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28 Mathieu Functions and Hill’s EquationComputation

§28.35 Tables


§28.35(i) Real Variables

  • Blanch and Clemm (1962) includes values of Mcn(1)(x,q) and Mcn(1)(x,q) for n=0(1)15 with q=0(.05)1, x=0(.02)1. Also Msn(1)(x,q) and Msn(1)(x,q) 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 Mcn(2)(x,q), Mcn(2)(x,q) for n=0(1)7, x=0(.02)1; n=8(1)15, x=0(.01)1. Also Msn(2)(x,q), Msn(2)(x,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.

  • Blanch and Rhodes (1955) includes Ben(t), Bon(t), t=12q, 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: Ben(t)=an(q)+2q-(4n+2)q, Bon(t)=bn(q)+2q-(4n-2)q.

  • Ince (1932) includes eigenvalues an, bn, and Fourier coefficients for n=0 or 1(1)6, q=0(1)10(2)20(4)40; 7D. Also cen(x,q), sen(x,q) for q=0(1)10, x=1(1)90, corresponding to the eigenvalues in the tables; 5D. Notation: an=ben-2q, bn=bon-2q.

  • Kirkpatrick (1960) contains tables of the modified functions Cen(x,q), Sen+1(x,q) 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 an(q), bn(q) 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 cen(x,q) and sen(x,q) for n=0(1)15, n=1(1)15, respectively, and various values of q in the interval [0,100]; joining factors ge,n(q), fe,n(q) for n=0(1)15 with q=0(.5 to 10)100 (but in a different notation). Also, eigenvalues for large values of q. Precision is generally 8D.

  • Stratton et al. (1941) includes bn, bn, and the corresponding Fourier coefficients for Sen(c,x) and Son(c,x) for n=0 or 1(1)4, c=0(.1or.2)4.5. Precision is mostly 5S. Notation: c=2q, bn=an+2q, bn=bn+2q, and for Sen(c,x), Son(c,x) see §28.1.

  • Zhang and Jin (1996, pp. 521–532) includes the eigenvalues an(q), bn+1(q) 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 cen(x,10), sen+1(x,10), n=0(1)7. Mathieu functions cen(x,10), sen+1(x,10), and their first x-derivatives for n=0(1)4, x=0(5)90. Modified Mathieu functions Mcn(j)(x,10), Msn+1(j)(x,10), 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 an(q), bn(q) for q=ρeiϕ, ρ=0(.5)25, ϕ=5(5)90, n=0(1)15; 4D. Also an(q) and bn(q) for q=iρ, ρ=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 Mcn(2)(x,q), Mcn(2)(x,q) for n=0,1, and Msn(2)(x,q), Msn(2)(x,q) for n=1,2, with q=0(.05)1; 7D.

  • Ince (1932) includes the first zero for cen, sen 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 cen(x,10), sen(x,10) for n=1(1)10, and the first 5 zeros of Mcn(j)(x,10), Msn(j)(x,10) 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).