auto correct button and 1 to inﬁnity (4.12.02)

A. Prashanth

I wish to bring the following two points to the notice of the maple user community. please elucidate what these observations of mine imply (from a maple viewpoint and also from the theory viewpoint).

ﬁrst, maple outputs unity when i launch the command:

> 1^infinity;

while analysis tells us this entity is actually an indeterminate form.

second, when i use the ’auto correct the syntax of the expression’ button, the resulting syntax produces an error as its not an acceptable maple syntax as the following commands list shows:

produces a correct answer; but when the command at the prompt is incorrect:

and i use the auto correct button, maple 7 produces the following which is incorrect syntax as far as the series and BesselJ functions of the package are concerned:

Carl Devore

[ first, maple outputs unity ...

Given a function f(x) deﬁned on the real numbers, it is reasonable to deﬁne f(infinity) = limit(f(x), x= infinity) if this limit exists (or is +/- inﬁnity). In this case, f(x) = 1^x. It is reasonable because it gives the *unique* (not indeterminate) continuous extension of the function over the compactiﬁcation.

When a math book says that 1^infinity is an indeterminate form, they mean that if f(x) or g(x) are *unknown* functions, except that we know limit(f(x), x=a) = 1 and limit(g(x), x=a) = infinity, then we still don’t have enough information to compute limit(f(x)^g(x), x= a).

If both f(x) and g(x) are known, then to say that the answer is indeterminate is simply a failure to give an answer.

[ second, when i use the 'auto correct' ...

It seems reasonable to me to expect an auto-correct feature to attempt to balance your parentheses, which is what is happening here. I think it is asking too much for it to check the number of arguments in each function call. Consider all that could happen:

- Procedures can have variable number of arguments.

- You could have redefined BesselJ to take three arguments.

- The number of arguments needed could be a function of the first argument.

You need to make a distinction between syntax and semantics. series(BesselJ(3,x,x)) is correct Maple syntax.

Stephen Forrest (4.12.02)

[ first, maple outputs unity ...

In what sense is this an indeterminate form? If you consider a sequence of complex numbers $$\{a_n\}$$ going to inﬁnity along any path, then $$1^(a_n)=1$$ for any $$n$$. So the limit is deﬁned and equal to 1.

As an example of something that _is_ an indeterminate form, take $$(-1)^\infty$$, for which Maple returns undefined + undefined*I.

[second, when i use the 'auto correct' ...

In the ﬁrst case, with "series(BesselJ(3,x,x)", the error was a missing parenthesis. In the second case, the error was an omitted argument.

It’s too hard for Maple to check that the number of arguments to a function is correct, so it doesn’t try. It just makes sure that all the parentheses match, that you have an ”end if” for every ”if”, etc.

Laurent Bernardin (5.12.02)

While Maple returns 1 by default for $$1^\infty$$

However, the fact that this is an invalid operation has been detected:

As with all numeric events, the default behaviour can be changed by installing a diﬀerent event handler:

(This is in Maple 6 and higher)

Robert Israel (5.12.02)

[first, maple outputs unity ...

Not a bug. In general, Maple is literal-minded, and you should not use 1 or inﬁnity (or, for that matter, any other constant) when what you mean is some quantity that is approaching that constant. If what you want is a limit, then use the ”limit” function.

[ second, when i use the 'auto correct' ...

In fact the syntax of the command

is correct, as far as Maple’s parser is concerned; it just happens that the BesselJ function in Maple 7 does not allow 3 arguments. I would call this a problem of semantics rather than of syntax, since in Maple it is in general impossible to tell how many arguments a function will accept without looking at the code.