value at infinity
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11: 4.31 Special Values and Limits
§4.31 Special Values and Limits
► …12: 3.6 Linear Difference Equations
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►If, as , the wanted solution grows (decays) in magnitude at least as fast as any solution of the corresponding homogeneous equation, then forward (backward) recursion is stable.
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►The least value of that satisfies (3.6.9) is found to be 16.
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►For a difference equation of order (),
…Typically conditions are prescribed at the beginning of the range, and conditions at the end.
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13: 16.2 Definition and Analytic Properties
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►The branch obtained by introducing a cut from to on the real axis, that is, the branch in the sector , is the principal branch (or principal
value) of ; compare §4.2(i).
Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at
, and .
Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values.
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►In general the series (16.2.1) diverges for all nonzero values of .
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►(However, except where indicated otherwise in the DLMF we assume that when
at least one of the is a nonpositive integer.)
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14: 15.6 Integral Representations
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►In (15.6.2) the point lies outside the integration contour, and assume their principal values where the contour cuts the interval , and
at
.
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15: 10.25 Definitions
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§10.25(ii) Standard Solutions
… ►In particular, the principal branch of is defined in a similar way: it corresponds to the principal value of , is analytic in , and two-valued and discontinuous on the cut . … ►It has a branch point at for all . The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in , and two-valued and discontinuous on the cut . … ►Except where indicated otherwise it is assumed throughout the DLMF that the symbols and denote the principal values of these functions. …16: 3.11 Approximation Techniques
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►A sufficient condition for to be the minimax polynomial is that attains its maximum at
distinct points in and changes sign at these consecutive maxima.
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►There exists a unique solution of this minimax problem and there are at least
values
, , such that , where
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►is called a Padé approximant at zero of if
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►A general procedure is to approximate by a rational function (vanishing at infinity) and then approximate by .
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►Given distinct points in the real interval , with ()(), on each subinterval , , a low-degree polynomial is defined with coefficients determined by, for example, values
and of a function and its derivative at the nodes and .
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17: 5.9 Integral Representations
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►(The fractional powers have their principal values.)
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►where the contour begins at
, circles the origin once in the positive direction, and returns to .
has its principal value where crosses the positive real axis, and is continuous.
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►where and the inverse tangent has its principal value.
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5.9.18
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18: 10.2 Definitions
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►This differential equation has a regular singularity at
with indices , and an irregular singularity at
of rank ; compare §§2.7(i) and 2.7(ii).
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►The principal branch of corresponds to the principal value of (§4.2(iv)) and is analytic in the -plane cut along the interval .
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►Each solution has a branch point at
for all .
The principal branches correspond to principal values of the square roots in (10.2.5) and (10.2.6), again with a cut in the -plane along the interval .
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19: 6.16 Mathematical Applications
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►Hence, if is fixed and , then , , or according as , , or ; compare (6.2.14).
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►The first maximum of for positive occurs at
and equals ; compare Figure 6.3.2.
Hence if and , then the limiting value of overshoots by approximately 18%.
Similarly if , then the limiting value of undershoots by approximately 10%, and so on.
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20: 13.2 Definitions and Basic Properties
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►This equation has a regular singularity at the origin with indices and , and an irregular singularity at infinity of rank one.
…In effect, the regular singularities of the hypergeometric differential equation at
and coalesce into an irregular singularity at
.
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►In general, has a branch point at
.
The principal branch corresponds to the principal value of in (13.2.6), and has a cut in the -plane along the interval ; compare §4.2(i).
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