Instinctively, having had a fair amount of mathematics, theanswer should be no. However, it seems often in math we take . . without making any assertions or assumptions that x can notequal zero. The reason I ask is because I\'m in Diff. Eq. dealingwith regular and irregular singular points and it seems we arereally playing with the whole dividing by things that arepotentially zero in a way I\'m not entirely comfortable. Forexample, take the function p(x) to equal as follows... . . Initially, when I considered p(2i), Ifigured the function to be undefined and non-analytic atthe said point since plugging it straight into theequation as is yields a zero in the denominator. However, if youbreak the equation down a bit further you get . . Which of course leaves you with . . which is analytic at x = 2i. This is the way the book/websiteapproached this problem and to me it just seems kind of sketchy.I\'ve also seen the same thing done with . . with x equalling zero based on the idea that . . Since so often over the many years of math I\'ve taken has itbeen driven into my head that \'ye shall not divide by zero,\'dividing these variables through knowing them to be zero is to saythe least, a bit counter-intuitive... Any ideas on why we suddenlyviolate this cardinal rule? Solution Ref: http://en.wikipedia.org/wiki/Analytic_function In mathematics, ananalytic function is a functionthat is locally given by a convergent powerseries or A function is analytic if it is equal to itsTaylor series insome neighborhood. ------------ I believe you\'re confusing the following: 1. limits as x approaches 0 or 2i 2. the expansion of an analyticfunction. 3. The value of an analytic function at a 0/0point For example, the first few terms of the taylor series for sinx/x is: 1-1/6*x^2+1/120*x^4-1/5040*x^6+O(x^7) or f(0) + f \'(0)x/1! + ... The f(0) term is calculated in the limit I believe (needs to be checked) that the inclusion of acountable number of points that is analytically reachable does notchange the value of an analytic function Hope this helps Hope this helps..

1. Instinctively, having had a fair amount of mathematics, theanswer should be no. However, it
seems often in math we take
.
.
without making any assertions or assumptions that x can notequal zero. The reason I ask is
because I'm in Diff. Eq. dealingwith regular and irregular singular points and it seems we
arereally playing with the whole dividing by things that arepotentially zero in a way I'm not
entirely comfortable. Forexample, take the function p(x) to equal as follows...
.
.
Initially, when I considered p(2i), Ifigured the function to be undefined and non-analytic atthe
said point since plugging it straight into theequation as is yields a zero in the denominator.
However, if youbreak the equation down a bit further you get
.
.
Which of course leaves you with
.
.
which is analytic at x = 2i. This is the way the book/websiteapproached this problem and to me it
just seems kind of sketchy.I've also seen the same thing done with
.
.
with x equalling zero based on the idea that
.
.
Since so often over the many years of math I've taken has itbeen driven into my head that 'ye
shall not divide by zero,'dividing these variables through knowing them to be zero is to saythe
least, a bit counter-intuitive... Any ideas on why we suddenlyviolate this cardinal rule?
Solution
Ref: http://en.wikipedia.org/wiki/Analytic_function
In mathematics, ananalytic function is a functionthat is locally given by a convergent
powerseries or A function is analytic if it is equal to itsTaylor series insome neighborhood.
------------

2. I believe you're confusing the following:
1. limits as x approaches 0 or 2i
2. the expansion of an analyticfunction.
3. The value of an analytic function at a 0/0point
For example, the first few terms of the taylor series for sinx/x is:
1-1/6*x^2+1/120*x^4-1/5040*x^6+O(x^7) or
f(0) + f '(0)x/1! + ...
The f(0) term is calculated in the limit
I believe (needs to be checked) that the inclusion of acountable number of points that is
analytically reachable does notchange the value of an analytic function
Hope this helps
Hope this helps.