Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PRELIMINARY ALGEBRA

the well-known result that the angle subtended by a diameter at any point on a

circle is a right angle.

Taking the diameter to be the line joiningQ=(−a,0)andR=(a,0)and the pointPto

be any point on the circlex^2 +y^2 =a^2 , prove that angleQP Ris a right angle.

IfPis the point (x, y), the slope of the lineQPis

m 1 =

y− 0

x−(−a)

=

y

x+a

.

That ofRPis

m 2 =

y− 0

x−(a)

=

y

x−a

.

Thus

m 1 m 2 =

y^2

x^2 −a^2

.

But, sincePis on the circle,y^2 =a^2 −x^2 and consequentlym 1 m 2 =−1. From result (1.24)
this implies thatQPandRPare orthogonal and thatQP Ris therefore a right angle. Note
that this is true foranypointPon the circle.

1.4 Partial fractions

In subsequent chapters, and in particular when we come to study integration

in chapter 2, we will need to express a functionf(x) that is the ratio of two

polynomials in a more manageable form. To remove some potential complexity

from our discussion we will assume that all the coefficients in the polynomials

are real, although this is not an essential simplification.

The behaviour off(x) is crucially determined by the location of the zeros of

its denominator, i.e. iff(x) is written asf(x)=g(x)/h(x) where bothg(x)and

h(x) are polynomials,§thenf(x) changes extremely rapidly whenxis close to

those valuesαithat are the roots ofh(x) = 0. To make such behaviour explicit,

we writef(x) as a sum of terms such asA/(x−α)n,inwhichAis a constant,αis

one of theαithat satisfyh(αi)=0andnis a positive integer. Writing a function

in this way is known as expressing it inpartial fractions.

Suppose, for the sake of definiteness, that we wish to express the function

f(x)=

4 x+2

x^2 +3x+2

§It is assumed that the ratio has been reduced so thatg(x)andh(x) do not contain any common

factors, i.e. there is no value ofxthat makes both vanish at the same time. We may also assume

without any loss of generality that the coefficient of the highest power ofxinh(x) has been made

equal to unity, if necessary, by dividing both numerator and denominator by the coefficient of this

highest power.