joining factors

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1: 28.22 Connection Formulas

… ►The joining factors in the above formulas are given by ►

28.22.5 $$g_{e,2m}(h)={(-1)}^m\sqrt{\frac{2}{\pi }}\frac{{\mathrm{ce}}_{2m}(\frac{1}{2}\pi ,h^2)}{A_0^{2m}(h^2)},$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $h$: parameter, $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.22.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

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28.22.9 $$f_{e,m}(h)=-\sqrt{\pi /2}{g}_{e,m}(h){\mathrm{Mc}}_m^{(2)}(0,h),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors and $f_{e,n}(h)$, $f_{o,n}(h)$: joining factors

A&S Ref:

20.8.12 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.22.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

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28.22.10 $$f_{o,m}(h)=-\sqrt{\pi /2}{g}_{o,m}(h)\mathrm{Ms}_m^{(2)}{}_{}{}^{\prime }(0,h),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors and $f_{e,n}(h)$, $f_{o,n}(h)$: joining factors

A&S Ref:

20.8.12 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.22.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

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$${\mathrm{ge}}_m(0,h^2)=\frac{1}{2}\pi S_m(h^2){\left(g_{o,m}(h)\right)}^2{\mathrm{se}}_m^{\prime }(0,h^2).$$

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2: 28.1 Special Notation

… ►The notation for the joining factors is … ►

$f_{o,n}(h).$

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3: 28.35 Tables

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National Bureau of Standards (1967) includes the eigenvalues $a_n\left(q\right)$, $b_n\left(q\right)$ for $n=0(1)3$ with $q=0(.2)20(.5)37(1)100$, and $n=4(1)15$ with $q=0(2)100$; Fourier coefficients for ${\mathrm{ce}}_n(x,q)$ and ${\mathrm{se}}_n(x,q)$ for $n=0(1)15$, $n=1(1)15$, respectively, and various values of $q$ in the interval $[0,100]$; joining factors $g_{e,n}(\sqrt{q})$, $f_{e,n}(\sqrt{q})$ for $n=0(1)15$ with $q=0(.5\text{ to }10)100$ (but in a different notation). Also, eigenvalues for large values of $q$. Precision is generally 8D.

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4: 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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5: 23.20 Mathematical Applications

… ►The addition law states that to find the sum of two points, take the third intersection with $C$ of the chord joining them (or the tangent if they coincide); then its reflection in the $x$-axis gives the required sum. … ►

§23.20(iii) Factorization

►§27.16 describes the use of primality testing and factorization in cryptography. …