Big O notation (for calculus, not computer science)

1. Jul 21, 2010

kungal

I understand the formal definition for big O notation but is there an intuitive interpretation?. For example, if

Code (Text):
f(x) = O(x[SUP]-1/4[/SUP])
is it reasonable to say that for large n f(x) grows at the same rate as
Code (Text):
n[SUP]-1/4[/SUP]
?

2. Jul 21, 2010

arildno

An O-notation requires a limit to which the independent variable is supposed to go.

3. Jul 21, 2010

kungal

The O's should actually be Op's (my mistake) and x is a set of random variables. I don't think this changes the nature of the question drastically.

Note that the result f(x) = Op(x-1/4) is quite common so it can't be a meaningless statement.

4. Jul 21, 2010

arildno

Yes it is.

Is it meant that f(x) is O(x^(-1/4) ) as x goes to zero, or as x goes to, say, infinity?
That is two entirely different situations, and needs, therefore, to be specified.
Hence, the meaninglessness of your first line.

5. Jul 21, 2010

HallsofIvy

I hate to disagree with Arildno but it is common practice (though perhaps "abuse of notation") to use just O(f(x)) to mean "as x goes to infinity".

6. Jul 21, 2010

arildno

Well, if that is the general default notation, I'll make a note of that.

In my own applied maths books, they dutifully make explicit what limiting operation we are speaking about

7. Jul 21, 2010

kungal

I'm glad we've cleared that up but is it reasonable to say that
if f = Op(n-1/4) then for large n f grows at the same rate as n-1/4?

8. Jul 21, 2010

arildno

Not at all.

f might, for example, become more and more strongly oscillatory as x goes to infinty, even though f's magnitude will be bounded by some constant multiplied by x^(-1/4).

For example, let
$$f(x)=Ax^{-\frac{1}{4}}\cos(x^{2})$$

This f is definitely O(x^(-1/4)), but its rate of growth will, be:
$$\frac{df}{dx}\to{-2A}x^{\frac{3}{4}}\sin(x^{2}), x\to\infty$$

9. Jul 21, 2010

kungal

Thanks

10. Jul 22, 2010

aprillove20

Well, O's should actually be Op's (my mistake) and x is a set of random variables.