HSN Hyperbolic Sine

HSN.1 Introduction

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

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

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

sinh(0) = 0,
\frac{∂ sinh(x)}{∂x}(0) = 1.

(HSN.1.2)

Related function: Hyperbolic Cosine

HSN.2 Series and asymptotic expansions

HSN.2.1 Taylor expansion at 0. HSN.2.1.1 First terms.

\begin{aligned}sinh(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} (HSN.2.1.1.1)

HSN.2.1.2 General form.

\begin{aligned}& sinh(x) & = ∑_{n=0}^{∞} u(n)x^{n}. \\ (HSN.2.1.2.1) \end{aligned}

The coefficients u(n) satisfy the recurrence

\begin{aligned}& −u(n) + (n^{2} + 3n + 2)u(n + 2) & = 0. \\ (HSN.2.1.2.2) \end{aligned}

Initial conditions of HSN.2.1.2.2 are given by

\begin{aligned}u(0) & = 0, \\ u(1) & = 1. \end{aligned} ; (HSN.2.1.2.3)

The recurrence HSN.2.1.2.2 has the closed form solution

\begin{aligned}u(2n + 1) & = \frac{1}{Γ(2n + 2)}, \\ u(2n) & = 0. \end{aligned} ; (HSN.2.1.2.4)

HSN.2.2 Asymptotic expansion at ∞.

2 HSN HYPERBOLIC SINE

HSN.2.2.1 Exact form.

sinh(x) = \frac{e^{x}}{2} − \frac{e^{(−x)}}{2} .

HSN.3 Graphs

HSN.3.1 Real axis.

HSN.3.2 Complex plane.