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11: 11.6 Asymptotic Expansions

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11.6.1 $${𝐊}_{\nu }\left(z\right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(k+\frac{1}{2}\right){(\frac{1}{2}z)}^{\nu -2k-1}}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}-k\right)},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐊}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.1.29

Referenced by:

§11.6(i), §11.6(i), §11.6(i), §11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(i), §11.6 and Ch.11

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

11.6.5 $${𝐇}_{\nu }\left(z\right),{𝐋}_{\nu }\left(z\right)\sim \frac{z}{\pi \nu \sqrt{2}}{\left(\frac{\mathrm{e}z}{2\nu }\right)}^{\nu },$$ $|\mathrm{ph}\nu |\le \pi -\delta$.

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Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order and $\delta$: arbitrary small positive constant

Referenced by:

(11.11.7)

Permalink:

http://dlmf.nist.gov/11.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(ii), §11.6 and Ch.11

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$c_3(\lambda )=20{\lambda }^{-6}-4{\lambda }^{-4},$

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11.6.9 $${𝐋}_{\nu }\left(\lambda \nu \right)\sim I_{\nu }\left(\lambda \nu \right),$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$,

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{ph}$: phase, $\nu$: real or complex order, $\delta$: arbitrary small positive constant and $\lambda$: parameter

Referenced by:

§11.6(iii)

Permalink:

http://dlmf.nist.gov/11.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(iii), §11.6 and Ch.11

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12: 20 Theta Functions

Chapter 20 Theta Functions

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13: 18.28 Askey–Wilson Class

… ►The $q$-Racah polynomials form a system of OP’s $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots },N$, that are orthogonal with respect to a weight function on a sequence $\{q^{-y}+c{q}^{y+1}\}$, $y=0,1,\mathrm{\dots },N$, with $c$ a constant. … ►With $x=q^{-y}+\gamma \delta q^{y+1}$, … ►

18.28.20 $$\sum _{y=0}^N{R}_n(q^{-y}+\gamma \delta q^{y+1})R_m(q^{-y}+\gamma \delta q^{y+1}){\omega }_y=h_n{\delta }_{n,m},$$ $n,m=0,1,\mathrm{\dots },N$,

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant, $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer, $h_n$ and $R_n(x)$

Permalink:

http://dlmf.nist.gov/18.28.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(viii), §18.28 and Ch.18

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18.28.21 $${\omega }_y=\frac{{(\alpha q,\beta \delta q,\gamma q,\gamma \delta q;q)}_y}{{(q,\frac{\gamma \delta }{\alpha }q,\frac{\gamma }{\beta }q,\delta q;q)}_y}\frac{1-\gamma \delta q^{2y+1}}{{\left(\alpha \beta q\right)}^y},$$

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Symbols:

${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $y$: real variable, $\delta$: arbitrary small positive constant and $q$: real variable

Referenced by:

§18.28(viii), Erratum (V1.2.0) Section 18.28

Permalink:

http://dlmf.nist.gov/18.28.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(viii), §18.28 and Ch.18

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18.28.34 $$\underset{q\to 1}{lim}{R}_n(q^{-y}+q^{y+\gamma +\delta +1};q^{\alpha },q^{\beta },q^{\gamma },q^{\delta }|q)=R_n(y(y+\gamma +\delta +1);\alpha ,\beta ,\gamma ,\delta ).$$

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Symbols:

$R_n(x;\alpha ,\beta ,\gamma ,\delta )$: Racah polynomial, $R_n(x;\alpha ,\beta ,\gamma ,\delta |q)$: $q$-Racah polynomial, $y$: real variable, $\delta$: arbitrary small positive constant, $q$: real variable and $n$: nonnegative integer

Source:

Koekoek et al. (2010, (14.2.24))

Referenced by:

§18.28(x), Erratum (V1.2.0) Section 18.28

Permalink:

http://dlmf.nist.gov/18.28.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

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14: 17.12 Bailey Pairs

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17.12.1 $$\sum _{n=0}^{\mathrm{\infty }}{\alpha }_n{\gamma }_n=\sum _{n=0}^{\mathrm{\infty }}{\beta }_n{\delta }_n,$$

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Symbols:

$n$: nonnegative integer, ${\alpha }_n$: part of Bailey pair and ${\beta }_n$: part of Bailey pair

Permalink:

http://dlmf.nist.gov/17.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.12, §17.12 and Ch.17

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${\gamma }_n={\displaystyle \sum _{j=n}^{\mathrm{\infty }}}{\delta }_j{u}_{j-n}{v}_{j+n}.$

… ►A sequence of pairs of rational functions of several variables $({\alpha }_n,{\beta }_n)$, $n=0,1,2,\mathrm{\dots }$, is called a Bailey pair provided that for each $n\geqq 0$ … ►When (17.12.5) is iterated the resulting infinite sequence of Bailey pairs is called a Bailey Chain. …

15: Bibliography L

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Algorithm 917: complex double-precision evaluation of the Wright $\omega$ function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§4.13, 2nd item, §4.48(iv).

Encodings:

BibTeX

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An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.11.

Encodings:

BibTeX

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Integral and series representations of the Dirac delta function. Commun. Pure Appl. Anal. 7 (2), pp. 229–247.

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External Links:

ISSN 1534-0392, MathReview Entry, ZentralBlatt

Referenced by:

§1.17(iv), (1.18.57).

Encodings:

BibTeX

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Monotonic sequences related to zeros of Bessel functions. Numer. Algorithms 49 (1-4), pp. 221–233.

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External Links:

ISSN 1017-1398, Document, FullText, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§10.21(iv), §10.21(iv).

Encodings:

BibTeX

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16: 1.9 Calculus of a Complex Variable

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§1.9(v) Infinite Sequences and Series

… ►This sequence converges pointwise to a function $f(z)$ if … ►

§1.9(vii) Inversion of Limits

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Double Sequences and Series

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17: 2.11 Remainder Terms; Stokes Phenomenon

… ►uniformly when $\theta \in [-\pi +\delta ,\pi -\delta ]$ ($\delta >0$) and $|\alpha |$ is bounded. … ►For the sector $-3\pi +\delta \le \mathrm{ph}z\le \pi -\delta$ the conjugate result applies. … ►We now compute the forward differences ${\mathrm{\Delta }}^j$, $j=0,1,2,\mathrm{\dots }$, of the moduli of the rounded values of the first 6 neglected terms: … ►Similar improvements are achievable by Aitken’s ${\mathrm{\Delta }}^2$-process, Wynn’s $ϵ$-algorithm, and other acceleration transformations. … ►For example, using double precision $d_{20}$ is found to agree with (2.11.31) to 13D. …

18: 4.4 Special Values and Limits

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4.4.16 $$\underset{z\to \mathrm{\infty }}{lim}{z}^a{\mathrm{e}}^{-z}=0,$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$ (),

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $a$: real or complex constant, $z$: complex variable and $\delta$: constant

A&S Ref:

4.2.20

Permalink:

http://dlmf.nist.gov/4.4.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(iii), §4.4 and Ch.4

►where $a$ ($\in ℂ$) and $\delta$ ($\in (0,\frac{1}{2}\pi ]$) are constants. … ►

4.4.19 $$\underset{n\to \mathrm{\infty }}{lim}\left(\left(\sum _{k=1}^n\frac{1}{k}\right)-\ln n\right)=\gamma =\mathrm{0.57721\; 56649\; 01532\; 86060}\mathrm{\dots },$$

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Symbols:

$\gamma$: Euler’s constant, $\ln z$: principal branch of logarithm function, $k$: integer and $n$: integer

A&S Ref:

4.1.32 (with 10D value)

Notes:

For more digits see OEIS Sequence A001620; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/4.4.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(iii), §4.4 and Ch.4

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19: 18.25 Wilson Class: Definitions

… ►For the Wilson class OP’s $p_n(x)$ with $x=\lambda (y)$: if the $y$-orthogonality set is $\{0,1,\mathrm{\dots },N\}$, then the role of the differentiation operator $d/dx$ in the Jacobi, Laguerre, and Hermite cases is played by the operator ${\mathrm{\Delta }}_y$ followed by division by ${\mathrm{\Delta }}_y(\lambda (y))$, or by the operator ${\nabla }_y$ followed by division by ${\nabla }_y(\lambda (y))$. Alternatively if the $y$-orthogonality interval is $(0,\mathrm{\infty })$, then the role of $d/dx$ is played by the operator ${\delta }_y$ followed by division by ${\delta }_y(\lambda (y))$. … ►Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials $W_n(x;a,b,c,d)$, continuous dual Hahn polynomials $S_n(x;a,b,c)$, Racah polynomials $R_n(x;\alpha ,\beta ,\gamma ,\delta )$, and dual Hahn polynomials $R_n(x;\gamma ,\delta ,N)$. … ►

… ►The first four sets imply $\gamma +\delta >-2$, and the last four imply . …

20: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►An inner product space $V$ is called a Hilbert space if every Cauchy sequence $\{v_n\}$ in $V$ (i. … ►of the Dirac delta distribution. … ►, for each $v\in V$ there is a sequence $\{v_n\}$ in $𝒟(T)$ such that $\Vert v_n-v\Vert \to 0$ as $n\to \mathrm{\infty }$. … ►Applying the representation (1.17.13), now symmetrized as in (1.17.14), as $\frac{1}{x}\delta \left(x-y\right)=\frac{1}{\sqrt{xy}}\delta \left(x-y\right)$, … ►These latter results also correspond to use of the $\delta \left(x-y\right)$ as defined in (1.17.12_1) and (1.17.12_2). …