asymptotic approximation

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31: 23.12 Asymptotic Approximations

§23.12 Asymptotic Approximations

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32: 19.38 Approximations

§19.38 Approximations

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33: 26.7 Set Partitions: Bell Numbers

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§26.7(iv) Asymptotic Approximation

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34: 28.8 Asymptotic Expansions for Large $q$

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Barrett’s Expansions

►Barrett (1981) supplies asymptotic approximations for numerically satisfactory pairs of solutions of both Mathieu’s equation (28.2.1) and the modified Mathieu equation (28.20.1). … ►

Dunster’s Approximations

►Dunster (1994a) supplies uniform asymptotic approximations for numerically satisfactory pairs of solutions of Mathieu’s equation (28.2.1). … ►

35: 8.22 Mathematical Applications

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8.22.3 $${\zeta }_x(s)=\sum _{k=1}^{\mathrm{\infty }}{k}^{-s}P(s,kx),$$ $\mathrm{\Re }s>1$.

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Symbols:

$P(a,z)$: normalized incomplete gamma function, $\mathrm{\Re }$: real part, $x$: real variable, $k$: nonnegative integer and ${\zeta }_x(s)$: incomplete Riemann zeta function

Permalink:

http://dlmf.nist.gov/8.22.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.22(ii), §8.22 and Ch.8

►For further information on ${\zeta }_x(s)$, including zeros and uniform asymptotic approximations, see Kölbig (1970, 1972a) and Dunster (2006). …

36: 10.72 Mathematical Applications

… ►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. … ►If $f(z)$ has a double zero $z_0$, or more generally $z_0$ is a zero of order $m$, $m=2,3,4,\mathrm{\dots }$, then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order $1/(m+2)$. … ►Then for large $u$ asymptotic approximations of the solutions $w$ can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on $u$ and $\alpha$). …

37: 36.15 Methods of Computation

… ►Close to the bifurcation set but far from $𝐱=\mathrm{𝟎}$, the uniform asymptotic approximations of §36.12 can be used. … ►This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of $\mathrm{\Phi }$, with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …

38: 8.12 Uniform Asymptotic Expansions for Large Parameter

§8.12 Uniform Asymptotic Expansions for Large Parameter

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$c_5(0)=-\frac{\mathrm{27\hspace{0.33em}45493}}{\mathrm{81517\hspace{0.33em}36320}}.$

… ►For other uniform asymptotic approximations of the incomplete gamma functions in terms of the function $\mathrm{erfc}$ see Paris (2002b) and Dunster (1996a). ►

Inverse Function

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39: 18.15 Asymptotic Approximations

§18.15 Asymptotic Approximations

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§18.15(i) Jacobi

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§18.15(ii) Ultraspherical

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§18.15(iii) Legendre

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§18.15(iv) Laguerre

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40: 10.57 Uniform Asymptotic Expansions for Large Order

§10.57 Uniform Asymptotic Expansions for Large Order

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