The arguments for value of 0^0 , at least in isolation, are presented in
http://mathforum.org/dr.math/faq/faq.0.to.0.power.html
where the value 1 is suggested.
Sometimes a distinction is made between 0.0^x and 0^x and between
types x=integer and x=real. (see link).
Dieter Kaiser wrote:
> I think the simplification of 0^a is not implemented as complete as
> possible. This might be a complete definition:
>
> (1) 0^a -> 0 for realpart(a) > 0
> (2) 0^a -> infinity for realpart(a) < 0
> (3) 0^a -> und for realpart(a) = 0
>
> For a number all is correct. Case (1) gives zero and the cases (2) and
> (3) give a Maxima error.
>
> But for a symbol or a symbolic expression we always get zero as a
> result. We assume a to be a positive real value.
>
> (%i1) declare(z,complex,j,imaginary)$
> (%i2) assume(a>0)$
>
> (%i3) 0^a;
> (%o3) 0
>
> (%i4) 0^-a; /* This expression is known to be negative */
> (%o4) 0
>
> Furthermore, Maxima does not look at the realpart of the exponent. The
> following should simplify to zero:
>
> (%i5) 0^(a+%i); /* The symbol %i generates an error */
> 0 to a complex quantity has been generated.
> -- an error. To debug this try: debugmode(true);
>
> (%i6) 0^(a+j); /* A symbol declared to be imaginary gives no error */
> (%o6) 0
>
> (%i8) 0^-(a+j); /* Not correct */
> (%o8) 0
>
> (%i9) 0^-(a+z); /* Not correct */
> (%o9) 0
>
>
> I think the code in simpexpt.lisp which handles the above cases has to
> look more carefully at the sign of the expression.
>
> There is a problem if Maxima can not deduce the sign. We have three
> options:
>
> (1) Go on with the assumptions that the expression is positive.
> Now, Maxima does this for all symbolic expressions. Maxima might
> give
> incorrect answers.
> (2) Generate a Maxima Error if the sign is not known.
> Maxima gives only correct answers.
> (3) Stop and ask the User for the sign.
> I think it is very uncomfortable to stop the execution.
>
> I have tried an implementation for the case (2). These are the results:
>
> (%i1) declare(j,imaginary,z,complex)$
> (%i2) assume(a>0)$
>
> The sign of x is not known. A Maxima error:
>
> (%i3) 0^x;
> 0^(x) has been generated. The sign of x is not known.
> -- an error. To debug this try: debugmode(true);
>
> Now a positive sign:
>
> (%i4) 0^a;
> (%o4) 0
>
> An error if the exponent is negative:
>
> (%i5) 0^-a;
> Division by 0
> -- an error. To debug this try: debugmode(true);
>
> The realpart is positive. Again zero as a result:
>
> (%i6) 0^(a+%i*y);
> (%o6) 0
>
> A negative realpart gives a Maxima error:
>
> (%i7) 0^(-a+%i*y);
> 0 to a complex quantity has been generated.
> -- an error. To debug this try: debugmode(true);
>
> The example with a positive number as realpart:
>
> (%i8) 0^(2+%i*y);
> (%o8) 0
>
> This is an example which is generated by the code of the confluent
> hypergeomtric function:
>
> (%i5) hgfred([x], [], 1);
> 0^(- x) has been generated. The sign of - x is not known.
> -- an error. To debug this try: debugmode(true);
>
> a is assumed to be positive:
>
> (%i6) hgfred([a], [], 1);
> Division by 0
> -- an error. To debug this try: debugmode(true);
>
> (%i7) hgfred([-a], [], 1);
> (%o7) 0
>
> The testsuite will generate only three errors. It is interesting that
> these errors are related to the sqrt function.
>
> Dieter Kaiser
>
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