Gauss 2F1(-1) sum
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11: 3.5 Quadrature
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§3.5(v) Gauss Quadrature
… ►Gauss–Legendre Formula
… ►Gauss–Chebyshev Formula
… ►Gauss–Jacobi Formula
… ►Gauss–Laguerre Formula
…12: 4.38 Inverse Hyperbolic Functions: Further Properties
13: 4.13 Lambert -Function
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►The decreasing solution can be identified as .
…where .
is a single-valued analytic function on , real-valued when , and has a square root branch point at .
…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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►where for , for on the relevant branch cuts,
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14: 16.12 Products
15: 33.6 Power-Series Expansions in
16: 15.3 Graphics
17: 13.14 Definitions and Basic Properties
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►It has a regular singularity at the origin with indices , and an irregular singularity at infinity of rank one.
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►For example, if , then
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►If , where , then
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►In cases when , where is a nonnegative integer,
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►When is not an integer
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18: 15.8 Transformations of Variable
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Group 2 Group 3
… ►Group 2 Group 1
… ►Group 2 Group 4
… ►Group 4 Group 2
… ►provided that lies in the intersection of the open disks , or equivalently, . …19: 32.8 Rational Solutions
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►Then for
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►Then has a rational solution iff one of the following holds with and :
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(c)
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(d)
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►where , , , , and , with , , independently, and at least one of , , or is an integer.
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, , and , with even.
, , and , with even.
20: 19.36 Methods of Computation
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►where , and
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►The step from to is an ascending Landen transformation if (leading ultimately to a hyperbolic case of ) or a descending Gauss transformation if (leading to a circular case of ).
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►Descending Gauss transformations of (see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)).
This method loses significant figures in if and are nearly equal unless they are given exact values—as they can be for tables.
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►The function is computed by descending Landen transformations if is real, or by descending Gauss transformations if is complex (Bulirsch (1965b)).
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