Theory and Application of Mathieu Functions

2k cosh 2w₁t a₁ absolutely and uniformly absolutely convergent amplitude approximation arbitrary constants asymptotic expansions axis b₁ b₂ ban+1 Bessel functions C₁ C₂ calculated Ce₂n cean cem(z characteristic numbers coefficients complete solution computation continuous function corresponding cosh 2z)y cosh z cosh curves cylinder decimal places degenerates derived differential equation elliptical coordinates finite form of solution frequency frequency modulation functions of integral given gives H₂ Hence Im,r imaginary independent solutions integral equations integral order k₁ magnetic Mathieu functions Mathieu's equation mode modified Mathieu functions multiplier non-periodic nuclei obtain period periodic functions positive pulsatance recurrence relations region of Fig respectively satisfies second solution sem(z shown sinh solution of Mathieu's stable Substituting Table tends to zero tion uniformly convergent unstable region values of q vanishes w₁ wave guide write y₁ Y₁(z yields