$SPAD/src/input asec.input

Timothy Daly December 3, 2016

Abstract

Contents

2

1 ASC.1 Introduction

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

(2x^{2} − 1) \frac{∂y(x)}{∂x} + (x^{3} − x) \frac{∂^{2}y(x)}{∂x^{2}} = 0.

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

2 ASC.2 Series and asymptotic expansions

2.1 ASC.2.1 Asymptotic expansion at −1

2.1.1 ASC.2.1.1 First terms

asec(x) ≈ (π …) + \sqrt{x + 1} (−i\sqrt{2} − \frac{5i}{12}(x + 1)\sqrt{2} − \frac{43i}{160}(x + 1)^{2}\sqrt{2}

− \frac{177i}{896}(x + 1)^{3}\sqrt{2} − \frac{2867i}{18432}(x + 1)^{4}\sqrt{2}−
\frac{11531i}{90112}(x + 1)^{5}\sqrt{2} − \frac{92479i}{851968}(x + 1)^{6}\sqrt{2}−
\frac{74069i}{786432}(x + 1)^{7}\sqrt{2} − \frac{11857475i}{142606336}(x + 1)^{8}\sqrt{2} …) .

2.2 ASC.2.1 Asymptotic expansion at 0

2.2.1 ASC.2.2.1 First terms

\begin{matrix}asec(x) ≈ \\ (iln(2) + \frac{i}{4}x^{2} + \frac{3i}{32}x^{4} + \frac{5i}{96}x^{6} + \frac{35i}{1024}x^{8} + iln(x) …) . \end{matrix}

2.3 ASC.2.3 Asymptotic expansion at 1

2.3.1 ASC.2.3.1 First terms

asec(x) ≈ \sqrt{x − 1} (\sqrt{2} − ^{\frac{5\sqrt{2}(x−1)}{12}} + ^{\frac{43\sqrt{2}(x−1)^{2}}{160}} −

\frac{177\sqrt{2}(x−1)^{3}}{896} + ^{\frac{2867\sqrt{2}(x−1)^{4}}{18432}} − ^{\frac{11531\sqrt{2}(x−1)^{5}}{90112}} +
\frac{92479\sqrt{2}(x−1)^{6}}{851968} − ^{\frac{74069\sqrt{2}(x−1)^{7}}{786432}} +
\frac{11857475\sqrt{2}(x−1)^{8}}{142606336} − ^{\frac{47442055\sqrt{2}(x−1)^{9}}{637534208}} +
\frac{126527543\sqrt{2}(x−1)^{10}}{1879048192} − ^{\frac{1518418695\sqrt{2}(x−1)^{11}}{24696061952}} +
\frac{24295375159\sqrt{2}(x−1)^{12}}{429496729600} − ^{\frac{97182800711\sqrt{2}(x−1)^{13}}{1855425871872}} +
\frac{777467420263\sqrt{2}(x−1)^{14}}{15942918602752} − ^{\frac{3109879375897\sqrt{2}(x−1)^{15}}{68169720922112}} …) .

2.3.2 ASC.2.3.2 General form

asec(x) ≈ \sqrt{x − 1} ∑_{n=0}^{∞} u(n)(x − 1)^{n}

The coefficients u(n) satisfy the recurrence

\begin{matrix}2u(n) (\frac{1}{2} + n) n + u(n − 1) (−\frac{1}{2} + n) (−\frac{1}{2} + 3n) + u(n − 2) (−\frac{3}{2} + n) (−\frac{1}{2} + n) \\ = 0. \end{matrix}

Initial conditions of ASC.2.3.2.2 are given by

\begin{aligned}u(0) & = \sqrt{2}, \\ u(1) & = \frac{−5\sqrt{2}}{12}. \end{aligned}

As implemented within Axiom the asec function is

sec^{−1}(z) == cos^{−1} (\frac{1}{z})

— * —

)spool asec.output
)set message test off
)set message auto off
)set break resume
digits(22)
)clear all
--S 1 of 10
asec(-2.0)
--R
--R   (1)  2.0943951023 9319549230 8
--R                                                                  Type: Float
--E 1
--S 2 of 10
asec(-1.5)
--R
--R   (2)  2.3005239830 2186298268 6
--R                                                                  Type: Float
--E 2
--S 3 of 10
asec(-1.0)
--R
--R   (3)  3.1415926535 8979323846 3
--R                                                                  Type: Float
--E 3
--S 4 of 10
asec(-0.5)
--R
--R   >> Error detected within library code:
--R   acos: argument > 1 in magnitude

--R
--R   Continuing to read the file...
--R
--E 4
--S 5 of 10
asec(-0.0)
--R
--R   >> Error detected within library code:
--R   asec: no reciprocal
--R
--R   Continuing to read the file...
--R
--E 5
--S 6 of 10
asec(0.0)
--R
--R   >> Error detected within library code:
--R   asec: no reciprocal
--R
--R   Continuing to read the file...
--R
--E 6
--S 7 of 10
asec(0.5)
--R
--R   >> Error detected within library code:
--R   acos: argument > 1 in magnitude
--R
--R   Continuing to read the file...
--R
--E 7
--S 8 of 10
asec(1.0)
--R
--R   (4)  0.0
--R                                                                  Type: Float
--E 8
--S 9 of 10
asec(1.5)
--R
--R   (5)  0.8410686705 6793025577 652
--R                                                                  Type: Float
--E 9
--S 10 of 10

asec(2.0)
--R
--R   (6)  1.0471975511 9659774615 42
--R                                                                  Type: Float
--E 10
)spool
)lisp (bye)

———-

References

[1] The Encyclopedia of Special Functions http://algo.inria.fr/esf/function/ASC/ASC.html