# Big-Ordo and Little-ordo notation

///////

Related KMR pages:

///////

Other relevant sources of information:

///////

Big-Ordo and little-ordo

The symbols $\, O \,$ = “big Ordo” and $\, o \,$ = “little ordo” describe how fast functions are decreasing towards zero when their independent variable is approaching some fixed point where they take the value zero.

Let $\, f : \mathbb{R} \rightarrow \mathbb{R} \,$ be a function
and let its independent variable $\, x$ approach a fixed point $\, a \in \mathbb{R}$ where $\, f(a) = 0$.

Definition: $\, O_a(f) \stackrel {\mathrm{def}}{=} \{ g : \mathbb{R} \rightarrow \mathbb{R} \} \,$ such that, whenever $\, x \,$ is close enough to $\, a$,
we have $\, | \,\dfrac{g(x)}{f(x)} \, | < M \} \,$ for some fixed, positive constant $\, M \in \mathbb{R}$.

Definition: $\, o_a(f) \stackrel {\mathrm{def}}{=} \{ g : \mathbb{R} \rightarrow \mathbb{R} \} \,$ such that, whenever $\, x \,$ is close enough to $\, a$,
we have $\, | \,\dfrac{g(x)}{f(x)} \, | < M \} \,$ for every fixed, positive constant $\, M \in \mathbb{R}$.

Intuitively, this means that:

$\, O_a(f) \,$ denotes the set of functions that decrease at least as fast as $\, f \,$
when $\, x \,$ approaches $\, a$,
while
$\, o_a(f) \,$ denotes the set of functions that decrease faster than $\, f \,$
when $\, x \,$ approaches $\, a$.

NOTATION: When the point that is approached is clear from the context,
we write $\, O(f) \,$ and $\, o(f) \,$ instead of $\, O_a(f) \,$ and $\, o_a(f)$.

///////

Rules of computation for $\, O(f) \,$ :

1) $\, f \cdot O(g) = O(f \cdot g) \,$

2) $\, O(f) \cdot O(g) = O(f \cdot g) \,$

3) $\, O(O(f)) = O(f) \,$

In words:

1) A function $\, f \,$
times any function that decreases at least as fast as the function $\, g \,$
must decrease at least as fast as the function $\, f \cdot g$.

2) Any function that decreases at least as fast as the function $\, f \,$
times any function that decreases at least as fast as the function $\, g \,$
must decrease at least as fast as the function $\, f \cdot g$.

3) Any function that decreases at least as fast as
any function that decreases at least as fast as the function $\, f \,$
must decrease at least as fast as the function $\, f$.

///////

Rules of computation for $\, o(f) \,$ :

1) $\, f \cdot o(g) = o(f \cdot g) \,$

2) $\, o(f) \cdot o(g) = o(f \cdot g) \,$

3) $\, o(o(f)) = 0 \,$

In words:

1) A function $\, f \,$
times any function that decreases faster than the function $\, g \,$
must decrease faster than the function $\, f \cdot g$.

2) Any function that decreases faster than the function $\, f \,$
times any function that decreases faster than the function $\, g \,$
must decrease faster than the function $\, f \cdot g$.

3) Any function that decreases faster than
any function that decreases faster than the function $\, f \,$
must be identically zero in some neighborhood of the point that is approached.

///////

Example:

The interactive simulation that created this movie

The blue line intersects the red parabola tangentially at the point $\, P$.

The green line intersects the red parabola transversally at the point $\, P$.

The blue point $\, W \,$ approaches $\, P \,$ along the blue line,
which intersects the parabola tangentially at $\, P$.

The black point $\, Q \,$ approaches $\, P \,$ along the parabola itself.

The green point $\, R \,$ approaches $\, P \,$ along the green line,
which intersects the parabola transversally at $\, P$.

The horizontal distance between the green and the red vertical lines is $\, ∆ \,$.

TAKE AWAY MESSAGE:

$\, | \, QW \, | = o(∆)$
$\, | \, RW \, | = O(∆)$

In general:

Let the parabola be replaced by any smooth curve
and fix a point $\, P \,$ on this curve. Then the following is true:

Any point $\, Q \,$ that approaches $\, P \,$ along a curve
that intersects the given curve tangentially at the point $\, P \,$
behaves like the point $\, Q \,$ in the movie, i.e., we have $\, | \, QW \, | = o(∆)$.

Any point $\, R \,$ that approaches $\, P \,$ along a curve
that intersects the given curve transversally (= non-tangentially) at the point $\, P \,$
behaves like the point $\, R \,$ in the movie, i.e., we have $\, | \, RW \, | = O(∆)$.