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►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 .
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The following standard special functions: si, Si, ci, Ci, shi, Shi,
ce, Ce, se, Se, ln, Ln, Lommels, LommelS, Jacobiphi,
and the list is still growing.
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►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).
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►The superscript on the left-hand sides of (29.12.1)–(29.12.8) agrees with the number of -zeros of each Lamé polynomial in the interval , while is the number of -zeros in the open line segment from
to .
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►Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to as in the respective sectors , being an arbitrary small positive constant.
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