4.4 Some Inequalities

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It is important to be able to obtain rough estimates of path integrals, so as

to be able to decide whether you have got a reasonable sort of answer or have

made a blunder somewhere. For this reason, the following inequalities are

useful:

Proposition 4.4.1 If c : [0; 1] ! C is a smooth path in C

____

Z 1

0                     

c(t)dt____

_ Z 1

0 jc(t)jdt (4.3)

Proof:

If R 1

0 c(t)dt = Rei_, the left hand side of 4.3 is just R.

We have that

R = Z 1

0

e􀀀i_c(t) dt

and since the left hand side is real we have also:

R = Z 1

0 <[e􀀀i_c(t)] dt

But

Z 1

0 <[e􀀀i_c(t)]dt _ Z 1

0 je􀀀i_c(t)jdt

since for all t, and any function g, <(g(t)) _ jg(t)j.

Then since jzwj = jzjjwj and je􀀀i_j = 1 we have

R =____

Z 1

0

c(t)dt____

_ Z 1

0 jc(t)jdt

2

It is not necessary for the path c to be smooth, but it needs to be continuous.

Note that we are integrating the constant function 1 over the path.

We can strengthen this as follows:

120 CHAPTER 4. INTEGRATION

Proposition 4.4.2 Let c be a smooth path in C and f : C 􀀀! C a continuous

function. Let L be the length of the path and M be the maximum value

of jfj on c. Then

____

Zc

f(z)dz____

_ ML

Proof:

____

Zc

f(z)dz____

=____

Z 1

0

f(z) _ z dt____ By the preceding result we have:

____

Z 1

0

f(z) _ z dt____

_ Z 1

0 jf(z) _ z dtj = Z 1

0 jf(z)jjz_j dt

And

Z 1

0 jf(z)jjz_j dt _ M Z 1

0 jz_j dt = ML

2

This is a rather coarse inequality, and we can get better estimates by partitioning

c and looking for better bounds on the parts.

Example 4.4.1 Estimate the modulus of the integral of _z from 1􀀀i to 1+i

We have that the length is 2 and the maximum value of j_zj along the path is

p2 at the end points. So

____

Zc

_zdz____

_ 2p2

From an earlier example we know that the actual value is 2. 2

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