# §28.35 Tables

## §28.35(i) Real Variables

• Blanch and Clemm (1962) includes values of $\mathop{{\mathrm{Mc}^{(1)}_{n}}\/}\nolimits\!\left(x,\sqrt{q}\right)$ and $\mathop{{\mathrm{Mc}^{(1)}_{n}}\/}\nolimits'\!\left(x,\sqrt{q}\right)$ for $n=0(1)15$ with $q=0(.05)1$, $x=0(.02)1$. Also $\mathop{{\mathrm{Ms}^{(1)}_{n}}\/}\nolimits\!\left(x,\sqrt{q}\right)$ and $\mathop{{\mathrm{Ms}^{(1)}_{n}}\/}\nolimits'\!\left(x,\sqrt{q}\right)$ for $n=1(1)15$ with $q=0(.05)1$, $x=0(.02)1$. Precision is generally 7D.

• Blanch and Clemm (1965) includes values of $\mathop{{\mathrm{Mc}^{(2)}_{n}}\/}\nolimits\!\left(x,\sqrt{q}\right)$, $\mathop{{\mathrm{Mc}^{(2)}_{n}}\/}\nolimits'\!\left(x,\sqrt{q}\right)$ for $n=0(1)7$, $x=0(.02)1$; $n=8(1)15$, $x=0(.01)1$. Also $\mathop{{\mathrm{Ms}^{(2)}_{n}}\/}\nolimits\!\left(x,\sqrt{q}\right)$, $\mathop{{\mathrm{Ms}^{(2)}_{n}}\/}\nolimits'\!\left(x,\sqrt{q}\right)$ 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.

• Blanch and Rhodes (1955) includes $\mathit{Be}_{n}(t)$, $\mathit{Bo}_{n}(t)$, $t=\tfrac{1}{2}\sqrt{q}$, $n=0(1)15$; 8D. The range of $t$ is 0 to 0.1, with step sizes ranging from 0.002 down to 0.00025. Notation: $\mathit{Be}_{n}(t)=\mathop{a_{n}\/}\nolimits\!\left(q\right)+2q-(4n+2)\sqrt{q}$, $\mathit{Bo}_{n}(t)=\mathop{b_{n}\/}\nolimits\!\left(q\right)+2q-(4n-2)\sqrt{q}$.

• Ince (1932) includes eigenvalues $\mathop{a_{n}\/}\nolimits$, $\mathop{b_{n}\/}\nolimits$, and Fourier coefficients for $n=0$ or $1(1)6$, $q=0(1)10(2)20(4)40$; 7D. Also $\mathop{\mathrm{ce}_{n}\/}\nolimits\!\left(x,q\right)$, $\mathop{\mathrm{se}_{n}\/}\nolimits\!\left(x,q\right)$ for $q=0(1)10$, $x=1(1)90$, corresponding to the eigenvalues in the tables; 5D. Notation: $\mathop{a_{n}\/}\nolimits=\mathit{be}_{n}-2q$, $\mathop{b_{n}\/}\nolimits=\mathit{bo}_{n}-2q$.

• Kirkpatrick (1960) contains tables of the modified functions $\mathop{\mathrm{Ce}_{n}\/}\nolimits\!\left(x,q\right)$, $\mathop{\mathrm{Se}_{n+1}\/}\nolimits\!\left(x,q\right)$ for $n=0(1)5$, $q=1(1)20$, $x=0.1(.1)1$; 4D or 5D.

• National Bureau of Standards (1967) includes the eigenvalues $\mathop{a_{n}\/}\nolimits\!\left(q\right)$, $\mathop{b_{n}\/}\nolimits\!\left(q\right)$ 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 $\mathop{\mathrm{ce}_{n}\/}\nolimits\!\left(x,q\right)$ and $\mathop{\mathrm{se}_{n}\/}\nolimits\!\left(x,q\right)$ for $n=0(1)15$, $n=1(1)15$, respectively, and various values of $q$ in the interval $[0,100]$; joining factors $g_{\mathit{e},n}(\sqrt{q})$, $f_{\mathit{e},n}(\sqrt{q})$ for $n=0(1)15$ with $q=0(.5\mbox{ to }10)100$ (but in a different notation). Also, eigenvalues for large values of $q$. Precision is generally 8D.

• 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 $1(1)4$, $c=0(.1~{}\textrm{or}~{}.2)4.5$. 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.

• Zhang and Jin (1996, pp. 521–532) includes the eigenvalues $\mathop{a_{n}\/}\nolimits\!\left(q\right)$, $\mathop{b_{n+1}\/}\nolimits\!\left(q\right)$ 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 $\mathop{\mathrm{ce}_{n}\/}\nolimits\!\left(x,10\right)$, $\mathop{\mathrm{se}_{n+1}\/}\nolimits\!\left(x,10\right)$, $n=0(1)7$. Mathieu functions $\mathop{\mathrm{ce}_{n}\/}\nolimits\!\left(x,10\right)$, $\mathop{\mathrm{se}_{n+1}\/}\nolimits\!\left(x,10\right)$, and their first $x$-derivatives for $n=0(1)4$, $x=0(5^{\circ})90^{\circ}$. Modified Mathieu functions $\mathop{{\mathrm{Mc}^{(j)}_{n}}\/}\nolimits\!\left(x,\sqrt{10}\right)$, $\mathop{{\mathrm{Ms}^{(j)}_{n+1}}\/}\nolimits\!\left(x,\sqrt{10}\right)$, and their first $x$-derivatives for $n=0(1)4$, $j=1,2$, $x=0(.2)4$. Precision is mostly 9S.

## §28.35(ii) Complex Variables

• Blanch and Clemm (1969) includes eigenvalues $\mathop{a_{n}\/}\nolimits\!\left(q\right)$, $\mathop{b_{n}\/}\nolimits\!\left(q\right)$ for $q=\rho e^{\mathrm{i}\phi}$, $\rho=0(.5)25$, $\phi=5^{\circ}(5^{\circ})90^{\circ}$, $n=0(1)15$; 4D. Also $\mathop{a_{n}\/}\nolimits\!\left(q\right)$ and $\mathop{b_{n}\/}\nolimits\!\left(q\right)$ 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.

## §28.35(iii) Zeros

• Blanch and Clemm (1965) includes the first and second zeros of $\mathop{{\mathrm{Mc}^{(2)}_{n}}\/}\nolimits\!\left(x,\sqrt{q}\right)$, $\mathop{{\mathrm{Mc}^{(2)}_{n}}\/}\nolimits'\!\left(x,\sqrt{q}\right)$ for $n=0,1$, and $\mathop{{\mathrm{Ms}^{(2)}_{n}}\/}\nolimits\!\left(x,\sqrt{q}\right)$, $\mathop{{\mathrm{Ms}^{(2)}_{n}}\/}\nolimits'\!\left(x,\sqrt{q}\right)$ for $n=1,2$, with $q=0(.05)1$; 7D.

• Ince (1932) includes the first zero for $\mathop{\mathrm{ce}_{n}\/}\nolimits$, $\mathop{\mathrm{se}_{n}\/}\nolimits$ 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 $q$.

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

## §28.35(iv) Further Tables

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