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From: Jon Slaughter on 3 Dec 2009 23:06
Tim Little wrote:
> On 2009-12-03, Jon Slaughter <Jon_Slaughter(a)Hotmail.com> wrote:
>> LerchPhi is an infinite series...
>
> So is nearly everything else, including functions like f(x) = 1/(1-x),
> which can be expressed as an analytic extension of the power series
>
> f(x) = sum_(k=0 to inf) x^k.

But you can't say the converse. Can LerchPhi be reprsended as a simple
non-infinite expression?

Your straw man is kinda ridiculous. x = x + 0 + 0 + 0 + 0 + 0 ... but we
don't consider x non-elementary because it can be expressed as an infinite
sum.

We consider it elementary precisely because it can be expressed in an
"elementary" way. You cannot say the same for LerchPhi or exp(x). I call
exp(x) quasi-elementary. Lerchphi may or may not be elementary but it is on
the edge. If you ask 100k recent graduates what Lerchphi is you might get 10
that know it as to 100% who know exp(x).

Again though, elementary is not a democracy. It either is or isn't. To me,
Zeta is more elementary than LerchPhi since it is a generalization.

It seems you want to call some functions elementary simply because they
simplify certain things. For example, since sum(k^(-s)) = zeta(s) we can
"evaluate" the infinite sum an a "closed form" way simply by saying... hey,
thats just zeta(s). Whats sum(k^(-2))... oh, thats just Zeta(2).

I feel that what you are basically doing is calling anything elementary that
is too complex. "What the heck does is sum(k^(s!)/Gamma(s+k)*cos(exp(s*k)))?
Well I'll just call it the SEFD(s) function and say it's elementary."

I make that conclusion by your use of mathematica to determine what is
elementary or not. (as if mathematica is some oracle)




>
>
>> So you can call it closed if you want... it's up to you. I agree it
>> is relative but you are streching the truth a lot if you think that
>> it is useful in closed forms.
>
> It is quite useful as a closed form. A lot is known about that
> function, so that it can be manipulated symbolically much more easily
> than by treating it generically as an instance of a power series.
>
>

It seems to me the only use you get out of it an emotional one. You feel as
if you've accomplished something. To me, you are equating giving something a
symbolic name to making it elementary. If you define it then it is
elementary. I disagree. I understand it has to be in mathematica but...


>> Finite sums of simple things like exponential and cos functions,
>> sqrt's, etc are more "closed" than things involving infinite sums?
>
> How are you defining "simple" things like exponential and cos
> functions that do not involve infinite sums? Probably the most common
> definition of e^x is by the power series
>
> e^x = Sum_(k=0 to inf) x^k/k!.
>
> Other definitions include limits of infinite sequences such as
> (1+1/n)^n, or as a solution to an integral equation or differential
> equation - both of which also involve limits.
>
> Apart from familiarity, why do you call this a "simple" thing while
> LerchPhi is not?
>

well, I personally call them quasi-elementary. They are not elementary but
are effectively elementary. exp(x), cos(x), sin(x), sinh(x), etc... all
came from elementary mathematics and where "atomic". That is, they were not
generalization of similar things. (They may have been discovered in the
process of generalizing things but are not directly generalizations)

exp(x) = is just a number raised to another number. Nothing complex about
this... infinite series or not. It is an "elementary" idea or process.

the trig and hyperbolic functions, similarly, came from elementary analysis.

In some sense, they are just one step away from the real elementary
"functions".

LerchPhi, Zeta, Hypergeometric, Gamma, erf, the Mathieu functions and
zillions of others that come from differential equations...

All those functions are non-elementary in that they are either
generalizations of the trig, hyperbolic, exp(x), etc.. or simply series
solutions to differential equations that are "special" (because someone said
they were).

At some point we gotta stop saying everything is elementary because then the
term is useless. You can still write expressions involving your defined
functions even though they are not elementary. But calling something
elementary is not going to change it's mathematical properties. It may make
you feel good to think you've accomplished something by doing so but you
haven't done anything mathematically.


If a function is useful... even if it is not elementary, it will be used.
Suppose in the next 10 years it will be found that the LerchPhi function
will become much more useful than any of the most elementary functions I
discussed. Even if LerchPhi is not elementary it will still be used as
such.

I personally do not find LerchPhi any more elementary than any of the other
millions of functions. Why is LerchPhi better than Lommel function? What
makes it more elementary? Both are quite useful. What about Jacobi's Zeta
function? Whats wrong with it being elementary?

Do you agree that at some point calling everything elementary starts to
become useless?



From: Jon Slaughter on 3 Dec 2009 23:17
Let me say that I'm not specifically arguing that LerchPhi should not be
called elementary. In fact I don't know. It really ultimately depends on the
definition which I think is not well-defined.

My main point, and I probably haven't been very clear on, is that it seems
you guys are glossing over some of the subtle points of what it means to be
elementary. (I'm not saying I'm completely right in my understanding of it
either. I imagine the best answer is somewhat between our different
understandings)

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