28 Mathieu Functions and Hill’s Equation Computation 28.34 Methods of Computation 28.36 Software

§28.35 Tables

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Keywords:: Fourier coefficients, Mathieu functions, Mathieu’s equation, eigenvalues (or characteristic values), modified Mathieu functions, tables
Permalink:: http://dlmf.nist.gov/28.35
See also:: Annotations for Ch.28

§28.35(i) Real Variables
§28.35(ii) Complex Variables
§28.35(iii) Zeros
§28.35(iv) Further Tables

§28.35(i) Real Variables

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Keywords:: joining factors, modified Mathieu functions, tables
Permalink:: http://dlmf.nist.gov/28.35.i
See also:: Annotations for §28.35 and Ch.28

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Blanch and Clemm (1962) includes values of ${\operatorname{Mc}^{(1)}_{n}}\left(x,\sqrt{q}\right)$ and ${\operatorname{Mc}^{(1)}_{n}}'\left(x,\sqrt{q}\right)$ for with , . Also ${\operatorname{Ms}^{(1)}_{n}}\left(x,\sqrt{q}\right)$ and ${\operatorname{Ms}^{(1)}_{n}}'\left(x,\sqrt{q}\right)$ for with , . Precision is generally 7D.
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Blanch and Clemm (1965) includes values of ${\operatorname{Mc}^{(2)}_{n}}\left(x,\sqrt{q}\right)$ , ${\operatorname{Mc}^{(2)}_{n}}'\left(x,\sqrt{q}\right)$ for , ; , . Also ${\operatorname{Ms}^{(2)}_{n}}\left(x,\sqrt{q}\right)$ , ${\operatorname{Ms}^{(2)}_{n}}'\left(x,\sqrt{q}\right)$ for , ; , . In all cases . Precision is generally 7D. Approximate formulas and graphs are also included.
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Blanch and Rhodes (1955) includes $\mathit{Be}_{n}(t)$ , $\mathit{Bo}_{n}(t)$ , $t=\tfrac{1}{2}\sqrt{q}$ , ; 8D. The range of is 0 to 0.1, with step sizes ranging from 0.002 down to 0.00025. Notation: $\mathit{Be}_{n}(t)=a_{n}\left(q\right)+2q-(4n+2)\sqrt{q}$ , $\mathit{Bo}_{n}(t)=b_{n}\left(q\right)+2q-(4n-2)\sqrt{q}$ .
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Ince (1932) includes eigenvalues $a_{n}$ , $b_{n}$ , and Fourier coefficients for or , ; 7D. Also $\operatorname{ce}_{n}\left(x,q\right)$ , $\operatorname{se}_{n}\left(x,q\right)$ for , , corresponding to the eigenvalues in the tables; 5D. Notation: $a_{n}=\mathit{be}_{n}-2q$ , $b_{n}=\mathit{bo}_{n}-2q$ .
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Kirkpatrick (1960) contains tables of the modified functions $\operatorname{Ce}_{n}\left(x,q\right)$ , $\operatorname{Se}_{n+1}\left(x,q\right)$ for , , ; 4D or 5D.
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National Bureau of Standards (1967) includes the eigenvalues $a_{n}\left(q\right)$ , $b_{n}\left(q\right)$ for with , and with ; Fourier coefficients for $\operatorname{ce}_{n}\left(x,q\right)$ and $\operatorname{se}_{n}\left(x,q\right)$ for , , respectively, and various values of in the interval ; joining factors $g_{\mathit{e},n}(\sqrt{q})$ , $f_{\mathit{e},n}(\sqrt{q})$ for with $q=0(.5\mbox{ to }10)100$ (but in a different notation). Also, eigenvalues for large values of . Precision is generally 8D.
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Stratton et al. (1941) includes $b_{n}$ , $b_{n}^{\prime}$ , and the corresponding Fourier coefficients for $\mathrm{Se}_{n}(c,x)$ and $\mathrm{So}_{n}(c,x)$ for or , $c=0(.1~\textrm{or}~.2)4.5$ . Precision is mostly 5S. Notation: $c=2\sqrt{q}$ , $b_{n}=a_{n}+2q$ , $b^{\prime}_{n}=b_{n}+2q$ , and for $\mathrm{Se}_{n}(c,x)$ , $\mathrm{So}_{n}(c,x)$ see §28.1.
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Zhang and Jin (1996, pp. 521–532) includes the eigenvalues $a_{n}\left(q\right)$ , $b_{n+1}\left(q\right)$ for , ; (’s) or 19 (’s), . Fourier coefficients for $\operatorname{ce}_{n}\left(x,10\right)$ , $\operatorname{se}_{n+1}\left(x,10\right)$ , . Mathieu functions $\operatorname{ce}_{n}\left(x,10\right)$ , $\operatorname{se}_{n+1}\left(x,10\right)$ , and their first -derivatives for , $x=0(5^{\circ})90^{\circ}$ . Modified Mathieu functions ${\operatorname{Mc}^{(j)}_{n}}\left(x,\sqrt{10}\right)$ , ${\operatorname{Ms}^{(j)}_{n+1}}\left(x,\sqrt{10}\right)$ , and their first -derivatives for , , . Precision is mostly 9S.

§28.35(ii) Complex Variables

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Keywords:: Fourier coefficients, Mathieu functions, Mathieu’s equation, eigenvalues (or characteristic values), modified Mathieu functions, tables
Permalink:: http://dlmf.nist.gov/28.35.ii
See also:: Annotations for §28.35 and Ch.28

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Blanch and Clemm (1969) includes eigenvalues $a_{n}\left(q\right)$ , $b_{n}\left(q\right)$ for $q=\rho e^{\mathrm{i}\phi}$ , $\rho=0(.5)25$ , $\phi=5^{\circ}(5^{\circ})90^{\circ}$ , ; 4D. Also $a_{n}\left(q\right)$ and $b_{n}\left(q\right)$ for $q=\mathrm{i}\rho$ , $\rho=0(.5)100$ , and , respectively; 8D. Double points for ; 8D. Graphs are included.

§28.35(iii) Zeros

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Keywords:: Mathieu functions, modified Mathieu functions, tables, zeros
Permalink:: http://dlmf.nist.gov/28.35.iii
See also:: Annotations for §28.35 and Ch.28

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Blanch and Clemm (1965) includes the first and second zeros of ${\operatorname{Mc}^{(2)}_{n}}\left(x,\sqrt{q}\right)$ , ${\operatorname{Mc}^{(2)}_{n}}'\left(x,\sqrt{q}\right)$ for , and ${\operatorname{Ms}^{(2)}_{n}}\left(x,\sqrt{q}\right)$ , ${\operatorname{Ms}^{(2)}_{n}}'\left(x,\sqrt{q}\right)$ for , with ; 7D.
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Ince (1932) includes the first zero for $\operatorname{ce}_{n}$ , $\operatorname{se}_{n}$ for or 6, ; 4D. This reference also gives zeros of the first derivatives, together with expansions for small .
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Zhang and Jin (1996, pp. 533–535) includes the zeros (in degrees) of $\operatorname{ce}_{n}\left(x,10\right)$ , $\operatorname{se}_{n}\left(x,10\right)$ for , and the first 5 zeros of ${\operatorname{Mc}^{(j)}_{n}}\left(x,\sqrt{10}\right)$ , ${\operatorname{Ms}^{(j)}_{n}}\left(x,\sqrt{10}\right)$ for or , . Precision is mostly 9S.