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11: 19.36 Methods of Computation

… ►

§19.36(ii) Quadratic Transformations

►Complete cases of Legendre’s integrals and symmetric integrals can be computed with quadratic convergence by the AGM method (including Bartky transformations), using the equations in §19.8(i) and §19.22(ii), respectively. … ►As

n\to\infty

c_{n}

a_{n}

, and

t_{n}

converge quadratically to limits

0

M

, and

T

, respectively; hence … ►Computation of Legendre’s integrals of all three kinds by quadratic transformation is described by Cazenave (1969, pp. 128–159, 208–230). ►Quadratic transformations can be applied to compute Bulirsch’s integrals (§19.2(iii)). …

12: 15.8 Transformations of Variable

… ►

§15.8(iii) Quadratic Transformations

►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. … ►

§15.8(iv) Quadratic Transformations (Continued)

… ►This is a quadratic transformation between two cases in Group 1. … ►which is a quadratic transformation between two cases in Group 3. …

13: 19.31 Probability Distributions

… ►

R_{G}\left(x,y,z\right)

and

R_{F}\left(x,y,z\right)

occur as the expectation values, relative to a normal probability distribution in

{\mathbb{R}}^{2}

{\mathbb{R}}^{3}

, of the square root or reciprocal square root of a quadratic form. …

14: 19.8 Quadratic Transformations

§19.8 Quadratic Transformations

… ►showing that the convergence of

c_{n}

to 0 and of

a_{n}

and

g_{n}

M\left(a_{0},g_{0}\right)

is quadratic in each case. … ►Again,

p_{n}

and

\varepsilon_{n}

converge quadratically to

M\left(a_{0},g_{0}\right)

and 0, respectively, and

Q_{n}

converges to 0 faster than quadratically. …

15: 16.16 Transformations of Variables

… ►

16.16.10

{F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y\right)=\frac{\Gamma\left(% \gamma^{\prime}\right)\Gamma\left(\beta-\alpha\right)}{\Gamma\left(\gamma^{% \prime}-\alpha\right)\Gamma\left(\beta\right)}(-y)^{-\alpha}{F_{4}}\left(% \alpha,\alpha-\gamma^{\prime}+1;\gamma,\alpha-\beta+1;\frac{x}{y},\frac{1}{y}% \right)+\frac{\Gamma\left(\gamma^{\prime}\right)\Gamma\left(\alpha-\beta\right% )}{\Gamma\left(\gamma^{\prime}-\beta\right)\Gamma\left(\alpha\right)}(-y)^{-% \beta}{F_{4}}\left(\beta,\beta-\gamma^{\prime}+1;\gamma,\beta-\alpha+1;\frac{x% }{y},\frac{1}{y}\right).

ⓘ

Symbols:: ${F_{4}}\left(\NVar{\alpha},\NVar{\beta};\NVar{\gamma},\NVar{\gamma^{\prime}};% \NVar{x},\NVar{y}\right)$ : fourth Appell function and $\Gamma\left(\NVar{z}\right)$ : gamma function
Permalink:: http://dlmf.nist.gov/16.16.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §16.16(ii), §16.16 and Ch.16

►For quadratic transformations of Appell functions see Carlson (1976). …

16: 32.7 Bäcklund Transformations

… ►

\mbox{P}_{\mbox{\scriptsize VI}}

also has quadratic and quartic transformations. …The quadratic transformation …

17: 1.11 Zeros of Polynomials

… ►

Quadratic Equations

…

18: 19.22 Quadratic Transformations

§19.22 Quadratic Transformations

… ►As

n\to\infty

p_{n}

and

\varepsilon_{n}

converge quadratically to

M\left(a_{0},g_{0}\right)

and 0, respectively, and

Q_{n}

converges to 0 faster than quadratically. …

19: 27.8 Dirichlet Characters

… ►If

k

is odd, then the real characters (mod

k

) are the principal character and the quadratic characters described in the next section.

20: 18.19 Hahn Class: Definitions

… ►

Wilson class (or quadratic lattice class). These are OP’s $p_{n}(x)=p_{n}(\lambda(y))$ ( $p_{n}(x)$ of degree $n$ in $x$ , $\lambda(y)$ quadratic in $y$ ) where the role of the differentiation operator is played by $\tfrac{\Delta_{y}}{\Delta_{y}(\lambda(y))}$ or $\tfrac{\nabla_{y}}{\nabla_{y}(\lambda(y))}$ or $\tfrac{\delta_{y}}{\delta_{y}(\lambda(y))}$ . The Wilson class consists of two discrete and two continuous families.

…