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1: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

§33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

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§33.5(i) Small $\rho$

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§33.5(iii) Small $|\eta |$

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2: 18.15 Asymptotic Approximations

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18.15.7

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Defines:

${\epsilon }_M(\rho ,\theta )$ (locally)

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\delta$: arbitrary small positive constant and $\rho$

Permalink:

http://dlmf.nist.gov/18.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(i), §18.15 and Ch.18

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3: 2.11 Remainder Terms; Stokes Phenomenon

… ►Owing to the factor ${\mathrm{e}}^{-\rho }$, that is, ${\mathrm{e}}^{-|z|}$ in (2.11.13), $F_{n+p}\left(z\right)$ is uniformly exponentially small compared with $E_p\left(z\right)$. … ►Hence from §7.12(i) $\mathrm{erfc}\left(\sqrt{\frac{1}{2}\rho }c(\theta )\right)$ is of the same exponentially-small order of magnitude as the contribution from the other terms in (2.11.15) when $\rho$ is large. On the other hand, when $\pi +\delta \le \theta \le 3\pi -\delta$, $c(\theta )$ is in the left half-plane and $\mathrm{erfc}\left(\sqrt{\frac{1}{2}\rho }c(\theta )\right)$ differs from 2 by an exponentially-small quantity. …

4: 33.12 Asymptotic Expansions for Large $\eta$

§33.12 Asymptotic Expansions for Large $\eta$

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§33.12(i) Transition Region

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§33.12(ii) Uniform Expansions

►With the substitution $\rho =2\eta z$, Equation (33.2.1) becomes … ►

5: 3.2 Linear Algebra

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$${\Vert 𝐀\Vert }_2=\sqrt{\rho (𝐀{𝐀}^{\mathrm{T}})},$$

►where $\rho (𝐀{𝐀}^{\mathrm{T}})$ is the largest of the absolute values of the eigenvalues of the matrix $𝐀{𝐀}^{\mathrm{T}}$; see §3.2(iv). … ►The sensitivity of the solution vector $𝐱$ in (3.2.1) to small perturbations in the matrix $𝐀$ and the vector $𝐛$ is measured by the condition number … ►If $𝐀$ is nondefective and $\lambda$ is a simple zero of $p_n(\lambda )$, then the sensitivity of $\lambda$ to small perturbations in the matrix $𝐀$ is measured by the condition number …

6: 33.23 Methods of Computation

§33.23 Methods of Computation

… ►Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. … ►

§33.23(vii) WKBJ Approximations

►WKBJ approximations (§2.7(iii)) for $\rho >{\rho }_{\mathrm{tp}}(\eta ,\mathrm{\ell })$ are presented in Hull and Breit (1959) and Seaton and Peach (1962: in Eq.  (12) $(\rho -c)/c$ should be $(\rho -c)/\rho$). …

7: 27.19 Methods of Computation: Factorization

… ►Type I probabilistic algorithms include the Brent–Pollard rho algorithm (also called Monte Carlo method), the Pollard $p-1$ algorithm, and the Elliptic Curve Method (ecm). …As of January 2009 the largest prime factors found by these methods are a 19-digit prime for Brent–Pollard rho, a 58-digit prime for Pollard $p-1$, and a 67-digit prime for ecm. … ►The snfs can be applied only to numbers that are very close to a power of a very small base. …

8: 2.5 Mellin Transform Methods

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§2.5(iii) Laplace Transforms with Small Parameters

… ►for any $\rho$ satisfying . … ►where $l$ ($\ge 2$) is an arbitrary integer and $\delta$ is an arbitrary small positive constant. … … ►To verify (2.5.48) we may use …

9: 28.35 Tables

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Blanch and Clemm (1969) includes eigenvalues $a_n\left(q\right)$, $b_n\left(q\right)$ for $q=\rho {\mathrm{e}}^{\mathrm{i}\varphi }$, $\rho =0(.5)25$, $\varphi =5^{\circ }(5^{\circ }){90}^{\circ }$, $n=0(1)15$; 4D. Also $a_n\left(q\right)$ and $b_n\left(q\right)$ for $q=\mathrm{i}\rho$, $\rho =0(.5)100$, $n=0(2)14$ and $n=2(2)16$, respectively; 8D. Double points for $n=0(1)15$; 8D. Graphs are included.

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Ince (1932) includes the first zero for ${\mathrm{ce}}_n$, ${\mathrm{se}}_n$ for $n=2(1)5$ or $6$, $q=0(1)10(2)40$; 4D. This reference also gives zeros of the first derivatives, together with expansions for small $q$.

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10: 13.7 Asymptotic Expansions for Large Argument

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

13.7.3 $$U(a,b,z)\sim z^{-a}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s{\left(a-b+1\right)}_s}{s!}{(-z)}^{-s},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$.

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $s$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

A&S Ref:

13.5.2

Referenced by:

§12.9(i), §13.12, §13.2(i)

Permalink:

http://dlmf.nist.gov/13.7.E3

Encodings:

pMML, png, TeX

See also:

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

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13.7.5 $$\left|{\epsilon }_n(z)\right|,{\beta }^{-1}\left|{\epsilon }_n^{\prime }(z)\right|\le 2\alpha C_n\left|\frac{{\left(a\right)}_n{\left(a-b+1\right)}_n}{n!z^{a+n}}\right|\exp \left(\frac{2\alpha \rho C_1}{|z|}\right),$$

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Defines:

${\epsilon }_n(z)$: function (locally)

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\exp z$: exponential function, $!$: factorial (as in $n!$), $n$: nonnegative integer, $z$: complex variable, $C_n$: coefficient, $\alpha$ , $\beta$ and $\rho$

Referenced by:

§13.2(i)

Permalink:

http://dlmf.nist.gov/13.7.E5

Encodings:

pMML, png, TeX

See also:

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

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$\rho =\frac{1}{2}\left|2a^2-2ab+b\right|+{\displaystyle \frac{\sigma (1+\frac{1}{4}\sigma )}{{(1-\sigma )}^2}}$ ,

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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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