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

\begin{aligned}& x\frac{∂y(x)}{∂x} + 2\frac{∂^{2}y(x)}{∂x^{2}} + x\frac{∂^{3}y(x)}{∂x^{3}} & = 0. \\ (CI.1.1) \end{aligned}

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.

\begin{aligned}& Ci(x) ≈ (\frac{x^{2}}{4} − \frac{x^{4}}{96} + \frac{x^{6}}{4320} − \frac{x^{8}}{322560} − ln(x) + γ …). \\ (CI.2.1.1.1) \end{aligned}

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.

Ci(x) ≈ e^{(− \frac{RootOf_{ξ,1}(1+ξ^{2})}{x})} xy_{0}(x)+

e^{(− \frac{RootOf_{ξ,2}(1+ξ^{2})}{x})} xy_{1}(x),

where

y_{0}(x) = \frac{i}{2} + \frac{i}{2} RootOf_{ξ,1}(1 + ξ^{2})x+

(\frac{i}{4} + \frac{5i}{4} RootOf_{ξ,1}(1 + ξ^{2})^{2})x^{2} + (i RootOf_{ξ,1}(1 + ξ^{2})+

4 RootOf_{ξ,1}(1 + ξ^{2})(\frac{i}{4} + \frac{5i}{4} RootOf_{ξ,1}(1 + ξ^{2})^{2}))x^{3} + 2 …

y_{1}(x) = \frac{−i}{2} − \frac{i}{2} RootOf_{ξ,2}(1 + ξ^{2})x−

− (\frac{−i}{4} − \frac{5i}{4} RootOf_{ξ,2}(1 + ξ^{2})^{2})x^{2} − −(−i RootOf_{ξ,2}(1 + ξ^{2})+

4 RootOf_{ξ,2}(1 + ξ^{2})(\frac{−i}{4} − \frac{5i}{4} RootOf_{ξ,2}(1 + ξ^{2})^{2}))x^{3}+

2 …

CI.2.2.2 General form. CI.2.2.2.1 Auxiliary function y_{0}(x). The coefficients u(n) of y_{0}(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

\begin{aligned}u(1) & = \frac{i}{2} RootOf_{ξ,1}(1 + ξ^{2}) \\ u(0) & = \frac{i}{2} \end{aligned}

CI.2.2.2.2 Auxiliary function y_{1}(x). The coefficients u(n) of y_{1}(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

\begin{aligned}u(1) & = −\frac{i}{2} RootOf_{ξ,2}(1 + ξ^{2}) \\ u(0) & = \frac{−i}{2} \end{aligned}