inversion

(0.001 seconds)

31—40 of 167 matching pages

31: Brian D. Sleeman

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

32: 19.37 Tables

… ►Tabulated for

\operatorname{arcsin}k=0(1^{\circ})90^{\circ}

to 6D by Byrd and Friedman (1971) and to 15D by Abramowitz and Stegun (1964, Chapter 17). … ►Tabulated for

\operatorname{arcsin}k=0(1^{\circ})90^{\circ}

to 6D by Byrd and Friedman (1971) and to 15D by Abramowitz and Stegun (1964, Chapter 17). … ►Tabulated for

\phi=0(5^{\circ})90^{\circ}

\operatorname{arcsin}k=0(1^{\circ})90^{\circ}

to 6D by Byrd and Friedman (1971), for

\phi=0(5^{\circ})90^{\circ}

\operatorname{arcsin}k=0(2^{\circ})90^{\circ}

and

5^{\circ}(10^{\circ})85^{\circ}

to 8D by Abramowitz and Stegun (1964, Chapter 17), and for

\phi=0(10^{\circ})90^{\circ}

\operatorname{arcsin}k=0(5^{\circ})90^{\circ}

to 9D by Zhang and Jin (1996, pp. 674–675). … ►Tabulated (with different notation) for

\phi=0(15^{\circ})90^{\circ}

\alpha^{2}=0(.1)1

\operatorname{arcsin}k=0(15^{\circ})90^{\circ}

to 5D by Abramowitz and Stegun (1964, Chapter 17), and for

\phi=0(15^{\circ})90^{\circ}

\alpha^{2}=0(.1)1

\operatorname{arcsin}k=0(15^{\circ})90^{\circ}

to 7D by Zhang and Jin (1996, pp. 676–677). …

36: 19.2 Definitions

… ►In (19.2.18)–(19.2.22) the inverse trigonometric and hyperbolic functions assume their principal values (§§4.23(ii) and 4.37(ii)). When

x

and

y

are positive,

R_{C}\left(x,y\right)

is an inverse circular function if

x<y

and an inverse hyperbolic function (or logarithm) if

x>y

: ►

19.2.18

R_{C}\left(x,y\right)=\frac{1}{\sqrt{y-x}}\operatorname{arctan}\sqrt{\frac{y-x% }{x}}=\frac{1}{\sqrt{y-x}}\operatorname{arccos}\sqrt{x/y},

0\leq x<y

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\operatorname{arccos}\NVar{z}$ : arccosine function and $\operatorname{arctan}\NVar{z}$ : arctangent function
Referenced by:: §19.2(iv)
Permalink:: http://dlmf.nist.gov/19.2.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §19.2(iv), §19.2 and Ch.19

►

19.2.19

R_{C}\left(x,y\right)=\frac{1}{\sqrt{x-y}}\operatorname{arctanh}\sqrt{\frac{x-% y}{x}}=\frac{1}{\sqrt{x-y}}\ln\frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{y}},

0<y<x

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\operatorname{arctanh}\NVar{z}$ : inverse hyperbolic tangent function and $\ln\NVar{z}$ : principal branch of logarithm function
Referenced by:: §19.26(i), Figure 19.3.2, Figure 19.3.2, §19.36(ii)
Permalink:: http://dlmf.nist.gov/19.2.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §19.2(iv), §19.2 and Ch.19

… ►

19.2.20

R_{C}\left(x,y\right)=\sqrt{\frac{x}{x-y}}R_{C}\left(x-y,-y\right)=\frac{1}{% \sqrt{x-y}}\operatorname{arctanh}\sqrt{\frac{x}{x-y}}=\frac{1}{\sqrt{x-y}}\ln% \frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{-y}},

y<0\leq x

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\operatorname{arctanh}\NVar{z}$ : inverse hyperbolic tangent function and $\ln\NVar{z}$ : principal branch of logarithm function
Referenced by:: §19.2(iv), §19.20(iii), §19.25(i), Figure 19.3.2, Figure 19.3.2, §19.36(i)
Permalink:: http://dlmf.nist.gov/19.2.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §19.2(iv), §19.2 and Ch.19

…

$x, y$	real variables.
…
${\mathbf{A}}^{-1}$	inverse of the square matrix $\mathbf{A}$
…

… ►Inverse Laplace transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §5.19), Oberhettinger and Badii (1973, §2.18), and Prudnikov et al. (1992b, §3.35). …Inverse Mellin transforms are given in Erdélyi et al. (1954a, §7.5). …

39: 19.12 Asymptotic Approximations

… ►

19.12.4

(1-\alpha^{2})\Pi\left(\alpha^{2},k\right)=\left(\ln\frac{4}{k^{\prime}}\right% )\left(1+O\left({k^{\prime}}^{2}\right)\right)-\alpha^{2}R_{C}\left(1,1-\alpha% ^{2}\right),

-\infty<\alpha^{2}<1

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $\alpha^{2}$ : real or complex parameter
Referenced by:: §19.12
Permalink:: http://dlmf.nist.gov/19.12.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §19.12 and Ch.19

►

19.12.5

(1-\alpha^{2})\Pi\left(\alpha^{2},k\right)=\left(\ln\left(\frac{4}{k^{\prime}}% \right)-R_{C}\left(1,1-\alpha^{-2}\right)\right)\*\left(1+O\left({k^{\prime}}^% {2}\right)\right),

1<\alpha^{2}<\infty

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $\alpha^{2}$ : real or complex parameter
Referenced by:: §19.12
Permalink:: http://dlmf.nist.gov/19.12.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.12 and Ch.19

… ►

19.12.6

R_{C}\left(x,y\right)=\frac{\pi}{2\sqrt{y}}-\frac{\sqrt{x}}{y}\left(1+O\left(% \sqrt{\frac{x}{y}}\right)\right),

x/y\to 0

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $\pi$ : the ratio of the circumference of a circle to its diameter
Referenced by:: §19.12
Permalink:: http://dlmf.nist.gov/19.12.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §19.12 and Ch.19

►

19.12.7

R_{C}\left(x,y\right)=\frac{1}{2\sqrt{x}}\left(\left(1+\frac{y}{2x}\right)\ln% \left(\frac{4x}{y}\right)-\frac{y}{2x}\right)\*(1+O\left(y^{2}/x^{2}\right)),

y/x\to 0

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $\ln\NVar{z}$ : principal branch of logarithm function
Referenced by:: §19.12, §19.2(iv)
Permalink:: http://dlmf.nist.gov/19.12.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §19.12 and Ch.19

40: 36.13 Kelvin’s Ship-Wave Pattern

… ►

\theta_{+}(\phi)=\tfrac{1}{2}(\operatorname{arcsin}\left(3\sin\phi\right)-\phi),

►

\theta_{-}(\phi)=\tfrac{1}{2}(\pi-\phi-\operatorname{arcsin}\left(3\sin\phi% \right)).

… ►

36.13.5

|\phi|=\phi_{c}=\operatorname{arcsin}\left(\tfrac{1}{3}\right)=19^{\circ}.47122.

ⓘ

Symbols:: $\operatorname{arcsin}\NVar{z}$ : arcsine function
Permalink:: http://dlmf.nist.gov/36.13.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §36.13 and Ch.36

…

inversion

31—40 of 167 matching pages

31: Brian D. Sleeman

32: 19.37 Tables

33: 35.2 Laplace Transform

Inversion Formula

34: 19.17 Graphics

35: 24.5 Recurrence Relations

§24.5(iii) Inversion Formulas

36: 19.2 Definitions

37: 1.1 Special Notation

38: 15.14 Integrals

39: 19.12 Asymptotic Approximations

40: 36.13 Kelvin’s Ship-Wave Pattern