CS Cosine

CS.1 Introduction

Let x be a complex variable of ℂ \ {∞}.The function Cosine (noted cos) is defined by the following second order differential equation

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

The initial conditions of CS.1.1 are given at 0 by

\begin{aligned}cos(0) & = 1, \\ \frac{∂ cos(x)}{∂x}(0) & = 0. \end{aligned} (CS.1.2)

Related function: Sine

CS.2 Series and asymptotic expansions

CS.2.1 Asymptotic expansion at ∞. CS.2.1.1 Exact form.

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

CS.2.2 Taylor expansion at 0. CS.2.2.1 First terms.

\begin{aligned}cos(x) & = 1 − \frac{1}{2}x^{2} + \frac{1}{24}x^{4} − \frac{1}{720}x^{6} + \frac{1}{40320}x^{8} − \frac{1}{3628800}x^{10}+ \\ & \frac{1}{479001600}x^{12} − \frac{1}{87178291200}x^{14} + O(x^{16}). \end{aligned} ; (CS.2.2.1.1)

CS.2.2.2 General form.

\begin{aligned}& cos(x) & = ∑_{n=0}^{∞} u(n)x^{n}. \\ (CS.2.2.2.1) \end{aligned}

The coefficients u(n) satisfy the recurrence

\begin{aligned}& u(n) + (n^{2} + 3n + 2)u(n + 2) & = 0. \\ (CS.2.2.2.2) \end{aligned}

Initial conditions of CS.2.2.2.2 are given by

\begin{aligned}u(1) & = 0, \\ u(0) & = 1. \end{aligned} ; (CS.2.2.2.3)

2 CS COSINE

The recurrence CS.2.2.2.2 has the closed form solution

\begin{aligned}u(2n + 1) & = 0, \\ u(2n) & = \frac{(−1)^{n}}{Γ(2n + 1)}. \end{aligned} ; (CS.2.2.2.4)

CS.3 Graphs

CS.3.1 Real axis.

CS.3.2 Complex plane.