§28.35 Tables

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  Blanch and Clemm (1962) includes values of ${\mathrm{Mc}}_n^{(1)}(x,\sqrt{q})$ and $\mathrm{Mc}_n^{(1)}{}_{}{}^{\prime }(x,\sqrt{q})$ for $n=0(1)15$ with $q=0(.05)1$, $x=0(.02)1$. Also ${\mathrm{Ms}}_n^{(1)}(x,\sqrt{q})$ and $\mathrm{Ms}_n^{(1)}{}_{}{}^{\prime }(x,\sqrt{q})$ for $n=1(1)15$ with $q=0(.05)1$, $x=0(.02)1$. Precision is generally 7D.

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  Blanch and Clemm (1965) includes values of ${\mathrm{Mc}}_n^{(2)}(x,\sqrt{q})$, $\mathrm{Mc}_n^{(2)}{}_{}{}^{\prime }(x,\sqrt{q})$ for $n=0(1)7$, $x=0(.02)1$; $n=8(1)15$, $x=0(.01)1$. Also ${\mathrm{Ms}}_n^{(2)}(x,\sqrt{q})$, $\mathrm{Ms}_n^{(2)}{}_{}{}^{\prime }(x,\sqrt{q})$ for $n=1(1)7$, $x=0(.02)1$; $n=8(1)15$, $x=0(.01)1$. In all cases $q=0(.05)1$. Precision is generally 7D. Approximate formulas and graphs are also included.

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  Blanch and Rhodes (1955) includes ${\mathrm{𝐵𝑒}}_n(t)$, ${\mathrm{𝐵𝑜}}_n(t)$, $t=\frac{1}{2}\sqrt{q}$, $n=0(1)15$; 8D. The range of $t$ is 0 to 0.1, with step sizes ranging from 0.002 down to 0.00025. Notation: ${\mathrm{𝐵𝑒}}_n(t)=a_n\left(q\right)+2q-(4n+2)\sqrt{q}$, ${\mathrm{𝐵𝑜}}_n(t)=b_n\left(q\right)+2q-(4n-2)\sqrt{q}$.

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  Ince (1932) includes eigenvalues $a_n$, $b_n$, and Fourier coefficients for $n=0$ or $1(1)6$, $q=0(1)10(2)20(4)40$; 7D. Also ${\mathrm{ce}}_n(x,q)$, ${\mathrm{se}}_n(x,q)$ for $q=0(1)10$, $x=1(1)90$, corresponding to the eigenvalues in the tables; 5D. Notation: $a_n={\mathrm{𝑏𝑒}}_n-2q$, $b_n={\mathrm{𝑏𝑜}}_n-2q$.

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  Kirkpatrick (1960) contains tables of the modified functions ${\mathrm{Ce}}_n(x,q)$, ${\mathrm{Se}}_{n+1}(x,q)$ for $n=0(1)5$, $q=1(1)20$, $x=0.1(.1)1$; 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 $n=0(1)3$ 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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  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 $n=0$ or $1(1)4$, $c=0(.1\text{or}.2)4.5$. Precision is mostly 5S. Notation: $c=2\sqrt{q}$, $b_n=a_n+2q$, $b_n^{\prime }=b_n+2q$, 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 $n=0(1)4$, $q=0(1)50$; $n=0(1)20$ ($a$’s) or 19 ($b$’s), $q=1,3,5,10,15,25,50(50)200$. Fourier coefficients for ${\mathrm{ce}}_n(x,10)$, ${\mathrm{se}}_{n+1}(x,10)$, $n=0(1)7$. Mathieu functions ${\mathrm{ce}}_n(x,10)$, ${\mathrm{se}}_{n+1}(x,10)$, and their first $x$-derivatives for $n=0(1)4$, $x=0(5^{\circ }){90}^{\circ }$. Modified Mathieu functions ${\mathrm{Mc}}_n^{(j)}(x,\sqrt{10})$, ${\mathrm{Ms}}_{n+1}^{(j)}(x,\sqrt{10})$, and their first $x$-derivatives for $n=0(1)4$, $j=1,2$, $x=0(.2)4$. Precision is mostly 9S.

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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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  Blanch and Clemm (1965) includes the first and second zeros of ${\mathrm{Mc}}_n^{(2)}(x,\sqrt{q})$, $\mathrm{Mc}_n^{(2)}{}_{}{}^{\prime }(x,\sqrt{q})$ for $n=0,1$, and ${\mathrm{Ms}}_n^{(2)}(x,\sqrt{q})$, $\mathrm{Ms}_n^{(2)}{}_{}{}^{\prime }(x,\sqrt{q})$ for $n=1,2$, with $q=0(.05)1$; 7D.

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  Ince (1932) includes the first zero for ${\mathrm{ce}}_n$, ${\mathrm{se}}_n$ for $n=2(1)5$ or $6$, $q=0(1)10(2)40$; 4D. This reference also gives zeros of the first derivatives, together with expansions for small $q$.

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  Zhang and Jin (1996, pp. 533–535) includes the zeros (in degrees) of ${\mathrm{ce}}_n(x,10)$, ${\mathrm{se}}_n(x,10)$ for $n=1(1)10$, and the first 5 zeros of ${\mathrm{Mc}}_n^{(j)}(x,\sqrt{10})$, ${\mathrm{Ms}}_n^{(j)}(x,\sqrt{10})$ for $n=0$ or $1(1)8$, $j=1,2$. Precision is mostly 9S.

For other tables prior to 1961 see Fletcher et al. (1962, §2.2) and Lebedev and Fedorova (1960, Chapter 11).