Equivalence of inverse hyperbolic functions and logarithms

maxima-bounces at math.utexas.edu wrote on 12/08/2008 02:42:04 PM:


> There is an equivalence between inverse hyperbolic functions and 
logarithms,
> eg:
> 
> arcsinh(x) = ln(x + sqrt(1 + x^2))
> 
> Is there a way to tell Maxima use one form over the other
> when calculating? Say when integrating ? So that the
> result is expressed in terms of inverse hyperbolic functions
> or of logarithms ?

Yes, the option variable logarc controls this; to read the user 
documentation
on logarg, type "? logarc" on a command line; a few examples:

Globally set logarc to true (asinh is the Maxima name for the inverse
hyperbolic sine function)

 (%i1) logarc : true$

 (%i2) asinh(x);
 (%o2) log(sqrt(x^2+1)+x)

 (%i3) logarc : false$
 (%i4) asinh(x);

There is also a function logarc

 (%o4) asinh(x)
 (%i5) logarc(%);

 (%o5) log(sqrt(x^2+1)+x)

If this isn't enough, you can locally set logarc to true for a single 
command:

 (%i6) asinh(x), logarc;
 (%o6) log(sqrt(x^2+1)+x)

> I know it's trivial to implement it, as a simplifier,
> just wondering whether there exists something already.

A nice general way to do things like is to substitute a lambda form for
a function:

 (%i8) subst('asinh = lambda([s], log(s + sqrt(1+s^2))), asinh(x + 42));
 (%o8) log(sqrt((x+42)^2+1)+x+42)

You can even do conditional substitutions.

Welcome to Maxima.

Barton

P.S. I would be great if our user documentation had a search mechanism
that could find things like this: for example: a search on "how to I
convert the inverse hyperbolics to log form" would find "logarc." The
Macsyma help system could do something like this. Many users will
guess "diff" and "int", but few will guess "logarc."