Talk:arccoth
Misnomers for the Inverse Hyperbolic Functions
[edit]There exists a lot of confusion as to what the actual names of the inverse of the hyperbolic functions are. For example, abbreviations such as Mathematica's ArcSinh and Maple's arcsinh commands are quite simply misnomers--indeed, mistakes--brought about due to inattention to proper terminology. I'm not saying they originated these modern corruptions, but they are helping to perpetuate them.
Using "arc" for the inverse hyperbolic functions--in analogy with arc sine, etc.--is simply an error and is mathematically incorrect. These functions do not describe an arc as do the inverse trigonometric functions--rather, they indeed decribe an area. The great irony is that using "area" for the inverse trigonometric functions would be correct. To quote Simo K. Kivelä of the Helsinki University of Technology Institute of Mathematics:
The latin names of the functions are 'area sinus hyperbolicus' etc. where 'area' refers to the area of a sector bounded by the unit hyperbola. In the trigonometric case, 'arc' is correct because the value of the function represents the length of an arc. (It could also be considered as area of a sector and therefore, 'ar' would in principle be correct also here, but it has never been used.) In the hyperbolic case, there is no arc, and the use of 'arc' should be considered as a mistake.[1]
See also the discussion on this at Wikipedia: [2].
Because this is a dictionary (and this applies also to Wikipedia), it's important that we get these terms right. It might be alright to have entries on misnomers such as "arcsinh," but at least the entry should point out that it is a misnomer and corruption of area hyperbolic sine, from the original Latin area sinus hyperbolicus.
To that end, below is a list of all of these functions, along with their proper names and their original Latin names. Listed first are the trigonometric functions, then the hyperbolic functions. Listed also are one (or sometimes two) correct abbreviations (keeping in mind that there are a number of ways to correctly abbreviate them; although, as I pointed out, "arc" in the inverse hyperbolic functions isn't correct). Further listed are the Maxima commands for these functions (which don't make the mistake of using "arc" for the inverse hyperbolic functions). About half of these commands also work in PARI/GP, with PARI/GP not having commands for the rest of the functions (although the one command difference is that cot is cotan in PARI/GP).
Trigonometric Functions:
Name | Abbreviation | Maxima Command
sine | sin | sin
cosine | cos | cos
tangent | tan | tan
cotangent | cot | cot
secant | sec | sec
cosecant | csc (or cosec) | csc
----------
Inverse Trigonometric Functions:
Name | Abbreviation | Maxima Command
arc sine | arcsin | asin
arc cosine | arccos | acos
arc tangent | arctan | atan
arc cotangent | arccot | acot
arc secant | arcsec | asec
arc cosecant | arccsc (or arccosec) | acsc
The original Latin names, respectively:
arcus sinus
arcus cosinus
arcus tangens
arcus cotangens
arcus secans
arcus cosecans
##########
Hyperbolic Functions:
Name | Abbreviation | Maxima Command
hyperbolic sine | sinh | sinh
hyperbolic cosine | cosh | cosh
hyperbolic tangent | tanh | tanh
hyperbolic cotangent | coth | coth
hyperbolic secant | sech | sech
hyperbolic cosecant | csch (or cosech) | csch
----------
Inverse Hyperbolic Functions:
Name | Abbreviation | Maxima Command
area hyperbolic sine | arsinh | asinh
area hyperbolic cosine | arcosh | acosh
area hyperbolic tangent | artanh | atanh
area hyperbolic cotangent | arcoth | acoth
area hyperbolic secant | arsech | asech
area hyperbolic cosecant | arcsch (or arcosech) | acsch
The original Latin names, respectively:
area sinus hyperbolicus
area cosinus hyperbolicus
area tangens hyperbolicus
area cotangens hyperbolicus
area secans hyperbolicus
area cosecans hyperbolicus