What is the correct graph of arccot x?

[Murray]

Hi all

The recent discussion about the way math software interprets 2x/2y and 2*x/2*y reminded me of this issue.

What is the correct graph of arccot x?

Mathematica, Matlab and Mathcad do not all see eye to eye about this. Here's what I found (back in May 2011):

http://www.intmath.com/blog/which-is-the-correct-graph-of-arccot-x/6009

Regards

Murray

[mok...@earthtreasury.org]

Many things in mathematics are matters of choice and definition, not
proof, correctness, and truth. This is one of them.

The arccot function is not a single function, and thus does not have a
"correct" graph. Taken just as the inverse of cot, it is
infinitely-valued, and can be redefined as a single-valued function with
an infinite set of different choices of "principal value". Each choice has
different properties that one might reasonably prefer for a particular
purpose. Choosing a range that does not include a pole, so that the
function is continuous, has a value. Defining all of the trigonometric
inverses to have the same range, regardless of poles, has a value,
although it means that they will not all be single-valued. Using a range
centered on 0, or with one edge at 0, has a value. It depends on your
purpose. So

I am right, and you are right, and all is right as right can be.

One widely-accepted set of definitions for the complex trig functions is
in the ANSI/ISO APL programming language standard, which I participated
in, both in the standards process and the first full implementation of the
standard in I-APL. These definitions have been taken over whole in Ada,
Common LISP, and elsewhere. A full understanding of the reasons for these
definitions requires a grasp of the Riemann surface structure of the
functions and their inverses in the complex plane.

If anyone wants to go into that further, I will be happy to oblige. With
pointers to pictures.

On Tue, January 29, 2013 3:18 am, Murray wrote:
> Hi all
>
> The recent discussion about the way math software interprets 2x/2y and
> 2*x/2*y reminded me of this issue.

This is a well-known problem with the vagaries of function precedence in
different programming languages and math software. What you got is the
equivalent of

(2x)/(2y), which is the same as x/y

(2*x/2)*y, which is the same as x*y

When in doubt, parenthesize. I prefer software that has no function
precedence, like APL (strict right to left evaluation) or FORTH (RPN), but
others find them too hard.

Everything should be as simple as possible, but _no simpler_.--Attributed,
apparently incorrectly, to Albert Einstein

> What is the correct graph of arccot x?
>
> Mathematica, Matlab and Mathcad do not all see eye to eye about this.
> Here's what I found (back in May 2011):
>
> http://www.intmath.com/blog/which-is-the-correct-graph-of-arccot-x/6009
>
> Regards
> Murray
>

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