Docstoc

CI Cosine Integral

Document Sample
CI Cosine Integral Powered By Docstoc
					                             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

				
DOCUMENT INFO