About integrable function

[subharpe]

Can someone tell me which function is integrable (analytically, not numerically) and which function is not and how to trace from the expression of the function?

 

[kalyanram]

Hi,
As far as I can remember any function that has "finite" number of discontinuties is integrable.
The idea behind integration is to compute the area swept under the function now suppose that the required function has discontinuties at say n points in the range
of integration then we can patch these points up as their contribution to area can be considered equivalent to n rectangles of finite length, but breadth of point thickness. Hence the contribute nothing in terms of area.

However a function with infinite discontinuties is not integral as that could be explained in the current context as

This I agree is vague but thats how analytical it can get I'm sure.

~Kalyan.

 

[jcy]

I think kalyanram is slightly in error to say that a function with an infinite number of discontinuities is not integrable. Actually, to be precise, you first need to specify which type of integral you are considering and the domain/range space. For the Riemann integral over a finite (closed?) domain, I think the requirement for the integral to exist is that the number of discontinuities is "countable". This could be a finite number or a countably infinite number. Another way to state this is that the number of discontinuities has "measure zero". However, if the domain is infinite (for exampe -∞ to ∞), I would expect that this condition is not sufficient to state whether the function is integrable.

For a more precise statement of when a function is integrable, you can loop up the term "absolute continuity", which gives a precise statement about the class of integrable functions. These theorems are usually proved in the context of a Lebesgue integral.

John

 

[bomo321]

Can someone tell me which function is integrable (analytically, not numerically) and which function is not and how to trace from the expression of the function?

The answers above seem to answer the question: when is a function integrable?
One answer is e.g. that all continuos functions are Riemann integrable on a closed interval. If we use another definition of integral e.g. Lebesgue integral then we will get a another set of integrable functions.

I think your actually asking the question: how can I tell if a function has an
integral function consisting of elementary functions (which is a different question)? An elementary functions is a functions that is an alebraic expression of polynomials, trig functions, log and exponential function. E.g. 1/tan(x) or log(x³-1)/cos(x).

One thing to note about the definition of elementary function is that its a bit arbitrary. There is really no reason why we should call this set 'the elementary functions'.

An example of an integral that does not have an anti-derivative made up of elementary functions is e^(-t²) (this can proved but you have to read a bit of math to understand it). So we can define a new function

f(x) = integral from 0 to x of e^(-t²)

which is not an elementary function. But there is really no reason why this function should be regarded as non elementary, other than that it might be a bit more unfamiliar than sin(x), cos(x) ,tan(x) and so on.

Anyway it's not an easy matter telling if a function has an analytical expression made up of elementary functions.