FullSimplify[ArcCos[x]+I ArcCosh[x]]

Questions
1

Hi!
My Mathematica does not understand that

FullSimplify[ArcCos[x]+I ArcCosh[x]]

is zero. How can I make simplifications like this? Or it is just I am using older version, isn’t it, and a new version already works with such expressions, does it?
Best,
Alex

2

Because in general it is not:

ArcCos[x] + I ArcCosh[x]
% /. x -> -1.0 - 1.0 I

gives:

4.47407 + 2.12255 I
3

Only works in range -Infinity < x < 1

FullSimplify[ArcCos[x] + I ArcCosh[x], Assumptions -> -Infinity < x < 1]
(*  0  *)
Plot[ArcCos[x] + I ArcCosh[x], {x, -2, 2}, PlotRange -> {{-2, 2}, {-1, 1}}, 
PlotStyle -> {Black, Thickness[0.02]}]

aqaq.png
aqaq.png379×237 1.13 KB

  NumberLinePlot[{-Infinity < x < 1}, {x, -2, 2}, PlotTheme -> "Detailed"]

lala.png

4

Thank you, I understand now. -1<x<1 helped for my problem. Thanks again!