}+\operatorname{arcsin}\left(\frac{c_{n}}{a% _{n}}\sin\phi_{n}\right)\right),

ⓘ

Symbols:: $\operatorname{arcsin}\NVar{z}$ : arcsine function, $\sin\NVar{z}$ : sine function, $a_{n}$ : numbers, $c_{n}$ : numbers, $n$ : positive and $\phi_{N}$ : approximations
Referenced by:: §22.20(ii)
Permalink:: http://dlmf.nist.gov/22.20.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §22.20(ii), §22.20 and Ch.22

►and the inverse sine has its principal value (§4.23(ii)). …This formula for

\operatorname{dn}

becomes unstable near

x=K

. … ►

§22.20(v) Inverse Functions

…

30: 22.21 Tables

… ►Spenceley and Spenceley (1947) tabulates

\operatorname{sn}\left(Kx,k\right)

\operatorname{cn}\left(Kx,k\right)

\operatorname{dn}\left(Kx,k\right)

\operatorname{am}\left(Kx,k\right)

\mathcal{E}\left(Kx,k\right)

for

\operatorname{arcsin}k=1^{\circ}(1^{\circ})89^{\circ}

and

x=0\left(\tfrac{1}{90}\right)1

to 12D, or 12 decimals of a radian in the case of

\operatorname{am}\left(Kx,k\right)

. … ►Tables of theta functions (§20.15) can also be used to compute the twelve Jacobian elliptic functions by application of the quotient formulas given in §22.2.