kindly tell me the scope of research in computer virus model with stochastic term and what are the limitation.

Real elementary functions are indefinitely differentiable( have derivatives of order "n" on their domain, for all n>=1). So, their power series are Taylor series. Consequently, if f:D(contained in R)->R is a real elementary function and a belongs to D, then f(x)=sumn=0 to infinity{[f(n)(a)/n!](x-a)n} for all x belongs to interval I=(a-r,a+r) contained in D. r>0 is the convergence radius of series sumn=0 to infinity{[f(n)(a)/n!](x-a)n} and r can be equal to infinity.

The coefficients of such series are cn=f(n)(a)/n!. If you are interested on convergence of series sumn=0 to infinitycn(series of coefficients from Taylor series), we can observe that such series are convergent and:

sumn=0 to infinitycn = sumn=0 to infinity[f(n)(a)/n!] = f(a+1), if a+1 belongs to (a-r,a+r).

For example:

1. ex=exp(x)=sumn=0 to infinity[xn/n!]. a is zero, r is infinity and the series of coefficients are exp(1)=e=sumn=0 to infinity[1/n!].

2. sin(x)=sumn=0 to infinity[(-1)nx2n+1/(2n+1)!]. a is zero, r is infinity and the series of coefficients are sin1=sumn=0 to infinity[(-1)n/(2n+1)!]

But we have also special cases. The elementary function 1/1-x=sumn=0 to infinityxn for all x belongs to (-1,1). In this case the series of coefficients do not converge because cn=1 for all n.

To compute the convergence radius r of power series sumn=0 to infinity[cn(x-a)n], we have the Cauchy-Hadamard formula: r=1/{limn->infinity(/cn/)1/n}

Dinu Teodorescu My question was given a convergent Taylor series (for a certain interval) how can we tell if it represents an elementary function or not.

We don't have mathematical methods to decide! Moreover, the class of "real elementary functions" is not clearly defined, in the sense that not all authors of math books or articles understand the same thing regarding elementary functions.

I was taking 'elementary' as in the Wikipedia entry: an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, including possibly their inverse functions.

Yes, this is the most used definition and, the most correct, I consider. But, as I said, we can not decide mathematically if a convergent Taylor series comes from an elementary function or not!

Dear Clarence, excuse me for question, but why is important for you to decide on above fact? Has that some importance for your research?

Perhaps you can yourself to introduce an appropriate definition for that, considering the neccessity of your research problem.

It is not related directly to my current research. But I think it would be a very interesting result if we could prove that there is no decision procedure to determine whether a convergent Taylor series comes from an elementary function or not.

Maybe this could be proved first for convergent Taylor series with computable coeficients.

This means: for a function f there is a Turing machine T_f which given input n and m gives as output the coeficent a_n of the Taylor series up to the mth decimal place.

As Turing machines can be codified by integers, computable convergent Taylor series are codified by a subset C of N.

It seems that all elementary functions have computable Taylor coeficients.

Thus they determine a subset E of C.

We would then show that E is not recursive in C.

Few words to the following comment by @Dinu Theodorescu

>

I wouldn't call it 'special case'. It is rather a typical case of a DIFFERENT class of cases. Indeed, the number

a+1 = 0+1 = 1

is off the domain of convergence of the Taylor series for 1/(1-x) wrt a=0.

Let ELEM be the problem: given a convergent Taylor series, does it represent an elementary function ?

Let INT be the problem: given an elementary function does it have a primitive/indefinite integral which is an elementary function ?

An observation that can be made is that if ELEM is decidable then so too is INT.

Given an elementary function write down its Taylor series and integrate each term. Then apply the decision procedure for ELEM (of course we must be more precise here, this is just the general idea).

Thus to show that ELEM is undecidable it suffices to show that INT is.

Clarence Lewis Protin Dear Clarence, I think all two problems, ELEM and INT, are undecidable. The problem INT seems to be easier as approach, but really, it is not!

The elementary function f:R->R, f(x)=ex^2 is continuous on R, then f has primitives(antiderivatives), but their explicit form cannot be found...it doesn't exist an explicit form for primitives of f. So, we can't decide if primitives of f are elementary functions or not!

On your place I would start with the inspection of hypergeometric series. But if you want to describe only elementary functions, this will definitely lead to Differential Galois Theory. With hypergeometric series the problem looks more realistic. However, I am not sure that this is possible.

Sergey Khrushchev ,

there is the really nice class of holomonic functions which can be characterised either as:

1) Being solutions of a homogenous linear differential equation with polynomial coeficients

2) Having Taylor series coeficients given by polynomial recurrence relations

There is an algorithm to pass between the presentations 1) and 2)

This includes elementary functions, the hypergeometric functions, Bessel function, etc.

The question arises: given a sequence of real numbers, is it decidable if they obey a polynormial recurrence relation ?

I think it might be interesting to reframe these questions in an arbitrary topos. For instance the effective topos or big or small toposes of sheaves over topological spaces. The 'reals' split into generally distinct concepts: Dedekind reals, Cauchy reals, Baire space N^N. Classical topology and analysis yields unexpected insights in a computable or constructive universe.

I think the notion of definable real number important, which is more general than computable real number. So we should consider definable analytic functions DU as well - those who coeficients can be defined by a first-order expression \phi(x). What properties does DU have ?