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31—40 of 129 matching pages

31: 5.11 Asymptotic Expansions

… ►For further information see Olver (1997b, pp. 293–295), and for other error bounds see Whittaker and Watson (1927, §12.33), Spira (1971), and Schäfke and Finsterer (1990). …

32: 8.20 Asymptotic Expansions of $E_{p}\left(z\right)$

… ►Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii). …

33: 18.38 Mathematical Applications

… ►The Askey–Gasper inequality … ►

§18.38(iii) Other OP’s

… ► … ► … ►A symmetric Laurent polynomial is a function of the form …

34: 23.15 Definitions

§23.15 Definitions

… ►The set of all bilinear transformations of this form is denoted by SL

(2,\mathbb{Z})

(Serre (1973, p. 77)). … ►If, in addition,

f(\tau)\to 0

q\to 0

, then

f(\tau)

is called a cusp form. … ►

35: 7.13 Zeros

… ►The other zeros of

\operatorname{erf}z

are

-z_{n}

\overline{z}_{n}

-\overline{z}_{n}

. … ►The other zeros of

\operatorname{erfc}z

are

\overline{z}_{n}

. …

36: 18.27 $q$ -Hahn Class

§18.27 $q$ -Hahn Class

… ►Together they form the $q$ -Askey scheme. … ►All these systems of OP’s have orthogonality properties of the form …Thus in addition to a relation of the form (18.27.2), such systems may also satisfy orthogonality relations with respect to a continuous weight function on some interval. … ►For other formulas, including

q

-difference equations, recurrence relations, duality formulas, special cases, and limit relations, see Koekoek et al. (2010, Chapter 14). …

37: 9.17 Methods of Computation

… ►However, in the case of

\operatorname{Ai}\left(z\right)

and

\operatorname{Bi}\left(z\right)

this accuracy can be increased considerably by use of the exponentially-improved forms of expansion supplied in §9.7(v). … ►As described in §3.7(ii), to ensure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows at least as fast as all other solutions of the differential equation. … ►For details, including the application of a generalized form of Gaussian quadrature, see Gordon (1969, Appendix A) and Schulten et al. (1979). …

38: 18.15 Asymptotic Approximations

… ►

§18.15(vi) Other Approximations

►The asymptotic behavior of the classical OP’s as

x\to\pm\infty

with the degree and parameters fixed is evident from their explicit polynomial forms; see, for example, (18.2.7) and the last two columns of Table 18.3.1. …

39: 1.10 Functions of a Complex Variable

… ►A function whose only singularities, other than the point at infinity, are poles is called a meromorphic function. … ►A cut neighborhood is formed by deleting a ray emanating from the center. … ►Then the value of

F(z)

at any other point is obtained by analytic continuation. … ►It should be noted that different branches of

(w-w_{0})^{1/\mu}

used in forming

(w-w_{0})^{n/\mu}

in (1.10.16) give rise to different solutions of (1.10.12). … ►Let

F(x,z)

have a converging power series expansion of the form …

40: 3.7 Ordinary Differential Equations

… ►Now suppose the path

\mathscr{P}

is such that the rate of growth of

w(z)

along

\mathscr{P}

is intermediate to that of two other solutions. … ►The remaining two equations are supplied by boundary conditions of the form … ►If

q(x)

C^{\infty}

on the closure of

(a,b)

, then the discretized form (3.7.13) of the differential equation can be used. … ►For further information, including other methods and examples, see Pryce (1993, §2.5.1). … ►An extensive literature exists on the numerical solution of ordinary differential equations by Runge–Kutta, multistep, or other methods. …