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21: 4.26 Integrals

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4.26.5 $$\int \sec xdx={\mathrm{gd}}^{-1}\left(x\right),$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\sec z$: secant function and $x$: real variable

A&S Ref:

4.3.117

Permalink:

http://dlmf.nist.gov/4.26.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(ii), §4.26 and Ch.4

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§4.26(iv) Inverse Trigonometric Functions

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4.26.14 $$\int \arcsin xdx=x\arcsin x+{(1-x^2)}^{1/2},$$ ,

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\arcsin z$: arcsine function and $x$: real variable

A&S Ref:

4.4.58 (modified)

Permalink:

http://dlmf.nist.gov/4.26.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(iv), §4.26 and Ch.4

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4.26.15 $$\int \arccos xdx=x\arccos x-{(1-x^2)}^{1/2},$$ .

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\arccos z$: arccosine function and $x$: real variable

A&S Ref:

4.4.59 (modified)

Permalink:

http://dlmf.nist.gov/4.26.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(iv), §4.26 and Ch.4

… ►Extensive compendia of indefinite and definite integrals of trigonometric and inverse trigonometric functions include Apelblat (1983, pp. 48–109), Bierens de Haan (1939), Gradshteyn and Ryzhik (2000, Chapters 2–4), Gröbner and Hofreiter (1949, pp. 116–139), Gröbner and Hofreiter (1950, pp. 94–160), and Prudnikov et al. (1986a, §§1.5, 1.7, 2.5, 2.7).

22: 16.20 Integrals and Series

… ►Extensive lists of Laplace transforms and inverse Laplace transforms of the Meijer $G$-function are given in Prudnikov et al. (1992a, §3.40) and Prudnikov et al. (1992b, §3.38). …

23: Bibliography X

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Prolate spheroidal wavefunctions, quadrature and interpolation. Inverse Problems 17 (4), pp. 805–838.

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Notes:

Special issue to celebrate Pierre Sabatier’s 65th birthday (Montpellier, 2000)

External Links:

ISSN 0266-5611, Document, MathReview (Thomas Sauer), ZentralBlatt

Referenced by:

§30.13(v).

Encodings:

BibTeX

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24: 18.40 Methods of Computation

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§18.40(ii) The Classical Moment Problem

… ►Having now directly connected computation of the quadrature abscissas and weights to the moments, what follows uses these for a Stieltjes–Perron inversion to regain $w(x)$. ►

Stieltjes Inversion via (approximate) Analytic Continuation

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Histogram Approach

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Derivative Rule Approach

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25: 4.32 Inequalities

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4.32.4 $$\arctan x\le \frac{1}{2}\pi \tanh x,$$ $x\ge 0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\tanh z$: hyperbolic tangent function, $\arctan z$: arctangent function and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.32 and Ch.4

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26: 7.22 Methods of Computation

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§7.22(i) Main Functions

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27: 19.6 Special Cases

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19.6.8 $$F(\varphi ,1)=(\sin \varphi )R_C(1,{\cos }^2\varphi )={\mathrm{gd}}^{-1}\left(\varphi \right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\sin z$: sine function and $\varphi$: real or complex argument

Referenced by:

§19.10(ii), §19.10(ii)

Permalink:

http://dlmf.nist.gov/19.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.6(ii), §19.6 and Ch.19

►For the inverse Gudermannian function ${\mathrm{gd}}^{-1}\left(\varphi \right)$ see §4.23(viii). …

28: 12.16 Mathematical Applications

… ►PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …

29: 26.22 Software

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Inverse Symbolic Calculator (website).

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30: Peter A. Clarkson

… ►His well-known book Solitons, Nonlinear Evolution Equations and Inverse Scattering (with M. …