Theory of Statistics - George Mason University

0.0.9.5 Working with Real-Valued Functions
0.0 Some Basic Mathematical Concepts 673
When dealing with general real-valued functions, it is often useful to decompose
the function into its nonnegative part and its negative part. In this way,
the function f is written as
f = f+ − f−.
An example of this technique is in the definition of the Lebesgue integral,
Definition 0.1.39.
0.0.9.6 Use of Transformations
Many problems are simplified by use of transformations of the variables. Some
useful transformations are those between trigonometric and exponential functions,
such as Euler’s formula,
e i(nx) = cos(nx) + i sin(nx), (0.0.78)
for integer n and real x.
Euler’s formula yields de Moivre’s formula for multiples of angles,
(cos(x) + i sin(x)) n = cos(nx) + i sin(nx), (0.0.79)
again for integer n and real x. (Note, in fact, that this formula does not hold
for non-integer n.) There are many other formulas among the trigonometric
functions that can be useful for transforming variables.
Another very useful class of transformations are those that take cartesian
coordinates into circular systems. In two dimensions, the “polar coordinates”
ρ and θ in terms of the cartesian coordinates x1 and x2 are
ρ = x2 1 + x2 ⎧ 2
⎨
θ = arcsin(x2/
⎩
x2 1 + x22 ) if x1 ≥ 0
π − arcsin(x2/ x2 1 + x22 ) if x1 < 0
0 if x1 = x2 = 0
(0.0.80)
The extension of these kinds of transformations to higher dimensions is called
“spherical coordinates”.
0.0.9.7 Expansion in a Taylor Series
One of the most useful tools in analysis is the Taylor series expansion of a
function about a point a. For a scalar-valued function of a scalar variable, it
is
f(x) = f(a) + (x − a)f ′ + 1
2! (x − a)2 f ′′ + · · · , (0.0.81)
Theory of Statistics c○2000–2013 James E. Gentle