Re: Bug in interpretation of mma Series[] command?

*Subject*: [mg2268] Re: Bug in interpretation of mma Series[] command?*From*: withoff (David Withoff)*Date*: Thu, 19 Oct 1995 05:34:11 GMT*Approved*: usenet@wri.com*Distribution*: local*Newsgroups*: wri.mathgroup*Organization*: Wolfram Research, Inc.*Sender*: daemon at wri.com ( )

In article <DGIy4H.I57 at wri.com> siegman at ee.stanford.edu (A. E. Siegman) writes: > Does the following make sense to anyone? > > In[6]:= > f[y] = Exp[-a y^2] * Series[1/(1 + y^2), {y,0,5}] > > Out[6]= > 2 > 2 a 4 6 > 1 + (-1 - a) y + (1 + a + --) y + O[y] > 2 > > This strikes me as a really counter-intuitive, perverse, > guaranteed-to-cause-trouble "gotcha". > > I have to agree that > > In[7]:= > f[y] = Exp[-a y^2] * Normal[Series[1/(1 + y^2), {y,0,5}]] > > Out[7]= > 2 4 > 1 - y + y > ----------- > 2 > a y > E > > does what's expected; but the first case should surely > either generate an error msg, or not be interpreted at > all. There's no way any sensible interpretation of > the input command (by normal users, that is) should lead > to series expansion of the Exp[] function also. > The design justification for this is identical to the justification for converting the exact number in the sum 1/4 + 5.27 to an inexact number 0.25 + 5.27 so that the two numbers can be combined. In the case of numbers, this sort of coercion is presumably not unexpected. It is mathematically correct, and is done automatically by most computer languages. The same thing is done with series, and for the same reasons. If a function for which you effectively know all of the terms in the series expansion Exp[-a y^2] (the analog of an exact number) is multiplied by a function for which you have only a few terms in the series expansion Series[1/(1 + y^2), {y,0,5}] ==> SeriesData[y, 0, {1, 0, -1, 0, 1}, 0, 6, 1] (the analog of an inexact number) the first function will be converted to a series expansion so that the two can be combined. Although it is useful in some formal manipulations to avoid these conversions (both with series expansions and with inexact numbers) there is no mathematical reason not to do it. Failure to do these conversions leads to a variety of practical difficulties. For example, in current versions of Mathematica, the input 2.7 Pi returns unchanged. Mathematically, this expression has all of the characteristics of an inexact number, but since the exact number is not coerced into an inexact number, your programs are obligated to figure that out for themselves, typically through careful use of N. (In the next version of Mathematica, this conversion will be done automatically.) Leaving something like Exp[-a y^2] * Series[1/(1 + y^2), {y,0,5}] as a product causes the same sort of trouble, only worse. This product has all of the mathematical characteristics of a series expansion, but unless Exp[-a y^2] is converted to a series expansion, it will remain a "disguised" series expansion, and will be difficult to deal with. I am sorry to hear that you are disappointed by this design, but the reasons for it are quite sound. If you have a good reason for avoiding these conversions, we would be happy to hear from you, and will do what we can to incorporate your suggestions into the design. Dave Withoff Research and Development Wolfram Research