HCS Hyperbolic Cosine

HCS.1 Introduction

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

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

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

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

(HCS.1.2)

Related function: Hyperbolic Sine

HCS.2 Series and asymptotic expansions

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

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

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

\begin{aligned}cosh(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} (HCS.2.2.1.1)

HCS.2.2.2 General form.

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

Initial conditions of HCS.2.2.2.2 are given by

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

2 HCS HYPERBOLIC COSINE

The recurrence HCS.2.2.2.2 has the closed form solution

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

HCS.3 Graphs

HCS.3.1 Real axis.

HCS.3.2 Complex plane.