"cannot be integrated" - what does it mean?

By Olof Sisask (P3033) on Thursday, March 8, 2001 - 07:11 pm :

On the topic of integration...

When people say that some functions cannot be integrated - do they mean that there's no 'method' for integrating them? Surely we will always be able to invent a function that gives the area under a given graph?

By Anonymous on Thursday, March 8, 2001 - 07:15 pm :

Is it because we don't know the simple equivalent version of the integral?

By Dan Goodman (Dfmg2) on Thursday, March 8, 2001 - 07:20 pm :

Hi Olof, there are two possible meanings to ''cannot be integrated''. One meaning is that it is a function which ''doesn't have an area''. A good example of a function which isn't Riemann integrable (which is the only sort of integral you will have come across so far) is the function

f

which satisfies

f (x) = 1

x

is a rational number and

f (x) = 0

x

is irrational. More likely, they mean it cannot be expressed in terms of ''elementary functions''. The elementary functions consist of

e^{z}

\log (z)

root (poly)

(the last one gives a solution to

poly (z) = 0

) and any combination of them. So,

\sqrt{x}

is elementary, since it is

root (X^{2} - x)

\sin (z)

and

\cos (z)

are also elementary, but you need to know a bit about complex numbers to see why. Also

e^{e^{z}}

e^{\sqrt[3]{z}}

\log (z^{23 / 51})

are elementary, etc. Some functions have integrals which are not expressible in terms of elementary functions, such as

\int x^{x} dx

\int \sin (z) / z dz

By Olof Sisask (P3033) on Thursday, March 8, 2001 - 07:34 pm :

Hi,
What specifically does Riemann integrable mean? Is there some other form of being integrable? That sounds decidedly like a University topic though!
I've come across the connection with exponentials and trigonometric functions (the 'formulas' for sin(z) and cos(z)) if those are what you were referring to?
Not wanting to ask too many questions, but how would u integrate x^x ?

Yet another question - what does integrating actually mean? Differentiation - finding the rate of change of a function at a point, or several points - that's fine, but I can't quite see how integration fits in, without talking about areas (which doesn't really make sense without graphs), or saying that it's the inverse of differentiation.

Regards,
Olof.

By Dan Goodman (Dfmg2) on Thursday, March 8, 2001 - 08:22 pm :

Hi Olof, Riemann integrable means that if you approximate the area of your function with rectangles everything works OK.

More accurately, a step function is a function g which looks like steps, i.e. g:[a,b]-> R is a function such that there are numbers a_i for i=0 to N with a₀ =a, a_N =b and a_i < a_i+1 and g constant on the interval [a_i ,a_i+1 ). If f is a function, let g and h be step functions so that g < = f < = h. You can define the integral of a step function by taking the areas of the rectangles, i.e.

$begin{displaymath}int\_a^b g dx = sum\_{i=0}^{N-1} g(a\_i)(a\_{i+1}-a\_i). end{displaymath}$

Now, if the maximum over all lower step function g (i.e. step functions g < = f) is equal to the minimum over all upper step functions h (i.e. step function f < = h) then we say that f is Riemann integrable with integral equal to the common value of the max and min of the g's and h's.

Exercise for the reader: why is the function f(x)=0 if x is irrational and 1 if x is rational not Riemann integrable?

Also, yes the formulas for sin(z) and cos(z) in terms of e^z are what I was referring to. And integration is defined by what I wrote above. You can use that definition to prove that (d/dx)integral gives you the original function, etc.

By Dave Sheridan (Dms22) on Friday, March 9, 2001 - 11:00 am :

Let's not forget that Riemann integration is only one form of integration. Measure theory gives a much better version, called Lesbegue integration, for which the function f quoted a while above is actually integrable, and its integral is equal to zero. Lesbegue integration is much more "natural" when you've seen it than Riemann integration, and anything which is Riemann integrable is also Lesbegue integrable with the same value of the integral. So Lesbegue integration is actually an extension of Riemann integration, although it comes about in a very different sense.

However, in order to even define Lesbegue integration you have to know quite a lot about measures (we're talking final year degree level stuff...) It's nice to know that it exists though, and anyone interested in learning about it should check the list archives on Probability and Measure Theory here , which discusses measures somewhat in the context of random walks. Measures and probability are very closely interwoven.

-Dave