inversion formula

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41: 18.17 Integrals

… ►Just as the indefinite integrals (18.17.1), (18.17.3) and (18.17.4), many similar formulas can be obtained by applying (1.4.26) to the differentiation formulas (18.9.15), (18.9.16) and (18.9.19)–(18.9.28). … ►Formulas (18.17.9), (18.17.10) and (18.17.11) are fractional generalizations of $n$-th derivative formulas which are, after substitution of (18.5.7), special cases of (15.5.4), (15.5.5) and (15.5.3), respectively. … ►Formulas (18.17.12) and (18.17.13) are fractional generalizations of the differentiation formulas given in (Erdélyi et al., 1953b, §10.9(15)). … ►Some of the resulting formulas are given below. … ►Formulas (18.17.45) and (18.17.49) are integrated forms of the linearization formulas (18.18.22) and (18.18.23), respectively. …

42: 15.12 Asymptotic Approximations

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$\mathrm{\Re }z>\frac{1}{2}$ and ${\alpha }_{-}-\frac{1}{2}\pi +\delta \le \mathrm{ph}c\le {\alpha }_{+}+\frac{1}{2}\pi -\delta$, where

15.12.1 $${\alpha }_{\pm }=\arctan \left(\frac{\mathrm{ph}z-\mathrm{ph}\left(1-z\right)\mp \pi }{\ln |1-z^{-1}|}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable and ${\alpha }_{\pm }$

Permalink:

http://dlmf.nist.gov/15.12.E1

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Annotations for §15.12(ii), §15.12 and Ch.15

with $z$ restricted so that $\pm {\alpha }_{\pm }\in [0,\frac{1}{2}\pi )$.

… ►

15.12.6 $$\zeta =\mathrm{arccosh}z.$$

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Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function, $z$: complex variable and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/15.12.E6

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Annotations for §15.12(iii), §15.12 and Ch.15

… ►

15.12.10 $$\zeta =\mathrm{arccosh}\left(\frac{1}{4}z-1\right),$$

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Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function, $z$: complex variable and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/15.12.E10

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Annotations for §15.12(iii), §15.12 and Ch.15

… ►By combination of the foregoing results of this subsection with the linear transformations of §15.8(i) and the connection formulas of §15.10(ii), similar asymptotic approximations for $F(a+e_1\lambda ,b+e_2\lambda ;c+e_3\lambda ;z)$ can be obtained with $e_j=\pm 1$ or $0$, $j=1,2,3$. …

43: Bibliography E

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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

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The Euler-Maclaurin formula revisited. J. Austral. Math. Soc. Ser. B 40 (E), pp. E27–E76 (electronic).

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External Links:

ISSN 0334-2700, FullText, MathReview (J. Németh), ZentralBlatt

Referenced by:

§2.10(i).

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A formula including Legendre’s $E{K}^{\prime }+K{E}^{\prime }-K{K}^{\prime }=\frac{1}{2}\pi$ . Messenger of Math. 33, pp. 31–32.

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Notes:

Errata on p. 45 of the same volume.

External Links:

ZentralBlatt

Referenced by:

§15.16.

Encodings:

BibTeX

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44: 10.68 Modulus and Phase Functions

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${\theta }_{\nu }\left(x\right)=\mathrm{Arctan}\left({\mathrm{bei}}_{\nu }x/{\mathrm{ber}}_{\nu }x\right),$

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${\varphi }_{\nu }\left(x\right)=\mathrm{Arctan}\left({\mathrm{kei}}_{\nu }x/{\ker }_{\nu }x\right).$

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45: 3.8 Nonlinear Equations

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Regula Falsi

… ►Inverse linear interpolation (§3.3(v)) is used to obtain the first approximation: … ►

§3.8(iv) Zeros of Polynomials

… ►Explicit formulas for the zeros are available if $n\le 4$; see §§1.11(iii) and 4.43. No explicit general formulas exist when $n\ge 5$. …

46: 36.7 Zeros

… ►More general asymptotic formulas are given in Kaminski and Paris (1999). … ►Near $z=z_n$, and for small $x$ and $y$, the modulus $|{\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)|$ has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose $z$ and $x$ repeat distances are given by …

47: 10.43 Integrals

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10.43.21 $${\int }_0^{\mathrm{\infty }}\sin \left(at\right)K_0\left(t\right)dt=\frac{\mathrm{arcsinh}a}{{(1+a^2)}^{\frac{1}{2}}},$$ .

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Symbols:

$dx$: differential of $x$ , $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\mathrm{\Im }$: imaginary part, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $\sin z$: sine function

A&S Ref:

11.4.15

Permalink:

http://dlmf.nist.gov/10.43.E21

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Annotations for §10.43(iv), §10.43 and Ch.10

… ►For asymptotic expansions of the direct transform (10.43.30) see Wong (1981), and for asymptotic expansions of the inverse transform (10.43.31) see Naylor (1990, 1996). …

48: 15.9 Relations to Other Functions

… ►with inverse … … ►The following formulas apply with principal branches of the hypergeometric functions, associated Legendre functions, and fractional powers. …

49: Bibliography D

… ►

Computation of the incomplete gamma function ratios and their inverses. ACM Trans. Math. Software 12 (4), pp. 377–393.

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External Links:

ISSN 0098-3500, Document, ZentralBlatt

Referenced by:

§8.12, §8.25(iii).

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Algorithm 654: Fortran subroutines for computing the incomplete gamma function ratios and their inverses. ACM Trans. Math. Software 13 (3), pp. 318–319.

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Notes:

Single-precision Fortran, maximum accuracy 14D.

External Links:

ISSN 0098-3500, Document, GAMS (module 654; package TOMS), ZentralBlatt

Referenced by:

1st item.

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A simple sum formula for Clebsch-Gordan coefficients. Lett. Math. Phys. 5 (3), pp. 207–211.

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External Links:

ISSN 0377-9017, Document, MathReview, ZentralBlatt

Referenced by:

§14.17(iv).

Encodings:

BibTeX

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Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results. SIAM J. Math. Anal. 9 (1), pp. 76–86.

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External Links:

Document, MathReview (R. C. Varma), ZentralBlatt

Referenced by:

§18.17(iii).

Encodings:

BibTeX

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50: 19.29 Reduction of General Elliptic Integrals

… ►Cubic cases of these formulas are obtained by setting one of the factors in (19.29.3) equal to 1. ►The advantages of symmetric integrals for tables of integrals and symbolic integration are illustrated by (19.29.4) and its cubic case, which replace the $8+8+12=28$ formulas in Gradshteyn and Ryzhik (2000, 3.147, 3.131, 3.152) after taking $x^2$ as the variable of integration in 3. …(19.29.7) subsumes all 72 formulas in Gradshteyn and Ryzhik (2000, 3.168), and its cubic cases similarly replace the $18+36+18=72$ formulas in Gradshteyn and Ryzhik (2000, 3.133, 3.142, and 3.141(1-18)). … ►The first choice gives a formula that includes the 18+9+18 = 45 formulas in Gradshteyn and Ryzhik (2000, 3.133, 3.156, 3.158), and the second choice includes the 8+8+8+12 = 36 formulas in Gradshteyn and Ryzhik (2000, 3.151, 3.149, 3.137, 3.157) (after setting $x^2=t$ in some cases). … ►The first formula replaces (19.14.4)–(19.14.10). …