numerically satisfactory solutions

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11: 12.2 Differential Equations

… ►All solutions are entire functions of $z$ and entire functions of $a$ or $\nu$. ►For real values of $z$ $(=x)$, numerically satisfactory pairs of solutions (§2.7(iv)) of (12.2.2) are $U(a,x)$ and $V(a,x)$ when $x$ is positive, or $U(a,-x)$ and $V(a,-x)$ when $x$ is negative. … ►In $ℂ$, for $j=0,1,2,3$, $U({(-1)}^{j-1}a,{(-\mathrm{i})}^{j-1}z)$ and $U({(-1)}^{j}a,{(-\mathrm{i})}^{j}z)$ comprise a numerically satisfactory pair of solutions in the half-plane $\frac{1}{4}(2j-3)\pi \le \mathrm{ph}z\le \frac{1}{4}(2j+1)\pi$. …

12: 9.12 Scorer Functions

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§9.12(iv) Numerically Satisfactory Solutions

… ►In $ℂ$, numerically satisfactory sets of solutions are given by …

13: Mathematical Introduction

… ►These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). …

14: 12.14 The Function $W(a,x)$

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§12.14(i) Introduction

… ► $W(a,x)$ and $W(a,-x)$ form a numerically satisfactory pair of solutions when . …

15: Bibliography G

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Numerically satisfactory solutions of hypergeometric recursions. Math. Comp. 76 (259), pp. 1449–1468.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview Entry, ZentralBlatt

Referenced by:

§15.19(iv).

Encodings:

BibTeX

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16: 28.8 Asymptotic Expansions for Large $q$

… ►Barrett (1981) supplies asymptotic approximations for numerically satisfactory pairs of solutions of both Mathieu’s equation (28.2.1) and the modified Mathieu equation (28.20.1). … ►Dunster (1994a) supplies uniform asymptotic approximations for numerically satisfactory pairs of solutions of Mathieu’s equation (28.2.1). …

17: 9.11 Products

… ►Numerically satisfactory triads of solutions can be constructed where needed on $ℝ$ or $ℂ$ by inspection of the asymptotic expansions supplied in §9.7. …

18: 10.45 Functions of Imaginary Order

… ►In consequence of (10.45.5)–(10.45.7), ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.45.1) when $x$ is large, and either ${\tilde{I}}_{\nu }\left(x\right)$ and $(1/\pi )\sinh \left(\pi \nu \right){\tilde{K}}_{\nu }\left(x\right)$, or ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$, comprise a numerically satisfactory pair when $x$ is small, depending whether $\nu \ne 0$ or $\nu =0$. …

19: 10.24 Functions of Imaginary Order

… ►In consequence of (10.24.6), when $x$ is large ${\tilde{J}}_{\nu }\left(x\right)$ and ${\tilde{Y}}_{\nu }\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.24.1); compare §2.7(iv). …

20: 9.13 Generalized Airy Functions

… ►The function on the right-hand side is recessive in the sector $-(2j-1)\pi /m\le \mathrm{ph}z\le (2j+1)\pi /m$, and is therefore an essential member of any numerically satisfactory pair of solutions in this region. …