real case
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1: 28.29 Definitions and Basic Properties
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§28.29(iii) Discriminant and Eigenvalues in the Real Case
…2: 14.32 Methods of Computation
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►In particular, for small or moderate values of the parameters and the power-series expansions of the various hypergeometric function representations given in §§14.3(i)–14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real.
In other cases recurrence relations (§14.10) provide a powerful method when applied in a stable direction (§3.6); see Olver and Smith (1983) and Gautschi (1967).
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3: 20.1 Special Notation
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, | integers. |
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the nome, , . Since is not a single-valued function of , it is assumed that is known, even when is specified. Most applications concern the rectangular case , , so that and and are uniquely related. | |
for (resolving issues of choice of branch). | |
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4: 10.2 Definitions
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►Table 10.2.1 lists numerically satisfactory pairs of solutions (§2.7(iv)) of (10.2.1) for the stated intervals or regions in the case
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5: 4.43 Cubic Equations
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4.43.2
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►Note that in Case (a) all the roots are real, whereas in Cases (b) and (c) there is one real root and a conjugate pair of complex roots.
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6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►Let be a self-adjoint extension of differential operator of the form (1.18.28) and assume has a complete set of eigenfunctions, , this latter being an appropriate sub-set of , or, in some cases
itself, with real eigenvalues .
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►See Titchmarsh (1962a, pp. 87–90) for a first principles derivation for the case
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7: 9.17 Methods of Computation
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►However, in the case of and this accuracy can be increased considerably by use of the exponentially-improved forms of expansion supplied in §9.7(v).
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►In the case of , for example, this means that in the sectors we may integrate along outward rays from the origin with initial values obtained from §9.2(ii).
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►In the case of the Scorer functions, integration of the differential equation (9.12.1) is more difficult than (9.2.1), because in some regions stable directions of integration do not exist.
An example is provided by on the positive real axis.
In these cases boundary-value methods need to be used instead; see §3.7(iii).
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