E.g.f. C(x) + S(x), where series C(x) and S(x) are given by
(0.a) C(x) + S(x) = Sum_{n>=0} a(n)*x^n/(n!)^3,
(0.b) C(x) = Sum_{n>=0} a(2*n)*x^(2*n)/(2*n)!^3,
(0.c) S(x) = Sum_{n>=0} a(2*n+1)*x^(2*n+1)/(2*n+1)!^3,
and satisfy the following relations.
(1.a) C(x)^2 - S(x)^2 = 1.
(1.b) C'(x)/S(x) = S'(x)/C(x) = 1/x * Integral 1/x * Integral C(x) dx dx.
(2.a) S(x) = Integral C(x)/x * Integral 1/x * Integral C(x) dx dx dx.
(2.b) C(x) = 1 + Integral S(x)/x * Integral 1/x * Integral C(x) dx dx dx.
(3.a) C(x) + S(x) = exp( Integral 1/x * Integral 1/x * Integral C(x) dx dx dx ).
(3.b) C(x) = cosh( Integral 1/x * Integral 1/x * Integral C(x) dx dx dx ).
(3.c) S(x) = sinh( Integral 1/x * Integral 1/x * Integral C(x) dx dx dx ).
Integration.
(4.a) S(x*y*z) = Integral C(x*y*z) * Integral Integral C(x*y*z) dx dy dz.
(4.b) C(x*y*z) = 1 + Integral S(x*y*z) * Integral Integral C(x*y*z) dx dy dz.
(4.c) S(x*y*z) = Integral C(x*y*z) * Integral Integral C(x*y*z) dz dy dx.
(4.d) C(x*y*z) = 1 + Integral S(x*y*z) * Integral Integral C(x*y*z) dz dy dx.
Exponential.
(5.a) C(x*y*z) + S(x*y*z) = exp( Integral Integral Integral C(x*y*z) dx dy dz ).
(5.b) C(x*y*z) = cosh( Integral Integral Integral C(x*y*z) dx dy dz ).
(5.c) S(x*y*z) = sinh( Integral Integral Integral C(x*y*z) dx dy dz ).
Derivatives.
(6.a) d/dx S(x*y*z) = C(x*y*z) * Integral Integral C(x*y*z) dy dz.
(6.b) d/dx C(x*y*z) = S(x*y*z) * Integral Integral C(x*y*z) dy dz.
(6.c) d/dy S(x*y*z) = C(x*y*z) * Integral Integral C(x*y*z) dx dz.
(6.d) d/dy C(x*y*z) = S(x*y*z) * Integral Integral C(x*y*z) dx dz.
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