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21: 23.19 Interrelations

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23.19.1 $$\lambda \left(\tau \right)=16{\left(\frac{{\eta }^2\left(2\tau \right)\eta \left(\frac{1}{2}\tau \right)}{{\eta }^3\left(\tau \right)}\right)}^8,$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\lambda \left(\tau \right)$: elliptic modular function and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.19 and Ch.23

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23.19.3 $$J\left(\tau \right)=\frac{g_2^3}{g_2^3-27g_3^2},$$

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Symbols:

$J\left(\tau \right)$: Klein’s complete invariant, $g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $𝕃$: lattice and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.19 and Ch.23

►where $g_2,g_3$ are the invariants of the lattice $𝕃$ with generators $1$ and $\tau$; see §23.3(i). … ►

23.19.4 $$\mathrm{\Delta }={(2\pi )}^{12}{\eta }^{24}\left(\tau \right).$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Delta }$: discriminant and $\tau$: complex variable

Referenced by:

§23.19

Permalink:

http://dlmf.nist.gov/23.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §23.19 and Ch.23

22: 11.7 Integrals and Sums

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11.7.3 $$\int z^{\nu }{𝐋}_{\nu -1}\left(z\right)dz=z^{\nu }{𝐋}_{\nu }\left(z\right),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(i), §11.7 and Ch.11

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11.7.4 $$\int z^{-\nu }{𝐋}_{\nu +1}\left(z\right)dz=z^{-\nu }{𝐋}_{\nu }\left(z\right)-\frac{2^{-\nu }z}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{3}{2}\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(i), §11.7 and Ch.11

… ►The following Laplace transforms of ${𝐇}_{\nu }\left(t\right)$ require $\mathrm{\Re }a>0$ for convergence, while those of ${𝐋}_{\nu }\left(t\right)$ require $\mathrm{\Re }a>1$. … ►

${\displaystyle {\int }_0^{\mathrm{\infty }}}{\mathrm{e}}^{-at}{𝐋}_1\left(t\right)dt$

… ►For integrals of ${𝐇}_{\nu }\left(x\right)$ and ${𝐋}_{\nu }\left(x\right)$ with respect to the order $\nu$, see Apelblat (1989). …

23: 11.13 Methods of Computation

… ►For a review of methods for the computation of ${𝐇}_{\nu }\left(z\right)$ see Zanovello (1975). For simple and effective approximations to ${𝐇}_0\left(z\right)$ and ${𝐇}_1\left(z\right)$ see Aarts and Janssen (2016). … ►Subsequently ${𝐇}_{\nu }\left(z\right)$ and ${𝐋}_{\nu }\left(z\right)$ are obtainable via (11.2.5) and (11.2.6). … ►Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that ${𝐇}_{\nu }\left(x\right)$ and ${𝐋}_{\nu }\left(x\right)$ can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. … ►Sequences of values of ${𝐇}_{\nu }\left(z\right)$ and ${𝐋}_{\nu }\left(z\right)$, with $z$ fixed, can be computed by application of the inhomogeneous difference equations (11.4.23) and (11.4.25). …

24: Bibliography

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Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument. ACM Trans. Math. Software 16 (2), pp. 178–182.

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Notes:

Double- and single-precision Fortran, maximum accuracy 18S or 14S.

External Links:

ISSN 0098-3500, Document, GAMS (module 683; package TOMS), MathReview, ZentralBlatt

Referenced by:

1st item, 1st item.

Encodings:

BibTeX

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Applications of basic hypergeometric functions. SIAM Rev. 16 (4), pp. 441–484.

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External Links:

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

Referenced by:

§17.16, Ch.17.

Encodings:

BibTeX

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Derivatives and integrals with respect to the order of the Struve functions ${𝐇}_{\nu }(x)$ and ${𝐋}_{\nu }(x)$ . J. Math. Anal. Appl. 137 (1), pp. 17–36.

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External Links:

ISSN 0022-247X, Document, MathReview (A. McD. Mercer), ZentralBlatt

Referenced by:

§11.4(vi), §11.7(iv).

Encodings:

BibTeX

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Note on the trivial zeros of Dirichlet $L$-functions. Proc. Amer. Math. Soc. 94 (1), pp. 29–30.

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External Links:

ISSN 0002-9939, FullText, MathReview (Jerzy Kaczorowski), ZentralBlatt

Referenced by:

(25.15.7), (25.15.8), §25.15(ii), §25.15.

Encodings:

BibTeX

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Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters. J. Math. Anal. Appl. 416 (1), pp. 52–80.

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External Links:

ISSN 0022-247X, Document, Link, MathReview (Vivek Sahai), ZentralBlatt

Referenced by:

§18.15(vi), §18.15(vi).

Encodings:

BibTeX

…

25: 19.9 Inequalities

… ►The perimeter $L(a,b)$ of an ellipse with semiaxes $a,b$ is given by …Almkvist and Berndt (1988) list thirteen approximations to $L(a,b)$ that have been proposed by various authors. …Ramanujan’s approximation and its leading error term yield the following approximation to $L(a,b)/(\pi (a+b))$: …Even for the extremely eccentric ellipse with $a=99$ and $b=1$, this is correct within 0. … ►where …

26: 11.2 Definitions

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11.2.2 $${𝐋}_{\nu }\left(z\right)=-\mathrm{i}{\mathrm{e}}^{-\frac{1}{2}\pi \mathrm{i}\nu }{𝐇}_{\nu }\left(\mathrm{i}z\right)={(\frac{1}{2}z)}^{\nu +1}\sum _{n=0}^{\mathrm{\infty }}\frac{{(\frac{1}{2}z)}^{2n}}{\mathrm{\Gamma }\left(n+\frac{3}{2}\right)\mathrm{\Gamma }\left(n+\nu +\frac{3}{2}\right)}.$$

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Defines:

${𝐋}_{\nu }\left(z\right)$: modified Struve function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\nu$: real or complex order and $n$: integer order

A&S Ref:

12.2.1

Referenced by:

§11.13(ii), §11.2(i), §11.4(i), §11.4(iii), §11.5(ii), §11.5(i), §11.6(ii), §11.6(ii)

Permalink:

http://dlmf.nist.gov/11.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §11.2(i), §11.2 and Ch.11

… ►The functions $z^{-\nu -1}{𝐇}_{\nu }\left(z\right)$ and $z^{-\nu -1}{𝐋}_{\nu }\left(z\right)$ are entire functions of $z$ and $\nu$. … ►

11.2.4 $${𝐋}_0\left(z\right)=\frac{2}{\pi }\left(z+\frac{z^3}{1^2\cdot 3^2}+\frac{z^5}{1^2\cdot 3^2\cdot 5^2}+\mathrm{\cdots }\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${𝐋}_{\nu }\left(z\right)$: modified Struve function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/11.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §11.2(i), §11.2 and Ch.11

… ►Unless indicated otherwise, ${𝐇}_{\nu }\left(z\right)$, ${𝐊}_{\nu }\left(z\right)$, ${𝐋}_{\nu }\left(z\right)$, and ${𝐌}_{\nu }\left(z\right)$ assume their principal values throughout the DLMF. … ►

11.2.10 $$w={𝐋}_{\nu }\left(z\right),{𝐌}_{\nu }\left(z\right).$$

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Symbols:

${𝐌}_{\nu }\left(z\right)$: modified Struve function, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §11.2(ii), §11.2(ii), §11.2 and Ch.11

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27: 18.7 Interrelations and Limit Relations

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18.7.19 $$H_{2n}\left(x\right)={(-1)}^n{2}^{2n}n!L_n^{(-\frac{1}{2})}\left(x^2\right),$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.40

Referenced by:

§18.16(v), §18.18(iii), §18.18(iv), §18.18(viii), §18.7(ii)

Permalink:

http://dlmf.nist.gov/18.7.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(ii), §18.7 and Ch.18

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18.7.20 $$H_{2n+1}\left(x\right)={(-1)}^n{2}^{2n+1}n!x{L}_n^{(\frac{1}{2})}\left(x^2\right).$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.41

Referenced by:

§18.16(v), §18.18(iii), §18.18(iv), §18.7(ii), §18.7(ii)

Permalink:

http://dlmf.nist.gov/18.7.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(ii), §18.7 and Ch.18

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18.7.21 $$\underset{\beta \to \mathrm{\infty }}{lim}{P}_n^{(\alpha ,\beta )}\left(1-\left(2x/\beta \right)\right)=L_n^{(\alpha )}\left(x\right).$$

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.15.5

Referenced by:

§18.10(ii), §18.7(iii)

Permalink:

http://dlmf.nist.gov/18.7.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

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18.7.22 $$\underset{\alpha \to \mathrm{\infty }}{lim}{P}_n^{(\alpha ,\beta )}\left((2x/\alpha )-1\right)={(-1)}^n{L}_n^{(\beta )}\left(x\right).$$

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.7(iii)

Permalink:

http://dlmf.nist.gov/18.7.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

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18.7.26 $$\underset{\alpha \to \mathrm{\infty }}{lim}{\left(\frac{2}{\alpha }\right)}^{\frac{1}{2}n}{L}_n^{(\alpha )}\left({(2\alpha )}^{\frac{1}{2}}x+\alpha \right)=\frac{{(-1)}^n}{n!}{H}_n\left(x\right).$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.7(iii)

Permalink:

http://dlmf.nist.gov/18.7.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

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28: 18.21 Hahn Class: Interrelations

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18.21.8 $$\underset{c\to 1}{lim}{M}_n({(1-c)}^{-1}x;\alpha +1,c)=\frac{L_n^{(\alpha )}\left(x\right)}{L_n^{(\alpha )}\left(0\right)}.$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $M_n(x;\beta ,c)$: Meixner polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii)

Permalink:

http://dlmf.nist.gov/18.21.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

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18.21.9 $$\underset{a\to \mathrm{\infty }}{lim}{(2a)}^{\frac{1}{2}n}{C}_n({(2a)}^{\frac{1}{2}}x+a;a)={(-1)}^n{H}_n\left(x\right).$$

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Symbols:

$C_n(x;a)$: Charlier polynomial, $H_n\left(x\right)$: Hermite polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii)

Permalink:

http://dlmf.nist.gov/18.21.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

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18.21.12 $$\underset{\varphi \to 0}{lim}{P}_n^{(\frac{1}{2}\alpha +\frac{1}{2})}(-{(2\varphi )}^{-1}x;\varphi )=L_n^{(\alpha )}\left(x\right).$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii), §18.24, §18.30(v)

Permalink:

http://dlmf.nist.gov/18.21.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

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18.21.13 $$n!\underset{\lambda \to \mathrm{\infty }}{lim}{\lambda }^{-n/2}{P}_n^{(\lambda )}(x{\lambda }^{1/2};\pi /2)=H_n\left(x\right).$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii), §18.21(ii), §18.30(v), Erratum (V1.2.0) Section 18.21

Permalink:

http://dlmf.nist.gov/18.21.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

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29: 3.2 Linear Algebra

… ►With $𝐲={[y_1,y_2,\mathrm{\dots },y_n]}^{\mathrm{T}}$ the process of solution can then be regarded as first solving the equation $𝐋𝐲=𝐛$ for $𝐲$ (forward elimination), followed by the solution of $𝐔𝐱=𝐲$ for $𝐱$ (back substitution). … ►Because of rounding errors, the residual vector $𝐫=𝐛-𝐀𝐱$ is nonzero as a rule. … ►(We are assuming that the matrix $𝐀$ is real; if not ${𝐀}^{\mathrm{T}}$ is replaced by ${𝐀}^{\mathrm{H}}$, the transpose of the complex conjugate of $𝐀$.) … ►In the case that the orthogonality condition is replaced by $𝐒$-orthogonality, that is, ${𝐯}_j^{\mathrm{T}}𝐒{𝐯}_k={\delta }_{j,k}$, $j,k=1,2,\mathrm{\dots },n$, for some positive definite matrix $𝐒$ with Cholesky decomposition $𝐒={𝐋}^{\mathrm{T}}𝐋$, then the details change as follows. … ►

${𝐯}_{j+1}={𝐋}^{-1}{\left({𝐋}^{-1}\right)}^{\mathrm{T}}𝐮/{\beta }_{j+1},$

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30: 11.3 Graphics

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Figure 11.3.1: ${𝐇}_{\nu }\left(x\right)$ for $0\le x\le 12$ and $\nu =0,\frac{1}{2},1,\frac{3}{2},2,3$. Magnify

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Figure 11.3.5: ${𝐇}_{\nu }\left(x\right)$ for $0\le x\le 8$ and $-4\le \nu \le 4$. Magnify 3D Help

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Figure 11.3.13: ${𝐋}_{\nu }\left(x\right)$ for and $\nu =0,\frac{1}{2},1,\frac{3}{2},2,3$. Magnify

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Figure 11.3.15: ${𝐋}_{\nu }\left(x\right)$ for and $\nu =-3,-2,-\frac{3}{2},-1,-\frac{1}{2}$. Magnify

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Figure 11.3.17: ${𝐋}_{\nu }\left(x\right)$ for $0\le x\le 5.6$ and $-4\le \nu \le 4$. Magnify 3D Help

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