values at z=0
(0.014 seconds)
11—20 of 144 matching pages
11: 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.
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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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►Except when each branch of is entire in and .
Unless specified otherwise, however, is assumed to have its principal value.
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12: 16.8 Differential Equations
13: 16.2 Definition and Analytic Properties
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►When the series (16.2.1) converges for all finite values of and defines an entire function.
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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 .
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►On the circle the series (16.2.1) is absolutely convergent if , convergent except at
if , and divergent if , where
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►In general the series (16.2.1) diverges for all nonzero values of .
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14: 10.72 Mathematical Applications
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►In regions in which the function has a simple pole at
and is analytic at
(the case in §10.72(i)), asymptotic expansions of the solutions of (10.72.1) for large can be constructed in terms of Bessel functions and modified Bessel functions of order , where is the limiting value of as .
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15: 28.5 Second Solutions ,
16: 28.2 Definitions and Basic Properties
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►
,
.
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17: 4.13 Lambert -Function
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►The other branches are single-valued analytic functions on , have a logarithmic branch point at
, and, in the case , have a square root branch point at
respectively.
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18: 1.10 Functions of a Complex Variable
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►If the path circles a branch point at
, times in the positive sense, and returns to without encircling any other branch point, then its value is denoted conventionally as .
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19: 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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20: 31.9 Orthogonality
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►Here is an arbitrary point in the interval .
The integration path begins at
, encircles once in the positive sense, followed by once in the positive sense, and so on, returning finally to .
…The branches of the many-valued functions are continuous on the path, and assume their principal values at the beginning.
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►The right-hand side may be evaluated at any convenient value, or limiting value, of in since it is independent of .
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►and the integration paths , are Pochhammer double-loop contours encircling distinct pairs of singularities , , .
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