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1 1

cos (x) = 1 2 x2 + 4! x4 (2.13)

; ;

1

x = 1 + x + 1 x2 + 1 x3 + = X 1 xn

e (2.14)

n=0 n!

2! 3!

1

X

1 = 1 + x + x2 + xn

= (2.15)

1x n=0

;

1 1

1) x2 3! ( 2) x3 +

(1 x) = 1 x + 2! ( 1) ( (2.16)

; ; ; ; ; ;

Up to this point the Taylor expansion was made around the point x = 0. However,

one can make a Taylor expansion around any arbitrary point x. The associated Taylor

series can be obtained by replacing the distance x that we move from the expansion point

by a distance h and by replacing the expansion point 0 by x. Making the replacements

x h and 0 x the expansion (2.11) is given by:

! !

1

X hn dnf

f(x + h) = n! dxn (x) (2.17)

n=0

The Taylor series can not only be used for functions of a single variable. As an example

consider a function f(x y) that depends on the variables x and y. The generalization of

the Taylor series (2.9) to functions of two variables is given by

1

X

cnmxnym :

f(x y) = (2.18)

n m=0

At this point the coe cients cnm are not yet known. They follow in the same way as the

coe cients of the Taylor series of a function that depends on a single variable by taking

the derivatives of the Taylor series and by evaluating the result in the point where the

expansion is made.

CHAPTER 2. SUMMATION OF SERIES

10

Problem e: Take suitable derivatives of (2.18) with respect to x and y and evaluate the

result in the expansion point x = y = 0 to show that up to second order the

Taylor expansion (2.18) is given by

f(x y) = f(0 0) + @f (0 0) x + @f (0 0) y

@x @y (2.19)

2f 2f 2

1 @ (0 0) x2 + @ (0 0) xy + 1 @ f (0 0) y2 +

+ 2 @x2 2 @y2

@x@y

Problem f: This is the Taylor expansion of f(x y) around the point x = y = 0. Make

suitable substitutions in this result to show that the Taylor expansion around an

arbitrary point (x y) is given by

f(x + hx y + hy ) = f(x y) + @f (x y) hx + @f (x y) hy

@x @y

2f 2f 2

1 @ (x y) h2 + @ (x y) hx hy + 1 @ f (x y) h2 +

+ 2 @x2 x @x@y y

2 @y2

(2.20)

Let us now return to the Taylor series (2.9) with the coe cients cm given by (2.10).

This series hides a very intriguing result. It follows from (2.9) and (2.10) that a function

f(x) is speci ed for all values of its argument x when all the derivatives are known at

a single point x = 0. This means that the global behavior of a function is completely

contained in the properties of the function at a single point. In fact, this is not always

true.

First, the series (2.9) is an in nite series, and the sum of in nitely many terms does not

necessarily lead to a nite answer. As an example look at the series (2.15). A series can

only converge when the terms go to zero as n because otherwise every additional

! 1,

term changes the sum. The terms in the series (2.15) are given by xn , these terms only go

to zero as n when < 1. In general, the Taylor series (2.9) only converges when

!1 jxj

x is smaller than a certain critical value called the radius of convergence. Details on the

criteria for the convergence of series can be found for example in Boas?? or Butkov??.

The second reason why the derivatives at one point do not necessarily constrain the

function everywhere is that a function may change its character over the range of parameter

values that is of interest. As an example let us return to a moving particle and consider

a particle with position x(t) that is at rest until a certain time t0 and that then starts

moving with a uniform velocity v = 0:

6

(

x(t) = x0 + v(t t ) for t > t0

t (2.21)

x0 0 for t 0

;

The motion of the particle is sketched in gure 2.2. A straightforward application of (2.8)

shows that all the coe cients cn of this function vanish except c0 which is given by x0 .

The Taylor series (2.7) is therefore given by x(t) = x0 which clearly di ers from (2.21).

The reason for this is that the function (2.21) changes its character at t = t0 in such a way

that nothing in the behavior for times t < t0 predicts the sudden change in the motion

at time t = t0 . Mathematically things go wrong because the higher derivatives of the

function do not exist at time t = t0 .

2.2. THE BOUNCING BALL 11

x(t)

t

Figure 2.2: The motion of a particle that suddenly changes character at time t0 .

Problem g: Compute the second derivative of x(t) at t = t0 .

The function (2.21) is said to be not analytic at the point t = t0 . The issue of analytic

functions is treated in more detail in the sections 12.1 and 13.1.

Problem h: Try to compute the Taylor series of the function x(t) = 1=t using (2.7) and

(2.8). Draw this function and explain why the Taylor series cannot be used for this

function.

Problem i: Do the same for the function x(t) = t. p

Frequently the result of a calculation can be obtained by summing a series. In section

2.2 this is used to study the behavior of a bouncing ball. The bounces are \natural" units

for analyzing the problem at hand. In section 2.3 the reverse is done when studying the

total re ection of a stack of re ective layers. In this case a series expansion actually gives

physical insight in a complex expression.

2.2 The bouncing ball

In this exercise we study a problem of a rubber ball that bounces on a at surface and

slowly comes to rest as sketched in gure (2.3). You will know from experience that the

ball bounces more and more rapidly with time. The question we address here is whether

the ball can actually bounce in nitely many times in a nite amount of time. This problem

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