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1—10 of 27 matching pages
1: 31.18 Methods of Computation
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►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)).
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2: About Color Map
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►Surface visualizations in the DLMF represent functions of the form by the height or the magnitude, , for complex functions, over the plane.
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►By painting the surfaces with a color that encodes the phase, , both the magnitude and phase of complex valued functions can be displayed.
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3: 6.12 Asymptotic Expansions
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►When the remainder is bounded in magnitude by the first neglected term, and has the same sign when .
When the remainder term is bounded in magnitude by times the first neglected term.
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►When , 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 .
When the remainders are bounded in magnitude by times the first neglected terms.
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4: 9.17 Methods of Computation
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►Since these expansions diverge, the accuracy they yield is limited by the magnitude of .
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5: 11.13 Methods of Computation
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►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.
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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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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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7: 13.29 Methods of Computation
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►Accuracy is limited by the magnitude of .
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►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.
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8: 15.19 Methods of Computation
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►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.
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9: 36.6 Scaling Relations
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Indices for -Scaling of Magnitude of or (Singularity Index)
…10: 7.12 Asymptotic Expansions
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►When the remainder terms are bounded in magnitude by the first neglected terms, and have the same sign as these terms when .
When the remainder terms are bounded in magnitude by times the first neglected terms.
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►When , and 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 .
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