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31: 8.11 Asymptotic Approximations and Expansions

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8.11.3 $$R_n(a,z)=O\left(z^{-n}\right),$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $a$: parameter, $n$: nonnegative integer, $\delta$: small positive constant and $R_n(a,z)$: remainder

Permalink:

http://dlmf.nist.gov/8.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(i), §8.11 and Ch.8

►where $\delta$ denotes an arbitrary small positive constant. … ►

8.11.10 $$P(a+1,x)=\frac{1}{2}\mathrm{erfc}\left(-y\right)-\frac{1}{3}\sqrt{\frac{2}{\pi a}}(1+y^2){\mathrm{e}}^{-y^2}+O\left(a^{-1}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $x$: real variable, $a$: parameter and $y$: change of variable

Permalink:

http://dlmf.nist.gov/8.11.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(iv), §8.11 and Ch.8

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8.11.11 $${\gamma }^{\ast }(1-a,-x)=x^{a-1}\left(-\cos \left(\pi a\right)+\frac{\sin \left(\pi a\right)}{\pi }\left(2\sqrt{\pi }F\left(y\right)+\frac{2}{3}\sqrt{\frac{2\pi }{a}}\left(1-y^2\right)\right){\mathrm{e}}^{y^2}+O\left(a^{-1}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $\sin z$: sine function, $x$: real variable, $a$: parameter and $y$: change of variable

Permalink:

http://dlmf.nist.gov/8.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(iv), §8.11 and Ch.8

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32: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

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13.21.1 $$M_{\kappa ,\mu }\left(x\right)=\sqrt{x}\mathrm{\Gamma }\left(2\mu +1\right){\kappa }^{-\mu }\left(J_{2\mu }\left(2\sqrt{x\kappa }\right)+\mathrm{env}{J}_{2\mu }\left(2\sqrt{x\kappa }\right)O\left({\kappa }^{-\frac{1}{2}}\right)\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{env}{J}_{\nu }\left(x\right)$: envelope of Bessel function $J_{\nu }\left(x\right)$ and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(i), §13.21 and Ch.13

… ►uniformly with respect to $x\in (0,A]$ in each case, where $A$ is an arbitrary positive constant. … ►

13.21.6 $$M_{-\kappa ,\mu }\left(4\kappa x\right)=\frac{2\mathrm{\Gamma }\left(2\mu +1\right)}{{\kappa }^{\mu -\frac{1}{2}}}{\left(\frac{x\zeta }{1+x}\right)}^{\frac{1}{4}}{I}_{2\mu }\left(4\kappa {\zeta }^{\frac{1}{2}}\right)\left(1+O\left({\kappa }^{-1}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind and $x$: real variable

Referenced by:

§13.21(i)

Permalink:

http://dlmf.nist.gov/13.21.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(i), §13.21 and Ch.13

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13.21.7 $$W_{-\kappa ,\mu }\left(4\kappa x\right)=\frac{\sqrt{8/\pi }{\mathrm{e}}^{\kappa }}{{\kappa }^{\kappa -\frac{1}{2}}}{\left(\frac{x\zeta }{1+x}\right)}^{\frac{1}{4}}{K}_{2\mu }\left(4\kappa {\zeta }^{\frac{1}{2}}\right)\left(1+O\left({\kappa }^{-1}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $x$: real variable

Referenced by:

§13.21(i)

Permalink:

http://dlmf.nist.gov/13.21.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(i), §13.21 and Ch.13

… ►uniformly with respect to $\mu \in [0,(1-\delta )\kappa ]$ and $x\in (0,(1-\delta )(2\kappa +2\sqrt{{\kappa }^2-{\mu }^2})]$, where $\delta$ again denotes an arbitrary small positive constant. …

33: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►Higher-order determinants are natural generalizations. The minor $M_{jk}$ of the entry $a_{jk}$ in the $n$th-order determinant $det[a_{jk}]$ is the ($n-1$)th-order determinant derived from $det[a_{jk}]$ by deleting the $j$th row and the $k$th column. …An $n$th-order determinant expanded by its $j$th row is given by …If all the elements of a row (column) of a determinant are multiplied by an arbitrary factor $\mu$, then the result is a determinant which is $\mu$ times the original. …

34: 13.7 Asymptotic Expansions for Large Argument

… ►Here $\delta$ denotes an arbitrary small positive constant. … ►where $m$ is an arbitrary nonnegative integer, and … ►

13.7.13

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $m$: integer, $n$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant and $R_n(a,b,z)$: remainder

Permalink:

http://dlmf.nist.gov/13.7.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §13.7(iii), §13.7 and Ch.13

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35: 32.2 Differential Equations

… ►with $\alpha$, $\beta$, $\gamma$, and $\delta$ arbitrary constants. … ►be a nonlinear second-order differential equation in which $F$ is a rational function of $w$ and $dw/dz$, and is locally analytic in $z$, that is, analytic except for isolated singularities in $ℂ$. … ►For arbitrary values of the parameters $\alpha$, $\beta$, $\gamma$, and $\delta$, the general solutions of ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ are transcendental, that is, they cannot be expressed in closed-form elementary functions. … ►

32.2.28 $$w(z;\alpha ,\beta ,\gamma ,\delta )=1+2ϵW(\zeta ;a),$$

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Symbols:

$z$: real, $\alpha$: arbitrary constant, $\beta$: arbitrary constant, $\gamma$: arbitrary constant, $\delta$: arbitrary constant and $W(\zeta )$: bilinear transformation

Permalink:

http://dlmf.nist.gov/32.2.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §32.2(vi), §32.2 and Ch.32

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32.2.32 $$w(z;\alpha ,\beta ,\gamma ,\delta )=1+ϵ\zeta W(\zeta ;a,b,c,d),$$

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Symbols:

$z$: real, $\alpha$: arbitrary constant, $\beta$: arbitrary constant, $\gamma$: arbitrary constant, $\delta$: arbitrary constant and $W(\zeta )$: bilinear transformation

Permalink:

http://dlmf.nist.gov/32.2.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §32.2(vi), §32.2 and Ch.32

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36: 13.20 Uniform Asymptotic Approximations for Large $\mu$

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13.20.1 $$M_{\kappa ,\mu }\left(z\right)=z^{\mu +\frac{1}{2}}\left(1+O\left({\mu }^{-1}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.20(i)

Permalink:

http://dlmf.nist.gov/13.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(i), §13.20 and Ch.13

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13.20.2 $$W_{\kappa ,\mu }\left(x\right)={\pi }^{-\frac{1}{2}}\mathrm{\Gamma }\left(\kappa +\mu \right){\left(\frac{1}{4}x\right)}^{\frac{1}{2}-\mu }\left(1+O\left({\mu }^{-1}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter and $x$: real variable

Referenced by:

§13.20(i)

Permalink:

http://dlmf.nist.gov/13.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(i), §13.20 and Ch.13

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13.20.4 $$M_{\kappa ,\mu }\left(x\right)=\sqrt{\frac{2\mu x}{X}}{\left(\frac{4{\mu }^{2}x}{2{\mu }^2-\kappa x+\mu X}\right)}^{\mu }{\left(\frac{2(\mu -\kappa )}{X+x-2\kappa }\right)}^{\kappa }{\mathrm{e}}^{\frac{1}{2}X-\mu }\left(1+O\left(\frac{1}{\mu }\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $x$: real variable and $X$

Permalink:

http://dlmf.nist.gov/13.20.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(ii), §13.20 and Ch.13

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13.20.5 $$W_{\kappa ,\mu }\left(x\right)=\sqrt{\frac{x}{X}}{\left(\frac{2{\mu }^2-\kappa x+\mu X}{(\mu -\kappa )x}\right)}^{\mu }{\left(\frac{X+x-2\kappa }{2}\right)}^{\kappa }{\mathrm{e}}^{-\frac{1}{2}X-\kappa }\left(1+O\left(\frac{1}{\mu }\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $x$: real variable and $X$

Permalink:

http://dlmf.nist.gov/13.20.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(ii), §13.20 and Ch.13

►uniformly with respect to $x\in (0,\mathrm{\infty })$ and $\kappa \in [0,(1-\delta )\mu ]$, where $\delta$ again denotes an arbitrary small positive constant. …

37: 32.11 Asymptotic Approximations for Real Variables

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32.11.6 $$w_k(x)=d{|x|}^{-1/4}\sin \left(\varphi (x)-{\theta }_0\right)+o\left({|x|}^{-1/4}\right),$$

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Symbols:

$o\left(x\right)$: order less than, $\sin z$: sine function, $x$: real, $k$: real, $\varphi (z)$: function, $d$: constant and ${\theta }_0$: constant

Permalink:

http://dlmf.nist.gov/32.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §32.11(ii), §32.11 and Ch.32

… ►with $d$ $(\ne 0)$ and $\chi$ arbitrary real constants. … ►

32.11.19 $$w(x)=\sigma \sqrt{\frac{1}{2}x}+\sigma \rho {(2x)}^{-1/4}\cos \left(\psi (x)+\theta \right)+O\left(x^{-1}\right),$$ $x\to +\mathrm{\infty }$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\cos z$: cosine function, $x$: real, $\sigma$: real constant, $\rho$: real constant, $\theta$: real constant and $\psi (z)$: function

Permalink:

http://dlmf.nist.gov/32.11.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §32.11(iii), §32.11 and Ch.32

… ►where $\lambda$ is an arbitrary constant such that , and …where $B$ and $\sigma$ are arbitrary constants such that $B\ne 0$ and . …

38: 18.16 Zeros

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18.16.8 $${\theta }_{n,m}={\varphi }_m+\left(\left({\alpha }^2-\frac{1}{4}\right)\frac{1-{\varphi }_m\cot {\varphi }_m}{2{\varphi }_m}-\frac{1}{4}({\alpha }^2-{\beta }^2)\tan \left(\frac{1}{2}{\varphi }_m\right)\right)\frac{1}{{\rho }^2}+{\varphi }_m^{2}O\left(\frac{1}{{\rho }^3}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\cot z$: cotangent function, $\tan z$: tangent function, $m$: nonnegative integer, $n$: nonnegative integer, $\rho$ , ${\theta }_{n,m}$: zeros and ${\varphi }_m$

Referenced by:

§18.16(ii)

Permalink:

http://dlmf.nist.gov/18.16.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.16(ii), §18.16(ii), §18.16 and Ch.18

►uniformly for $m=1,2,\mathrm{\dots },\lfloor cn\rfloor$, where $c$ is an arbitrary constant such that . … ►

18.16.14 $$x_{n,n-m+1}=\nu +2^{\frac{2}{3}}{a}_m{\nu }^{\frac{1}{3}}+\frac{1}{5}{2}^{\frac{4}{3}}{a}_m^2{\nu }^{-\frac{1}{3}}+O\left(n^{-1}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $a_k$: $k$th zero of Airy $\mathrm{Ai}$ , $m$: nonnegative integer, $n$: nonnegative integer, $\nu$ and $x$: real variable

Referenced by:

§18.16(iv)

Permalink:

http://dlmf.nist.gov/18.16.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §18.16(iv), §18.16(iv), §18.16 and Ch.18

… ►Arrange them in decreasing order: … ►

18.16.18 $${ϵ}_{n,m}=O\left(n^{-\frac{5}{6}}\right).$$

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Defines:

${ϵ}_{n,m}$ (locally)

Symbols:

$O\left(x\right)$: order not exceeding, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.16.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §18.16(v), §18.16 and Ch.18

…

39: 32.10 Special Function Solutions

… ►with $C_1$, $C_2$ arbitrary constants. … ►with $a=\alpha +\frac{1}{2}\epsilon$, and $C_1$, $C_2$ arbitrary constants. … ►where $C$ is an arbitrary constant and $\mathrm{erfc}$ is the complementary error function (§7.2(i)). … ►with $C_1$, $C_2$ arbitrary constants. … ►with $C_1$, $C_2$ arbitrary constants. …

40: Bibliography F

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Zeros of the Macdonald function of complex order. J. Comput. Appl. Math. 211 (2), pp. 223–231.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview (Eugenia N. Petropoulou), ZentralBlatt

Referenced by:

§10.42, §10.42.

Encodings:

BibTeX

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Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. II. J. Math. Anal. Appl. 7 (3), pp. 440–451.

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External Links:

Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.11(iii).

Encodings:

BibTeX

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Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. J. Math. Anal. Appl. 6 (3), pp. 394–403.

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External Links:

Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.11(iii).

Encodings:

BibTeX

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Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. III. J. Math. Anal. Appl. 12 (3), pp. 593–601.

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External Links:

Document, MathReview (M. J. O. Strutt), ZentralBlatt

Referenced by:

§16.11(iii).

Encodings:

BibTeX

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Algorithm 309. Gamma function with arbitrary precision. Comm. ACM 10 (8), pp. 511–512.

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External Links:

ISSN 0001-0782, Document

Referenced by:

§5.24(ii).

Encodings:

BibTeX

…