OFFSET
0,5
COMMENTS
FORMULA
E.g.f.: D(x,k) = Sum_{n>=0} Sum_{j=0..n} a(n,j) * x^(2*n)*k^(2*j)/(2*n)! along with the related functions C = C(x,k), S = S(x,k), D = D(x,k), and T = T(x,k) satisfy the following formulas.
Definition.
(1.a) (C + i*S) = exp(i*x*D).
(1.b) (D + i*k*T) = exp(i*k*x*C).
(2.a) C^2 + S^2 = 1.
(2.b) D^2 + k^2*T^2 = 1.
Circular functions.
(3.a) C = cos(x*D).
(3.b) S = sin(x*D).
(3.c) D = cos(k*x*C).
(3.d) T = (1/k) * sin(k*x*C).
(4.a) C = cos( x*cos(k*x*C) ).
(4.b) S = sin( x*cos(k*x*sqrt(1 - S^2)) ).
(4.c) D = cos( k*x*cos(x*D) ).
(4.d) T = (1/k) * sin( k*x*cos(x*sqrt(1 - k^2*T^2)) ).
(5.a) (C*D - k*S*T) = cos(x*D + k*x*C).
(5.b) (S*D + k*C*T) = sin(x*D + k*x*C).
Transformations.
(6.a) C(x, 1/k) = D(x/k, k).
(6.b) D(x, 1/k) = C(x/k, k).
(6.c) S(x, 1/k) = k * T(x/k, k).
(6.d) T(x, 1/k) = k * S(x/k, k).
(6.e) D(x, k) = C(k*x, 1/k).
(6.f) C(x, k) = D(k*x, 1/k).
(6.g) T(x, k) = (1/k) * S(k*x, 1/k).
(6.h) S(x, k) = (1/k) * T(k*x, 1/k).
Integrals.
(7.a) C = 1 - Integral S*D + x*S*D' dx.
(7.b) S = Integral C*D + x*C*D' dx.
(7.c) D = 1 - k^2 * Integral T*C + x*T*C' dx.
(7.d) T = Integral D*C + x*D*C' dx.
Derivatives (d/dx).
(8.a) C*C' = -S*S'.
(8.b) D*D' = -k^2*T*T'.
(9.a) C' = -S * (D + x*D').
(9.b) S' = C * (D + x*D').
(9.c) D' = -k^2 * T * (C + x*C').
(9.d) T' = D * (C + x*C').
(10.a) C' = -S * (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.b) S' = C * (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.c) D' = -k^2 * T * (C - x*S*D) / (1 - k^2*x^2*S*T).
(10.d) T' = D * (C - x*S*D) / (1 - k^2*x^2*S*T).
(11.a) (C + x*C') = (C - x*S*D) / (1 - k^2*x^2*S*T).
(11.b) (D + x*D') = (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
EXAMPLE
E.g.f.: D(x,k) = 1 - (k^2)*x^2/2! + (12*k^2 + k^4)*x^4/4! - (120*k^2 + 420*k^4 + k^6)*x^6/6! + (896*k^2 + 36400*k^4 + 10248*k^6 + k^8)*x^8/8! - (5760*k^2 + 2170560*k^4 + 4858560*k^6 + 196920*k^8 + k^10)*x^10/10! + (33792*k^2 + 102960000*k^4 + 1127738304*k^6 + 461126160*k^8 + 3247860*k^10 + k^12)*x^12/12! - (186368*k^2 + 4083183104*k^4 + 187282263168*k^6 + 340884800256*k^8 + 35248293080*k^10 + 48361404*k^12 + k^14)*x^14/14! + ...
where D(x,k) = cos( k*x*cos(x*D(x,k)) ).
This triangle of coefficients a(n,j) of x^(2*n)*k^(2*j)/(2*n)! in D(x,k) begins
1;
0, -1;
0, 12, 1;
0, -120, -420, -1;
0, 896, 36400, 10248, 1;
0, -5760, -2170560, -4858560, -196920, -1;
0, 33792, 102960000, 1127738304, 461126160, 3247860, 1;
0, -186368, -4083183104, -187282263168, -340884800256, -35248293080, -48361404, -1;
0, 983040, 141360128000, 25081621813248, 153279541958400, 76526954183680, 2290777550880, 669616080, 1; ...
PROG
(PARI) {a(n, j) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n)));
for(i=1, 2*n,
C = cos( x*cos(k*x*C +Ox) );
S = sin( x*cos(k*x*sqrt(1 - S^2 +Ox)) );
D = cos( k*x*cos(x*D +Ox));
T = (1/k)*sin( k*x*cos(x*sqrt(1 - k^2*T^2 +Ox))); );
(2*n)! * polcoeff(polcoeff(D, 2*n, x), 2*j, k)}
for(n=0, 10, for(j=0, n, print1( a(n, j), ", ")); print(""))
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Paul D. Hanna, Feb 19 2024
STATUS
approved