cp's OEIS Frontend

A293417 Decimal expansion of the minimum ripple factor for a reflectionless, Chebyshev filter, in the limit where the order approaches infinity.

%I A293417 #40 Sep 08 2022 08:46:19
%S A293417 2,1,9,4,8,6,9,3,0,8,7,6,8,1,3,9,1,6,8,9,4,5,8,8,3,4,4,8,7,6,6,0,7,1,
%T A293417 7,9,4,3,0,9,2,1,3,3,3,1,6,8,8,3,8,7,4,1,9,4,1,9,8,0,8,8,6,1,2,7,5,1,
%U A293417 0,0,4,6,9,4,6,8,7,0,8,2,4,5,2,8,3,7,3,5,5,2,5,1,5,5,2,4,0,5,0,7,4,4,7,5,9,6,8,7
%N A293417 Decimal expansion of the minimum ripple factor for a reflectionless, Chebyshev filter, in the limit where the order approaches infinity.
%C A293417 This is the smallest ripple factor (a constant) for which the prototype elements of the generalized reflectionless filter topology (see Morgan, 2017) needs no negative elements, where the order of the filter approaches infinity. It is also the ripple factor for which the first two and last two Chebyshev prototype parameters (of the canonical ladder, or Cauer, topology) are equal.
%C A293417 Other related sequences in the OEIS are the decimal and continued fraction expansions of the limiting ripple factors for third, fifth, seventh, and ninth order, as well as for the limiting case where the order diverges to infinity. As these ripple factors do approach a common limit very quickly, the sequences for the fifth- and higher-order constants share the same initial terms, to greater length as the order increases.
%C A293417 There are simple radical expressions for the third- and fifth-order constants (see formulas). Further, the third-order constant is a quadratic irrational, thus having a repeating continued fraction expansion. I do not know if such simple expressions or patterns exist for the higher-order constants or the limiting (infinite-order) constant.
%D A293417 M. Morgan, Reflectionless Filters, Norwood, MA: Artech House, pp. 129-132, January 2017.
%H A293417 Iain Fox, <a href="/A293417/b293417.txt">Table of n, a(n) for n = 0..20000</a>
%F A293417 Equals sqrt(exp(4*arctanh(exp(-Pi*sqrt(2))))-1).
%e A293417 0.2194869308...
%t A293417 RealDigits[Sqrt[Exp[4 ArcTanh[Exp[-(Pi Sqrt[2])]]] - 1],10,100][[1]]
%o A293417 (PARI) sqrt(exp(4*atanh(exp(-Pi*sqrt(2))))-1) \\ _Michel Marcus_, Oct 15 2017
%o A293417 (Magma) R:= RealField(); Sqrt(Exp(4*Argtanh(Exp(-Pi(R)*Sqrt(2))))-1); // _G. C. Greubel_, Feb 16 2018
%Y A293417 Decimal expansions (A020784, A293409, A293415, A293416, A293417) and continued fractions (A040021, A293768, A293769, A293770, A293882) for third-, fifth-, seventh-, ninth-order and the limiting "infinite-order" constant, respectively.
%K A293417 cons,easy,nonn
%O A293417 0,1
%A A293417 _Matthew A. Morgan_, Oct 15 2017