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31: 31.8 Solutions via Quadratures

… ►The variables $\lambda$ and $\nu$ are two coordinates of the associated hyperelliptic (spectral) curve $\mathrm{\Gamma }:{\nu }^2={\prod }_{j=1}^{2g+1}(\lambda -{\lambda }_j)$. …

32: 19.36 Methods of Computation

… ►The incomplete integrals $R_F(x,y,z)$ and $R_G(x,y,z)$ can be computed by successive transformations in which two of the three variables converge quadratically to a common value and the integrals reduce to $R_C$, accompanied by two quadratically convergent series in the case of $R_G$; compare Carlson (1965, §§5,6). …

33: 15.8 Transformations of Variable

… ►A quadratic transformation relates two hypergeometric functions, with the variable in one a quadratic function of the variable in the other, possibly combined with a fractional linear transformation. …

34: 18.38 Mathematical Applications

… ►This gives also new structures and results in the one-variable case, but the obtained nonsymmetric special functions can now usually be written as a linear combination of two known special functions. …

35: 4.24 Inverse Trigonometric Functions: Further Properties

… ►which requires $z$ $(=x+\mathrm{i}y)$ to lie between the two rectangular hyperbolas given by ►

4.24.6 $$x^2-y^2=-\frac{1}{2}.$$

ⓘ

Symbols:

$x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/4.24.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(i), §4.24 and Ch.4

… ►

4.24.7 $$\frac{d}{dz}\arcsin z={(1-z^2)}^{-1/2},$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\arcsin z$: arcsine function and $z$: complex variable

A&S Ref:

4.4.52

Permalink:

http://dlmf.nist.gov/4.24.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(ii), §4.24 and Ch.4

►

4.24.8 $$\frac{d}{dz}\arccos z=-{(1-z^2)}^{-1/2},$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\arccos z$: arccosine function and $z$: complex variable

A&S Ref:

4.4.53

Permalink:

http://dlmf.nist.gov/4.24.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(ii), §4.24 and Ch.4

►

4.24.9 $$\frac{d}{dz}\arctan z=\frac{1}{1+z^2}.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\arctan z$: arctangent function and $z$: complex variable

A&S Ref:

4.4.54

Permalink:

http://dlmf.nist.gov/4.24.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(ii), §4.24 and Ch.4

…

36: 31.17 Physical Applications

… ►The problem of adding three quantum spins $𝐬$, $𝐭$, and $𝐮$ can be solved by the method of separation of variables, and the solution is given in terms of a product of two Heun functions. …

37: 4.38 Inverse Hyperbolic Functions: Further Properties

… ►which requires $z$ $(=x+\mathrm{i}y)$ to lie between the two rectangular hyperbolas given by ►

4.38.8 $$x^2-y^2=\frac{1}{2}.$$

ⓘ

Symbols:

$x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/4.38.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(i), §4.38 and Ch.4

… ►

4.38.9 $$\frac{d}{dz}\mathrm{arcsinh}z={(1+z^2)}^{-1/2}.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{arcsinh}z$: inverse hyperbolic sine function and $z$: complex variable

A&S Ref:

4.6.37

Permalink:

http://dlmf.nist.gov/4.38.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(ii), §4.38 and Ch.4

… ►

4.38.11 $$\frac{d}{dz}\mathrm{arctanh}z=\frac{1}{1-z^2}.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{arctanh}z$: inverse hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.6.39

Permalink:

http://dlmf.nist.gov/4.38.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(ii), §4.38 and Ch.4

… ►

4.38.13 $$\frac{d}{dz}\mathrm{arcsech}z=-\frac{1}{z{(1-z^2)}^{1/2}}.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{arcsech}z$: inverse hyperbolic secant function and $z$: complex variable

A&S Ref:

4.6.41 (has an error.)

Permalink:

http://dlmf.nist.gov/4.38.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(ii), §4.38 and Ch.4

…

38: 1.13 Differential Equations

… ►

1.13.4

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Symbols:

$𝒲$: Wronskian, $det$: determinant, $z$: variable, $w_1(z)$: solution and $w_2(z)$: solution

Referenced by:

§1.13(i), Erratum (V1.0.25) for Section 1.13

Permalink:

http://dlmf.nist.gov/1.13.E4

Encodings:

pMML, png, TeX

Addition (effective with 1.0.25):

The determinant form of the two-argument Wronskian was added as an equality to this equation.

See also:

Annotations for §1.13(i), §1.13(i), §1.13 and Ch.1

… ► $u$ and $z$ belong to domains $U$ and $D$ respectively, the coefficients $f(u,z)$ and $g(u,z)$ are continuous functions of both variables, and for each fixed $u$ (fixed $z$) the two functions are analytic in $z$ (in $u$). …

39: 31.1 Special Notation

… ► ►►►

$x$, $y$	real variables.
$z$, $\zeta$, $w$, $W$	complex variables.
…

… ►Sometimes the parameters are suppressed.

40: Bibliography I

… ►

Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

ⓘ

Notes:

With two chapters by Walter Van Assche, With a foreword by Richard A. Askey, Corrected reprint of the 2005 original.

External Links:

ISBN 978-0-521-14347-9, MathReview Entry, ZentralBlatt

Referenced by:

§18.12, (18.15.4_5), (18.16.19), (18.16.20), (18.16.21), §18.17(ii), §18.18(vii), §18.19, (18.2.20), §18.2(vi), §18.2(xii), §18.2(xii), §18.2(x), §18.20(i), §18.20(ii), §18.21(ii), §18.22(i), §18.22(ii), §18.22(iii), §18.23, §18.25(i), §18.25(iii), §18.26(i), §18.26(iv), (18.27.4), §18.27(i), §18.27(ii), §18.27(iii), §18.27(v), §18.27(vi), §18.28(i), §18.28(ii), §18.28(iii), §18.28(v), §18.28(vi), §18.28(vii), §18.28(viii), §18.30(vi), §18.30(vii), §18.30(iii), §18.30(iv), §18.30, §18.30(vi), §18.34(i), §18.34(ii), §18.34(ii), §18.34(iii), (18.35.4), (18.35.5), (18.35.6), (18.35.6_2), (18.35.6_3), (18.35.6_4), (18.35.7), (18.35.8), §18.35(ii), §18.35(iii), §18.36(iii), §18.38(ii), §18.39(iv), §18.39(iv), §18.39(iv), (18.40.4), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.

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