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