# CI Cosine Integral

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```					                             CI Cosine Integral

CI.1 Introduction
Let x be a complex variable of C \ {0, ∞}.The function Cosine Integral (noted
Ci) is deﬁned by the following third order diﬀerential equation

∂y(x)    ∂ 2 y(x)    ∂ 3 y(x)
(CI.1.1)                 x         +2          +x          = 0.
∂x        ∂x2         ∂x3

The initial conditions of CI.1.1 at 0 are not simple to state, since 0 is a (regular)
singular point.
Related function: Sine Integral

CI.2 Series and asymptotic expansions
CI.2.1 Asymptotic expansion at 0.
CI.2.1.1 First terms.

x2   x4   x6   x8
(CI.2.1.1.1)   Ci(x) ≈       −    +    −      − ln(x) + γ . . . .
4    96 4320 322560

CI.2.1.2 General form. The general form of is not easy to state and requires
to exhibit the basis of formal solutions of ?? (coming soon).

CI.2.2 Asymptotic expansion at ∞.
CI.2.2.1 First terms.

RootOf ξ,1 (1+ξ2 )
−           x
Ci(x) ≈ e                                 xy0 (x)+
RootOf ξ,2 (1+ξ2 )
−           x
e                             xy1 (x),
1
2                                  CI COSINE INTEGRAL

where
i    i
y0 (x) =     + RootOf ξ,1 1 + ξ 2 x+
2 2
i   5i                    2
+ RootOf ξ,1 1 + ξ 2       x2 +   i RootOf ξ,1 1 + ξ 2 +
4   4

i  5i                     2
4 RootOf ξ,1 1 + ξ 2     + RootOf ξ,1 1 + ξ 2            x3 + 2 . . .
4  4
−i  i
y1 (x) =      − RootOf ξ,2 1 + ξ 2 x−
2   2
−i 5i                         2
−     − RootOf ξ,2 1 + ξ 2           x2 − − −i RootOf ξ,2 1 + ξ 2 +
4   4

−i 5i                         2
4 RootOf ξ,2 1 + ξ 2     − RootOf ξ,2 1 + ξ 2             x3 +
4  4
2...
CI.2.2.2 General form.
CI.2.2.2.1 Auxiliary function y0 (x). The coeﬃcients u(n) of y0 (x) satisfy the
following recurrence
2u(n)n + u(n − 1) −2 RootOf ξ,1 1 + ξ 2 −

5 RootOf ξ,1 1 + ξ 2 (n − 1) − 3(n − 1)2 RootOf ξ,1 1 + ξ 2         +
u(n − 2) 8 − 5n − 4(n − 2)2 − (n − 2)3 = 0
whose initial conditions are given by
i
u(1) = RootOf ξ,1 1 + ξ 2
2
i
u(0) =
2
CI.2.2.2.2 Auxiliary function y1 (x). The coeﬃcients u(n) of y1 (x) satisfy the
following recurrence
2u(n)n + u(n − 1) −2 RootOf ξ,2 1 + ξ 2 −

5 RootOf ξ,2 1 + ξ 2 (n − 1) − 3(n − 1)2 RootOf ξ,2 1 + ξ 2         +
u(n − 2) 8 − 5n − 4(n − 2)2 − (n − 2)3 = 0
whose initial conditions are given by
i
u(1) = − RootOf ξ,2 1 + ξ 2
2
−i
u(0) =
2

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