Re: Re: 0^0 = 1?

*To*: mathgroup at smc.vnet.net*Subject*: [mg95857] Re: [mg95796] Re: 0^0 = 1?*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Wed, 28 Jan 2009 06:44:04 -0500 (EST)*References*: <gl7211$c8r$1@smc.vnet.net> <gl9mua$ajr$1@smc.vnet.net> <200901271201.HAA23416@smc.vnet.net>

On 27 Jan 2009, at 13:01, Dave Seaman wrote: > There have been many discussions of this point in sci.math over the > years. The emerging consensus among mathematicians (not necessarily > my > own opinion) seems to be that it makes sense to observe that very > distinction, > namely, that 0^0 = 1 when the exponent is an integer, but that the > expression should be left undefined when the exponent is real or > complex. > > There is also considerable precedent in various programming > languages for > doing this. In fact, the value of 0^0 (or 0**0 or pow(0,0), or > whatever > the notation may be) is often defined to be 1 of the appropriate > base type, > provided the exponent is an integer, and undefined otherwise. Since you must know perfectly well that Mathematica does not distinguish between the integer 0 and the complex number 0 your entire argument is spurious in the context of the Mathematica language. You can't seriously expect that Mathematica should be completely rewritten to accommodate you on this. > >>> > >> One place I think it would it would cause problems is in >> Mathematica Limit >> and Series code. Things that do not evaluate to Indeterminate are >> often >> taken as "correct" limiting values. This is one reason that having >> values >> at discontinuities e.g. on branch cuts requires careful treatment, >> special >> code, is a source of bugs, etc. While I have not tried the >> experiment, I >> would venture to guess that making 0^0 evaluate to 1 would bring >> substantial new troubles to that part of the code base. As I alluded >> above, this sort of change incurs a development cost that can be >> rather >> steep, and for no gain I can discern. > > Mathematically, the definition of a limit makes no mention of > whether the > expression is defined at the point in question. It's unfortunate that > Mathematica code conflates two concepts that have nothing to do with > each > other. But of course they do have something to do with each other and that something is the concept of continuity of a function. If Mathematica follows the conventions that functions are defined only where they are continuous than it is reasonable, when computing a limit, to fist check if the function has a value at the point where the limit is being computer. If so, that value can be returned. This would give a substantial benefit in terms of performance, which is of primary importance for computer programs. > > > There are several benefits that I can see for defining 0^0 = 1. > > 1) The binomial theorem says > > (a+b)^n = Sum[Binomial[n,k] a^k b^(n-k),{k,0,n}] > > but in order for this to work for the case a=0 or b=0, we need 0^0 > = 1. > > 2) The derivative of x^n is D[x^n,x] = n x^(n-1), but in order for > this to work for the case n=1, we need 0^0 = 1. > > 3) The MacLaurin series for f[x] is given by > > Sum[D[f,{x,k}][0]/k! x^k,{k,0,Infinity}] > > but for this to work for x = 0 we need 0^0 = 1. The Exp[0] > argument that I mentioned previously is a special case. > > In each of these cases, the exponent is the integer 0. The base may > be > real or complex. All these are benefits only in respect of mathematical elegance. I cant see any benefit in any of the above in respect of computing performance - which is what counts in programs such as Mathematica. Andrzej Kozlowski

**References**:**Re: 0^0 = 1?***From:*Dave Seaman <dseaman@no.such.host>

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**Re: 0^0 = 1?**

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