approximants
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1: 18.13 Continued Fractions
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is the denominator of the th approximant to:
…and is the denominator of the th approximant to:
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is the denominator of the th approximant to:
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is the denominator of the th approximant to:
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is the denominator of the th approximant to:
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2: 3.11 Approximation Techniques
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►is called a Padé approximant at zero of if
…It is denoted by .
Thus if , then the Maclaurin expansion of (3.11.21) agrees with (3.11.20) up to, and including, the term in .
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►The array of Padé approximants
…Approximants with the same denominator degree are located in the same column of the table.
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3: 36.13 Kelvin’s Ship-Wave Pattern
4: 3.10 Continued Fractions
5: 8.10 Inequalities
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Padé Approximants
…6: 1.12 Continued Fractions
7: 10.72 Mathematical Applications
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►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter.
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8: 13.31 Approximations
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►For a discussion of the convergence of the Padé approximants that are related to the continued fraction (13.5.1) see Wimp (1985).
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9: Bibliography W
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A rational approximant for the digamma function.
Numer. Algorithms 33 (1-4), pp. 499–507.
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Some explicit Padé approximants for the function and a related quadrature formula involving Bessel functions.
SIAM J. Math. Anal. 16 (4), pp. 887–895.
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10: 2.4 Contour Integrals
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►For a coalescing saddle point and a pole see Wong (1989, Chapter 7) and van der Waerden (1951); in this case the uniform approximants are complementary error functions.
For a coalescing saddle point and endpoint see Olver (1997b, Chapter 9) and Wong (1989, Chapter 7); if the endpoint is an algebraic singularity then the uniform approximants are parabolic cylinder functions with fixed parameter, and if the endpoint is not a singularity then the uniform approximants are complementary error functions.
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