…
►As , and converge to a common limit called the AGM (Arithmetic-Geometric Mean) of and .
…showing that the convergence of to 0 and of and to is quadratic in each case.
…
►
Ince (1932) includes eigenvalues , , and Fourier coefficients
for or , ; 7D. Also
, for ,
, corresponding to the eigenvalues in the tables; 5D. Notation:
, .
National Bureau of Standards (1967) includes the eigenvalues , for
with , and with ; Fourier
coefficients for and for
, , respectively, and various values of in the
interval ; joining factors ,
for
with (but in a different notation). Also,
eigenvalues for large values of . Precision is generally 8D.
Zhang and Jin (1996, pp. 521–532) includes the eigenvalues
, for ,
; (’s) or 19 (’s), .
Fourier coefficients for , ,
. Mathieu functions ,
, and their first -derivatives for ,
. Modified Mathieu functions
, , and
their first -derivatives for , , . Precision is
mostly 9S.
Ince (1932) includes the first zero for ,
for or , ; 4D. This reference
also gives zeros of the first derivatives, together with expansions for small
.
…
►For real each of the functions , , , and has exactly zeros in .
…For the zeros of and approach asymptotically the zeros of , and the zeros of and approach asymptotically the zeros of .
…Furthermore, for
and also have purely imaginary zeros that correspond uniquely to the purely imaginary -zeros of (§10.21(i)), and they are asymptotically equal as and .
…
…
►For further information on , , and expansions of , in Fourier series or in series of , functions, see McLachlan (1947, Chapter VII) or Meixner and Schäfke (1954, §2.72).
…
►►►
…