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1: 31.18 Methods of Computation

… ►Care needs to be taken to choose integration paths in such a way that the wanted solution is growing in magnitude along the path at least as rapidly as all other solutions (§3.7(ii)). …

2: About Color Map

… ►Surface visualizations in the DLMF represent functions of the form $z=f(x,y)$ by the height $z$ or the magnitude, $|z|$, for complex functions, over the $x\times y$ plane. … ►By painting the surfaces with a color that encodes the phase, $\mathrm{ph}f$, both the magnitude and phase of complex valued functions can be displayed. …

3: 6.12 Asymptotic Expansions

… ►When $|\mathrm{ph}z|\le \frac{1}{2}\pi$ the remainder is bounded in magnitude by the first neglected term, and has the same sign when $\mathrm{ph}z=0$. When the remainder term is bounded in magnitude by $\csc \left(|\mathrm{ph}z|\right)$ times the first neglected term. … ►When $|\mathrm{ph}z|\le \frac{1}{4}\pi$, these remainders are bounded in magnitude by the first neglected terms in (6.12.3) and (6.12.4), respectively, and have the same signs as these terms when $\mathrm{ph}z=0$. When the remainders are bounded in magnitude by $\csc \left(2|\mathrm{ph}z|\right)$ times the first neglected terms. …

4: 9.17 Methods of Computation

… ►Since these expansions diverge, the accuracy they yield is limited by the magnitude of $|z|$. …

5: 11.13 Methods of Computation

… ►To insure stability the integration path must be chosen so that as we proceed along it the wanted solution grows in magnitude at least as rapidly as the complementary solutions. …

6: Bibliography N

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Algorithm 707: CONHYP: A numerical evaluator of the confluent hypergeometric function for complex arguments of large magnitudes. ACM Trans. Math. Software 18 (3), pp. 345–349.

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Notes:

Double-precision Fortran, minimum accuracy 9S, maximum accuracy 13S.

External Links:

ISSN 0098-3500, Document, GAMS (module 707; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Numerical evaluation of the confluent hypergeometric function for complex arguments of large magnitudes. J. Comput. Appl. Math. 39 (2), pp. 193–200.

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Notes:

Extended-precision Fortran evaluator, minimum accuracy 9S, maximum accuracy 13S.

External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§13.32(iii).

Encodings:

BibTeX

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7: 13.29 Methods of Computation

… ►Accuracy is limited by the magnitude of $|z|$. … ►As described in §3.7(ii), to insure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows in magnitude at least as fast as all other solutions of the differential equation. …

8: 15.19 Methods of Computation

… ►As noted in §3.7(ii), the integration path should be chosen so that the wanted solution grows in magnitude at least as fast as all other solutions. …

9: 36.6 Scaling Relations

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Indices for $k$-Scaling of Magnitude of ${\mathrm{\Psi }}_K$ or ${\mathrm{\Psi }}^{(\mathrm{U})}$ (Singularity Index)

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10: 7.12 Asymptotic Expansions

… ►When $|\mathrm{ph}z|\le \frac{1}{4}\pi$ the remainder terms are bounded in magnitude by the first neglected terms, and have the same sign as these terms when $\mathrm{ph}z=0$. When the remainder terms are bounded in magnitude by $\csc \left(2|\mathrm{ph}z|\right)$ times the first neglected terms. … ►When $|\mathrm{ph}z|\le \frac{1}{8}\pi$, $R_n^{(\mathrm{f})}(z)$ and $R_n^{(\mathrm{g})}(z)$ are bounded in magnitude by the first neglected terms in (7.12.2) and (7.12.3), respectively, and have the same signs as these terms when $\mathrm{ph}z=0$. …