asymptotic forms

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11: Bibliography S

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

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12: Bibliography T

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Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. Quart. Appl. Math. 43 (1), pp. 103–123.

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External Links:

ISSN 0033-569X, FullText, MathReview (I. Fenyő), ZentralBlatt

Referenced by:

§2.4(i).

Encodings:

BibTeX

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13: 36.15 Methods of Computation

… ►Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7. …

14: Bibliography C

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Asymptotics and closed form of a generalized incomplete gamma function. J. Comput. Appl. Math. 67 (2), pp. 371–379.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§8.16.

Encodings:

BibTeX

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15: 2.6 Distributional Methods

… ►To derive an asymptotic expansion of $𝒮f\left(z\right)$ for large values of $|z|$, with , we assume that $f(t)$ possesses an asymptotic expansion of the form … ►The distribution method outlined here can be extended readily to functions $f(t)$ having an asymptotic expansion of the form … ►The replacement of $f(t)$ by its asymptotic expansion (2.6.9), followed by term-by-term integration leads to convolution integrals of the form … ►If both $f$ and $g$ in (2.6.34) have asymptotic expansions of the form (2.6.9), then the distribution method can also be used to derive an asymptotic expansion of the convolution $f\ast g$; see Li and Wong (1994). …

16: 26.10 Integer Partitions: Other Restrictions

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§26.10(v) Limiting Form

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26.10.16 $$p(𝒟,n)\sim \frac{{\mathrm{e}}^{\pi \sqrt{n/3}}}{{(768n^3)}^{1/4}},$$ $n\to \mathrm{\infty }$.

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $p(\mathrm{condition},n)$: restricted number of partitions of $n$ and $n$: nonnegative integer

A&S Ref:

24.2.2

Permalink:

http://dlmf.nist.gov/26.10.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(v), §26.10 and Ch.26

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17: 27.11 Asymptotic Formulas: Partial Sums

… ►It is more fruitful to study partial sums and seek asymptotic formulas of the form …

18: 2.1 Definitions and Elementary Properties

… ►Asymptotic expansions of the forms (2.1.14), (2.1.16) are unique. …

19: 13.2 Definitions and Basic Properties

… ►It can be regarded as the limiting form of the hypergeometric differential equation (§15.10(i)) that is obtained on replacing $z$ by $z/b$, letting $b\to \mathrm{\infty }$, and subsequently replacing the symbol $c$ by $b$. … ►Although $M(a,b,z)$ does not exist when $b=-n$, $n=0,1,2,\mathrm{\dots }$, many formulas containing $M(a,b,z)$ continue to apply in their limiting form. … ►

13.2.6 $$U(a,b,z)\sim z^{-a},$$ $z\to \mathrm{\infty }$, $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

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Defines:

$U(a,b,z)$: Kummer confluent hypergeometric function

Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable and $\delta$: small positive constant

Referenced by:

§13.2(i), §13.2(iv)

Permalink:

http://dlmf.nist.gov/13.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

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§13.2(iii) Limiting Forms as $z\to 0$

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§13.2(iv) Limiting Forms as $z\to \mathrm{\infty }$

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20: 26.3 Lattice Paths: Binomial Coefficients

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26.3.4 $$\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)x^m=\frac{1}{{(1-x)}^{n+1}},$$ .

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $x$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.1 (in slightly different form)

Permalink:

http://dlmf.nist.gov/26.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(ii), §26.3 and Ch.26

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§26.3(v) Limiting Form

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26.3.12 $$\left(\genfrac{}{}{0.0pt}{}{2n}{n}\right)\sim \frac{4^n}{\sqrt{\pi n}},$$ $n\to \mathrm{\infty }$.

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Symbols:

$\sim$: asymptotic equality, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter and $n$: nonnegative integer

Referenced by:

§26.5(iv)

Permalink:

http://dlmf.nist.gov/26.3.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(v), §26.3 and Ch.26