A263184 E.g.f. is the series C(x) such that C(x) + i*S(x) = 1 + i * Integral C(x)/(C(x) + i*S(x)) dx, even powers only, where i^2 = -1.

1, 1, -11, 589, -73079, 16276921, -5689569731, 2870590000069, -1974092553870959, 1774769713597881841, -2020648226794749619451, 2841491193407098834197949, -4836474745329895366132510439, 9799325146545425451960283425961, -23306183195981226869509072955841971, 64295665973634629981724639566520575029, -203644190177923848088768870897628056746719

Offset

0,3

Links

Table of n, a(n) for n=0..16.

Formula

Let C = C(x) and S = S(x) satisfy

(*) C + i*S = 1 + i * Integral C/(C + i*S) dx

then C and S also satisfy

(1) C^2 - S^2 = 1

(2) C*C' - S*S' = 0

(3) C*S' + S*C' = C

(4) S' = C^2/(C^2 + S^2)

(5) C' = C*S/(C^2 + S^2)

(6) C*S = Integral C dx

(7) C + i*S = exp( i * Integral C/(C + i*S)^2 dx ).

...

E.g.f.: sqrt(1 + Series_Reversion(2*x - arctan(x))^2). - Paul D. Hanna, Oct 15 2015

Example

E.g.f.: C(x) = 1 + x^2/2! - 11*x^4/4! + 589*x^6/6! - 73079*x^8/8! + 16276921*x^10/10! - 5689569731*x^12/12! + 2870590000069*x^14/14! - 1974092553870959*x^16/16! + ...
RELATED SERIES:
(a) S(x) = x - 2*x^3/3! + 64*x^5/5! - 5648*x^7/7! + 975616*x^9/9! - 278461952*x^11/11! + 118706427904*x^13/13! - 70671453390848*x^15/15! + 56012750847410176*x^17/17! + ...
(b) C(x)^2 = 1 + 2*x^2/2! - 16*x^4/4! + 848*x^6/6! - 104704*x^8/8! + 23255552*x^10/10! - 8114200576*x^12/12! + 4088708507648*x^14/14! - 2809153285586944*x^16/16! + ...
(c) S(x)^2 = 2*x^2/2! - 16*x^4/4! + 848*x^6/6! - 104704*x^8/8! + 23255552*x^10/10! -+ ...
(d) sqrt(C(x)^2 + S(x)^2) = 1 + 2*x^2/2! - 28*x^4/4! + 1688*x^6/6! - 226672*x^8/8! + 53581472*x^10/10! - 19645025728*x^12/12! + 10314899562368*x^14/14! - 7340759012323072*x^16/16! + ...
(e) log(C(x) + i*S(x)) = i*x + 2*x^2/2! - 7*i*x^3/3! - 40*x^4/4! + 293*i*x^5/5! + 2768*x^6/6! - 30763*i*x^7/7! - 405760*x^8/8! + 6040937*i*x^9/9! + 102313472*x^10/10! -+ ...
(f) arccosh(C(x)) = x - 3*x^3/3! + 93*x^5/5! - 8127*x^7/7! + 1397337*x^9/9! - 397761243*x^11/11! + 169266767733*x^13/13! - 100648672656087*x^15/15! + ...

Prog

(PARI) {a(n) = local(CS=1); for(i=1, 2*n, CS = 1 + I*intformal( (CS+conj(CS))/2 / CS +O(x^(2*n+2))) ); (2*n)!*polcoeff(real(CS), 2*n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = (2*n)! * polcoeff( sqrt( 1 + serreverse(2*x - atan(x +O(x^(2*n+2))))^2), 2*n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A263185.

Sequence in context: A185203 A363460 A265978 * A370087 A288326 A260583

Adjacent sequences: A263181 A263182 A263183 * A263185 A263186 A263187

Keyword

sign

Author

Paul D. Hanna, Oct 11 2015

Status

approved