as z→0
(0.031 seconds)
21—30 of 616 matching pages
21: 19.21 Connection Formulas
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►Upper signs apply if , and lower signs if :
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19.21.4
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►If and , then as (19.21.6) reduces to Legendre’s relation (19.21.1).
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►Because is completely symmetric, can be permuted on the right-hand side of (19.21.10) so that if the variables are real, thereby avoiding cancellations when is calculated from and (see §19.36(i)).
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►Let be real and nonnegative, with at most one of them 0.
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22: 1.9 Calculus of a Complex Variable
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►and when ,
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►A function is continuous at a point if .
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►( may or may not belong to .)
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►A function is said to be analytic (holomorphic) at if it is complex differentiable in a neighborhood of .
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►where is an integer called the winding number of
with respect to
.
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23: 10.2 Definitions
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►This solution of (10.2.1) is an analytic function of , except for a branch point at when is not an integer.
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►For fixed
each branch of is entire in .
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►Whether or not is an integer has a branch point at .
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►For fixed
each branch of is entire in .
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►Each solution has a branch point at for all .
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24: 14.28 Sums
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►When , , , and ,
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14.28.1
►where the branches of the square roots have their principal values when and are continuous when .
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►
14.28.2
, ,
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25: 10.30 Limiting Forms
26: 23.9 Laurent and Other Power Series
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►Let be the nearest lattice point to the origin, and define
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►
23.9.2
,
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23.9.3
.
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►Also, Abramowitz and Stegun (1964, (18.5.25)) supplies the first 22 terms in the reverted form of (23.9.2) as .
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►
23.9.7
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27: 10.12 Generating Function and Associated Series
28: 3.8 Nonlinear Equations
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►If and , then is a simple zero of .
If and , then is a zero of of multiplicity
; compare §1.10(i).
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►An iterative method converges locally to a solution if there exists a neighborhood of such that whenever the initial approximation lies within .
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►The results for are given in Table 3.8.2.
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►For an arbitrary starting point , convergence cannot be predicted, and the boundary of the set of points that generate a sequence converging to a particular zero has a very complicated structure.
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