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ttw · 04-30-2017, 12:35 AM

There is an important distinction in the symbols 0^0, 0^0.0, 0.0^0 and 0.0^0.0. The first case (integer ^ integer) is the easiest. This should be 1. This the only value that shows up in combinatorics or other algebraic expressions. I have never seen a case where defining 0^0 to be other than 1 is a problem.

The second case (integer ^ real ) is tricky. I have never come across this case in "real life": it's only turned up in questions about computer arithmetic defaults or in academic examples. I'd probably make this one undefined. The difference between the cases is that x^integer can be reduced to repeated multiplication (where the repeat factor could be 0 or 1); the second case cannot. The power function over reals is defined in terms or principle branch cuts log functions. Log(0) (as either an integer or real) is undefined ( or - infinity which doesn't help except to delay error interrupts.) The real power function a^b is defined as Exp(b*log(a)).

The third case (real^integer) could be defined as 1.0 with no problem; the integer exponent 0 can be interpreted as an empty product and thus becomes 1.0 (because the things being multiplied are real). This definition fits smoothly into algebraic work. It does not fit the b*log(a) stuff as well.

The fourth case (real^real) is best defaulted to undefined. The b*log(a) has no nice interpretation as a and b approach zero along various paths. This one occurs now and then.

The cases with 0 exponent are easy; defining them to be 1 or 1.0 causes no problems (at least that I have seen and I looked for problems in for this expression.) It's equivalent to defining the sum of an empty list of numbers as zero or the product of an empty list as one. With these definitions, the commutative and associative laws continue to hold.

The cases with 0.0 as an exponent are troublesome. Using the exponential with complex numbers is interesting as one gets (an infinite number) of branch cuts. The limits of the exponential function with a complex number approaching 0 isn't well defined.