AHCS Inverse Hyperbolic Cosecant

AHCS.1 Introduction

Let x be a complex variable of ℂ\{0}.The function Inverse Hyperbolic Cosecant (noted arccsch) is defined by the following second order differential equation

\begin{aligned}& (2x^{2} + 1)\frac{∂y(x)}{∂x} + (x^{3} + x)\frac{∂^{2}y(x)}{∂x^{2}} & = 0. \\ (AHCS.1.1) \end{aligned}

The initial conditions of AHCS.1.1 at 0 are not simple to state, since 0 is a (regular) singular point.

AHCS.2 Series and asymptotic expansions

AHCS.2.1 Asymptotic expansion at −i.

2 AHCS INVERSE HYPERBOLIC COSECANT

AHCS.2.1.1 First terms.

arccsch(x) ≈ (\frac{i}{2}π …) + \sqrt{x − RootOf_{ξ,2}(1 + ξ^{2})}(1 − i+

(\frac{5}{12} − \frac{5i}{12}) RootOf_{ξ,2}(1 + ξ^{2})(x − RootOf_{ξ,2}(1 + ξ^{2})) + (1 − i)

(\frac{3}{40} + \frac{11 RootOf_{ξ,2}(1 + ξ^{2})^{2}}{32} ) (x − RootOf_{ξ,2}(1 + ξ^{2}))^{2}+

(1 − i)(\frac{25 RootOf_{ξ,2}(1 + ξ^{2})}{336}+

\frac{17 RootOf_{ξ,2}(1 + ξ^{2})(12 + 55 RootOf_{ξ,2}(1 + ξ^{2})^{2})}{2688})

(x − RootOf_{ξ,2}(1 + ξ^{2}))^{3} + (1 − i)(\frac{7}{384}+

\frac{385 RootOf_{ξ,2}(1 + ξ^{2})^{2}}{4608} +

\frac{23 RootOf_{ξ,2}(1 + ξ^{2})^{2}(404 + 935 RootOf_{ξ,2}(1 + ξ^{2})^{2})}{55296})

(x − RootOf_{ξ,2}(1 + ξ^{2}))^{4} + (1 − i)(\frac{15 RootOf_{ξ,2}(1 + ξ^{2})}{704}+

\frac{51 RootOf_{ξ,2}(1 + ξ^{2})(12 + 55 RootOf_{ξ,2}(1 + ξ^{2})^{2})}{28160} + 29 RootOf_{ξ,2}(1 + ξ^{2})

(1008 + 13912 RootOf_{ξ,2}(1 + ξ^{2})^{2}+ 21505 RootOf_{ξ,2}(1 + ξ^{2})^{4})/1351680)

(x − RootOf_{ξ,2}(1 + ξ^{2}))^{5} + (1 − i)(\frac{77}{13312}+

\frac{4235 RootOf_{ξ,2}(1 + ξ^{2})^{2}}{159744} +

\frac{253 RootOf_{ξ,2}(1 + ξ^{2})^{2}(404 + 935 RootOf_{ξ,2}(1 + ξ^{2})^{2})}{1916928}+

7 RootOf_{ξ,2}(1 + ξ^{2})^{2}(87408 + 538088 RootOf_{ξ,2}(1 + ξ^{2})^{2}+ 623645 RootOf_{ξ,2}(1 + ξ^{2})^{4})/7667712)

(x − RootOf_{ξ,2}(1 + ξ^{2}))^{6} + (1 − i)(\frac{13 RootOf_{ξ,2}(1 + ξ^{2})}{1792}+

\frac{221 RootOf_{ξ,2}(1 + ξ^{2})(12 + 55 RootOf_{ξ,2}(1 + ξ^{2})^{2})}{358400} + 377 RootOf_{ξ,2}(1 + ξ^{2})

(1008 + 13912 RootOf_{ξ,2}(1 + ξ^{2})^{2}+ 21505 RootOf_{ξ,2}(1 + ξ^{2})^{4})/51609600 + 41 RootOf_{ξ,2}(1 + ξ^{2})( 44352 + 1223984 RootOf_{ξ,2}(1 + ξ^{2})^{2}+ 4712836 RootOf_{ξ,2}(1 + ξ^{2})^{4} + 4365515 RootOf_{ξ,2}(1 + ξ^{2})^{6})/

247726080)(x − RootOf_{ξ,2}(1 + ξ^{2}))^{7} + (1 − i)(\frac{1155}{557056}+

\frac{21175 RootOf_{ξ,2}(1 + ξ^{2})^{2}}{2228224} +

\frac{1265 RootOf_{ξ,2}(1 + ξ^{2})^{2}(404 + 935 RootOf_{ξ,2}(1 + ξ^{2})^{2})}{26738688}

+ 35 RootOf_{ξ,2}(1 + ξ^{2})^{2}(87408+ 538088 RootOf_{ξ,2}(1 + ξ^{2})^{2} + 623645 RootOf_{ξ,2}(1 + ξ^{2})^{4})/ 106954752 + 47 RootOf_{ξ,2}(1 + ξ^{2})^{2}(36363456+ 418800176 RootOf_{ξ,2}(1 + ξ^{2})^{2}+ 1160708620 RootOf_{ξ,2}(1 + ξ^{2})^{4}+ 894930575 RootOf_{ξ,2}(1 + ξ^{2})^{6})/44920995840) (x − RootOf_{ξ,2}(1 + ξ^{2}))^{8} …).

(AHCS.2.1.1.1)

AHCS.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).

AHCS.2.2 Asymptotic expansion at i.

4 AHCS INVERSE HYPERBOLIC COSECANT

AHCS.2.2.1 First terms.

arccsch(x) ≈ (−\frac{i}{2}π …) + \sqrt{x − RootOf_{ξ,1}(1 + ξ^{2})}(1 + i+

(\frac{5}{12} + \frac{5i}{12}) RootOf_{ξ,1}(1 + ξ^{2})(x − RootOf_{ξ,1}(1 + ξ^{2})) + (1 + i)

(\frac{3}{40} + \frac{11 RootOf_{ξ,1}(1 + ξ^{2})^{2}}{32} ) (x − RootOf_{ξ,1}(1 + ξ^{2}))^{2}+

(1 + i)(\frac{25 RootOf_{ξ,1}(1 + ξ^{2})}{336}+

\frac{17 RootOf_{ξ,1}(1 + ξ^{2})(12 + 55 RootOf_{ξ,1}(1 + ξ^{2})^{2})}{2688})

(x − RootOf_{ξ,1}(1 + ξ^{2}))^{3} + (1 + i)(\frac{7}{384}+

\frac{385 RootOf_{ξ,1}(1 + ξ^{2})^{2}}{4608} +

\frac{23 RootOf_{ξ,1}(1 + ξ^{2})^{2}(404 + 935 RootOf_{ξ,1}(1 + ξ^{2})^{2})}{55296})

(x − RootOf_{ξ,1}(1 + ξ^{2}))^{4} + (1 + i)(\frac{15 RootOf_{ξ,1}(1 + ξ^{2})}{704}+

\frac{51 RootOf_{ξ,1}(1 + ξ^{2})(12 + 55 RootOf_{ξ,1}(1 + ξ^{2})^{2})}{28160} + 29 RootOf_{ξ,1}(1 + ξ^{2})

(1008 + 13912 RootOf_{ξ,1}(1 + ξ^{2})^{2}+ 21505 RootOf_{ξ,1}(1 + ξ^{2})^{4})/1351680)

(x − RootOf_{ξ,1}(1 + ξ^{2}))^{5} + (1 + i)(\frac{77}{13312}+

\frac{4235 RootOf_{ξ,1}(1 + ξ^{2})^{2}}{159744} +

\frac{253 RootOf_{ξ,1}(1 + ξ^{2})^{2}(404 + 935 RootOf_{ξ,1}(1 + ξ^{2})^{2})}{1916928}+

7 RootOf_{ξ,1}(1 + ξ^{2})^{2}(87408 + 538088 RootOf_{ξ,1}(1 + ξ^{2})^{2}+ 623645 RootOf_{ξ,1}(1 + ξ^{2})^{4})/7667712)

(x − RootOf_{ξ,1}(1 + ξ^{2}))^{6} + (1 + i)(\frac{13 RootOf_{ξ,1}(1 + ξ^{2})}{1792}+

\frac{221 RootOf_{ξ,1}(1 + ξ^{2})(12 + 55 RootOf_{ξ,1}(1 + ξ^{2})^{2})}{358400} + 377 RootOf_{ξ,1}(1 + ξ^{2})

(1008 + 13912 RootOf_{ξ,1}(1 + ξ^{2})^{2}+ 21505 RootOf_{ξ,1}(1 + ξ^{2})^{4})/51609600 + 41 RootOf_{ξ,1}(1 + ξ^{2})( 44352 + 1223984 RootOf_{ξ,1}(1 + ξ^{2})^{2}+ 4712836 RootOf_{ξ,1}(1 + ξ^{2})^{4} + 4365515 RootOf_{ξ,1}(1 + ξ^{2})^{6})/

247726080)(x − RootOf_{ξ,1}(1 + ξ^{2}))^{7} + (1 + i)(\frac{1155}{557056}+

\frac{21175 RootOf_{ξ,1}(1 + ξ^{2})^{2}}{2228224} +

\frac{1265 RootOf_{ξ,1}(1 + ξ^{2})^{2}(404 + 935 RootOf_{ξ,1}(1 + ξ^{2})^{2})}{26738688}

+ 35 RootOf_{ξ,1}(1 + ξ^{2})^{2}(87408+ 538088 RootOf_{ξ,1}(1 + ξ^{2})^{2} + 623645 RootOf_{ξ,1}(1 + ξ^{2})^{4})/ 106954752 + 47 RootOf_{ξ,1}(1 + ξ^{2})^{2}(36363456+ 418800176 RootOf_{ξ,1}(1 + ξ^{2})^{2}+ 1160708620 RootOf_{ξ,1}(1 + ξ^{2})^{4}+ 894930575 RootOf_{ξ,1}(1 + ξ^{2})^{6})/44920995840) (x − RootOf_{ξ,1}(1 + ξ^{2}))^{8} …).

(AHCS.2.2.1.1)

AHCS.2.2.2 General form. The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).

AHCS.2.3 Asymptotic expansion at 0. AHCS.2.3.1 First terms.

arccsch(x) ≈ (ln(2) − \frac{x^{2}}{4} + \frac{3x^{4}}{32} − \frac{5x^{6}}{96} + \frac{35x^{8}}{1024} + ln(x) …).

(AHCS.2.3.1.1)

AHCS.2.3.2 General form. The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).