SN Sine

SN.1 Introduction

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

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

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

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

Related function: Cosine

SN.2 Series and asymptotic expansions

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

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

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

\begin{aligned}sin(x) & = x − \frac{1}{6}x^{3} + \frac{1}{120}x^{5} − \frac{1}{5040}x^{7} + \frac{1}{362880}x^{9} − \frac{1}{39916800}x^{11}+ \\ & \frac{1}{6227020800}x^{13} − \frac{1}{1307674368000}x^{15} + O(x^{16}). \end{aligned} ; (SN.2.2.1.1)

SN.2.2.2 General form.

\begin{aligned}& sin(x) & = ∑_{n=0}^{∞} u(n)x^{n}. \\ (SN.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. \\ (SN.2.2.2.2) \end{aligned}

Initial conditions of SN.2.2.2.2 are given by

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

2 SN SINE

The recurrence SN.2.2.2.2 has the closed form solution

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

SN.3 Graphs

SN.3.1 Real axis.

SN.3.2 Complex plane.