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1: 12.18 Methods of Computation
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►Because PCFs are special cases of confluent hypergeometric functions, the methods of computation described in §13.29 are applicable to PCFs.
These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions.
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2: 6.6 Power Series
§6.6 Power Series
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6.6.1
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6.6.4
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6.6.5
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►The series in this section converge for all finite values of and .
3: 16.25 Methods of Computation
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►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations.
They are similar to those described for confluent hypergeometric functions, and hypergeometric functions in §§13.29 and 15.19.
There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations).
This occurs when the wanted solution is intermediate in asymptotic growth compared with other solutions.
In these cases integration, or recurrence, in either a forward or a backward direction is unstable.
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4: 10.8 Power Series
§10.8 Power Series
… ►When is not an integer the corresponding expansions for , , and are obtained by combining (10.2.2) with (10.2.3), (10.4.7), and (10.4.8). … ►
10.8.1
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►In particular,
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10.8.2
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5: 28.15 Expansions for Small
§28.15 Expansions for Small
►§28.15(i) Eigenvalues
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28.15.1
►Higher coefficients can be found by equating powers of
in the following continued-fraction equation, with :
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§28.15(ii) Solutions
…6: 12.4 Power-Series Expansions
7: 12.15 Generalized Parabolic Cylinder Functions
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►This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function.
See Faierman (1992) for power series and asymptotic expansions of a solution of (12.15.1).
8: 10.31 Power Series
§10.31 Power Series
… ►When is not an integer the corresponding expansion for is obtained from (10.25.2) and (10.27.4). … ►In particular, ►
10.31.2
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10.31.3