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

§28.35 Tables

Keywords:: Mathieu functions, Mathieu’s equation, modified Mathieu functions
Permalink:: http://dlmf.nist.gov/28.35

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

§28.35(i) Real Variables

Keywords:: modified Mathieu functions
Permalink:: http://dlmf.nist.gov/28.35.i

Blanch and Clemm (1962) includes values of $\mathop{{\mathrm{Mc}^{{(1)}}_{{n}}}\/}\nolimits\!\left(x,\sqrt{q}\right)$ and ${\mathop{{\mathrm{Mc}^{{(1)}}_{{n}}}\/}\nolimits^{{\prime}}}\!\left(x,\sqrt{q}\right)$ for with , . Also $\mathop{{\mathrm{Ms}^{{(1)}}_{{n}}}\/}\nolimits\!\left(x,\sqrt{q}\right)$ and ${\mathop{{\mathrm{Ms}^{{(1)}}_{{n}}}\/}\nolimits^{{\prime}}}\!\left(x,\sqrt{q}\right)$ for with , . Precision is generally 7D.
Blanch and Clemm (1965) includes values of $\mathop{{\mathrm{Mc}^{{(2)}}_{{n}}}\/}\nolimits\!\left(x,\sqrt{q}\right)$ , ${\mathop{{\mathrm{Mc}^{{(2)}}_{{n}}}\/}\nolimits^{{\prime}}}\!\left(x,\sqrt{q}\right)$ for , ; , . Also $\mathop{{\mathrm{Ms}^{{(2)}}_{{n}}}\/}\nolimits\!\left(x,\sqrt{q}\right)$ , ${\mathop{{\mathrm{Ms}^{{(2)}}_{{n}}}\/}\nolimits^{{\prime}}}\!\left(x,\sqrt{q}\right)$ for , ; , . In all cases . Precision is generally 7D. Approximate formulas and graphs are also included.
Blanch and Rhodes (1955) includes $\mathit{Be}_{n}(t)$ , $\mathit{Bo}_{n}(t)$ , $t=\tfrac{1}{2}\sqrt{q}$ , ; 8D. The range of is 0 to 0.1, with step sizes ranging from 0.002 down to 0.00025. Notation: $\mathit{Be}_{n}(t)=\mathop{a_{{n}}\/}\nolimits\!\left(q\right)+2q-(4n+2)\sqrt{q}$ , $\mathit{Bo}_{n}(t)=\mathop{b_{{n}}\/}\nolimits\!\left(q\right)+2q-(4n-2)\sqrt{q}$ .
Ince (1932) includes eigenvalues $\mathop{a_{{n}}\/}\nolimits$ , $\mathop{b_{{n}}\/}\nolimits$ , and Fourier coefficients for or , ; 7D. Also $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(x,q\right)$ , $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(x,q\right)$ for , , 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 , , ; 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 with , and with ; Fourier coefficients for $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(x,q\right)$ and $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(x,q\right)$ 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.
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.
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 , ; (’s) or 19 (’s), . Fourier coefficients for $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(x,10\right)$ , $\mathop{\mathrm{se}_{{n+1}}\/}\nolimits\!\left(x,10\right)$ , . Mathieu functions $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(x,10\right)$ , $\mathop{\mathrm{se}_{{n+1}}\/}\nolimits\!\left(x,10\right)$ , and their first -derivatives for , $x=0(5^{\circ})90^{\circ}$ . 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 -derivatives for , , . Precision is mostly 9S.

§28.35(ii) Complex Variables

Keywords:: Mathieu functions, Mathieu’s equation, modified Mathieu functions
Permalink:: http://dlmf.nist.gov/28.35.ii

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

§28.35(iii) Zeros

Keywords:: Mathieu functions, modified Mathieu functions
Permalink:: http://dlmf.nist.gov/28.35.iii

Blanch and Clemm (1965) includes the first and second zeros of $\mathop{{\mathrm{Mc}^{{(2)}}_{{n}}}\/}\nolimits\!\left(x,\sqrt{q}\right)$ , ${\mathop{{\mathrm{Mc}^{{(2)}}_{{n}}}\/}\nolimits^{{\prime}}}\!\left(x,\sqrt{q}\right)$ for , and $\mathop{{\mathrm{Ms}^{{(2)}}_{{n}}}\/}\nolimits\!\left(x,\sqrt{q}\right)$ , ${\mathop{{\mathrm{Ms}^{{(2)}}_{{n}}}\/}\nolimits^{{\prime}}}\!\left(x,\sqrt{q}\right)$ for , with ; 7D.
Ince (1932) includes the first zero for $\mathop{\mathrm{ce}_{{n}}\/}\nolimits$ , $\mathop{\mathrm{se}_{{n}}\/}\nolimits$ for or 6, ; 4D. This reference also gives zeros of the first derivatives, together with expansions for small .
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 , 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 or , . Precision is mostly 9S.