of noninteger order

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1: 28.12 Definitions and Basic Properties

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§28.12(ii) Eigenfunctions ${\mathrm{me}}_{\nu }(z,q)$

… ►For $q=0$, … ►

28.12.12 $${\mathrm{ce}}_{\nu }(z,q)=\frac{1}{2}\left({\mathrm{me}}_{\nu }(z,q)+{\mathrm{me}}_{\nu }(-z,q)\right),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, ${\mathrm{me}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

20.5.2 (in different form)

Permalink:

http://dlmf.nist.gov/28.12.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §28.12(iii), §28.12 and Ch.28

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28.12.13 $${\mathrm{se}}_{\nu }(z,q)=-\frac{1}{2}\mathrm{i}\left({\mathrm{me}}_{\nu }(z,q)-{\mathrm{me}}_{\nu }(-z,q)\right).$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, ${\mathrm{se}}_n(z,q)$: Mathieu function, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §28.12(iii), §28.12 and Ch.28

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2: 10.74 Methods of Computation

… ►In the case of $J_n\left(x\right)$, the need for initial values can be avoided by application of Olver’s algorithm (§3.6(v)) in conjunction with Equation (10.12.4) used as a normalizing condition, or in the case of noninteger orders, (10.23.15). …

3: 10.15 Derivatives with Respect to Order

§10.15 Derivatives with Respect to Order

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Noninteger Values of $\nu$

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4: Bibliography S

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The computation of eigenvalues and solutions of Mathieu’s differential equation for noninteger order. ACM Trans. Math. Software 19 (3), pp. 377–390.

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External Links:

ISSN 0098-3500, Document, ZentralBlatt

Referenced by:

item (e).

Encodings:

BibTeX

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Algorithm 721: MTIEU1 and MTIEU2: Two subroutines to compute eigenvalues and solutions to Mathieu’s differential equation for noninteger and integer order. ACM Trans. Math. Software 19 (3), pp. 391–406.

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Notes:

Double-precision Fortran, maximum accuracy 18D

External Links:

ISSN 0098-3500, Document, GAMS (module 721; package TOMS), ZentralBlatt

Referenced by:

3rd item, 5th item.

Encodings:

BibTeX

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5: 28.28 Integrals, Integral Representations, and Integral Equations

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§28.28(iii) Integrals of Products of Mathieu Functions of Noninteger Order

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6: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►The Coulomb functions given in this chapter are most commonly evaluated for real values of $\rho$, $r$, $\eta$, $ϵ$ and nonnegative integer values of $\mathrm{\ell }$, but they may be continued analytically to complex arguments and order $\mathrm{\ell }$ as indicated in §33.13. ►Examples of applications to noninteger and/or complex variables are as follows. …

7: 13.9 Zeros

… ►When and $b>0$ let ${\varphi }_r$, $r=1,2,3,\mathrm{\dots }$, be the positive zeros of $M(a,b,x)$ arranged in increasing order of magnitude, and let $j_{b-1,r}$ be the $r$th positive zero of the Bessel function $J_{b-1}\left(x\right)$ (§10.21(i)). … ►

13.9.8 $${\varphi }_r=\frac{j_{b-1,r}^2}{2b-4a}\left(1+\frac{2b(b-2)+j_{b-1,r}^2}{3{(2b-4a)}^2}\right)+O\left(\frac{1}{a^5}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $r=1,2,3,\mathrm{\dots }$ , ${\varphi }_r$: positive zeros and $j_{b,r}$: positive zero of Bessel

Referenced by:

§13.22, §13.9(i)

Permalink:

http://dlmf.nist.gov/13.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §13.9(i), §13.9 and Ch.13

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13.9.9 $$z=\pm (2n+a)\pi \mathrm{i}+\ln \left(-\frac{\mathrm{\Gamma }\left(a\right)}{\mathrm{\Gamma }\left(b-a\right)}{\left(\pm 2n\pi \mathrm{i}\right)}^{b-2a}\right)+O\left(n^{-1}\ln n\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.9(i)

Permalink:

http://dlmf.nist.gov/13.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.9(i), §13.9 and Ch.13

… ►Let $T(a,b)$ be the total number of zeros in the sector , $P(a,b)$ be the corresponding number of positive zeros, and $a$, $b$, and $a-b+1$ be nonintegers. … ►

13.9.16 $$a=-n-\frac{2}{\pi }\sqrt{zn}-\frac{2z}{{\pi }^2}+\frac{1}{2}b+\frac{1}{4}+\frac{z^2\left(\frac{1}{3}-4{\pi }^{-2}\right)+z-{(b-1)}^2+\frac{1}{4}}{4\pi \sqrt{zn}}+O\left(\frac{1}{n}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.9(ii), DLMF Update; Version 1.0.13, Erratum (V1.0.12) for Equation (13.9.16), Erratum (V1.0.13) for Equation (13.9.16), Erratum (V1.0.13) for Equation (13.9.16)

Permalink:

http://dlmf.nist.gov/13.9.E16

Encodings:

pMML, png, TeX

Errata (effective with 1.0.13):

In applying changes in Version 1.0.12 to this equation, an editing error was made; it has been corrected.

Reported 2016-09-12 by Adri Olde Daalhuis

Errata (effective with 1.0.12):

Originally this equation was expressed in terms of the asymptotic symbol $\sim$. As a consequence of the use of the $O$ order symbol on the right hand side, $\sim$ was replaced by $=$.

Reported 2016-07-11 by Rudi Weikard

See also:

Annotations for §13.9(ii), §13.9 and Ch.13

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