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21: 28.35 Tables

§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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22: 28.13 Graphics

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§28.13(i) Eigenvalues ${\lambda }_{\nu }\left(q\right)$ for General $\nu$

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23: 28.6 Expansions for Small $q$

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§28.6(i) Eigenvalues

… ►Higher coefficients in the foregoing series can be found by equating coefficients in the following continued-fraction equations: … ►

28.6.19 $$a-{(2n+2)}^2-\frac{q^2}{a-{(2n)}^2-}\frac{q^2}{a-{(2n-2)}^2-}\mathrm{\cdots }\frac{q^2}{a-2^2}=-\frac{q^2}{{(2n+4)}^2-a-}\frac{q^2}{{(2n+6)}^2-a-}\mathrm{\cdots },$$ $a=b_{2n+2}\left(q\right)$.

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Symbols:

$b_n\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter, $n$: integer and $a$: parameter

Referenced by:

item (e)

Permalink:

http://dlmf.nist.gov/28.6.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(i), §28.6 and Ch.28

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24: 2.7 Differential Equations

… ►where ${\lambda }_1$, ${\lambda }_2$ are the roots of the characteristic equation … ►See §2.11(v) for other examples. … ►The transformed differential equation either has a regular singularity at $t=\mathrm{\infty }$, or its characteristic equation has unequal roots. … ►This is characteristic of numerically satisfactory pairs. …

25: 28.1 Special Notation

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${\lambda }_{\nu }\left(q\right).$

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26: Bibliography M

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The characteristic numbers of the Mathieu equation with purely imaginary parameter. Phil. Mag. Series 7 8 (53), pp. 834–840.

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External Links:

Document, ZentralBlatt

Referenced by:

§28.7.

Encodings:

BibTeX

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Asymptotic expansions of oblate spheroidal wave functions and their characteristic numbers. J. Reine Angew. Math. 211, pp. 33–47.

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External Links:

MathReview (J. Meixner), ZentralBlatt

Referenced by:

§30.9(ii), §30.9(ii).

Encodings:

BibTeX

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Asymptotic expansions of prolate spheroidal wave functions and their characteristic numbers. J. Reine Angew. Math. 212, pp. 26–48.

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External Links:

MathReview (J. Meixner), ZentralBlatt

Referenced by:

§30.9(i), §30.9(i).

Encodings:

BibTeX

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Asymptotic expansions of ellipsoidal wave functions and their characteristic numbers. Math. Nachr. 31, pp. 89–101.

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External Links:

Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.11, §29.7(i), §29.7(ii).

Encodings:

BibTeX

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27: Bibliography N

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Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview, ZentralBlatt

Referenced by:

§28.1, §28.26(i), 6th item, Notations S, Notations S.

Encodings:

BibTeX

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28: 28.33 Physical Applications

… ►For points $(q,a)$ that are at intersections of $ℒ$ with the characteristic curves $a=a_n\left(q\right)$ or $a=b_n\left(q\right)$, a periodic solution is possible. …

29: 28.8 Asymptotic Expansions for Large $q$

§28.8 Asymptotic Expansions for Large $q$

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30: Bibliography B

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Mathieu’s Equation for Complex Parameters. Tables of Characteristic Values. U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Table of characteristic values of Mathieu’s equation for large values of the parameter. J. Washington Acad. Sci. 45 (6), pp. 166–196.

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External Links:

MathReview (L. Fox)

Referenced by:

3rd item.

Encodings:

BibTeX

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The double confluent Heun equation: Characteristic exponent and connection formulae. Methods Appl. Anal. 1 (3), pp. 348–370.

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External Links:

ISSN 1073-2772, FullText, MathReview (W. Balser), ZentralBlatt

Referenced by:

§31.12.

Encodings:

BibTeX

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