Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX VARIABLES

whereanis given byf(n)(z 0 )/n!. The Taylor expansion is valid inside the region

of analyticity and, for any particularz 0 , can be shown to be unique.

To prove Taylor’s theorem (24.50), we note that, sincef(z) is analytic inside

and onC, we may use Cauchy’s formula to writef(z)as

f(z)=

1

2 πi

∮

C

f(ξ)

ξ−z

dξ, (24.51)

whereξlies onC. Now we may expand the factor (ξ−z)−^1 as a geometric series

in (z−z 0 )/(ξ−z 0 ),

1

ξ−z

=

1

ξ−z 0

∑∞

n=0

(

z−z 0

ξ−z 0

)n

,

so (24.51) becomes

f(z)=

1

2 πi

∮

C

f(ξ)

ξ−z 0

∑∞

n=0

(

z−z 0

ξ−z 0

)n

dξ

=

1

2 πi

∑∞

n=0

(z−z 0 )n

∮

C

f(ξ)

(ξ−z 0 )n+1

dξ

=

1

2 πi

∑∞

n=0

(z−z 0 )n

2 πif(n)(z 0 )

n!

, (24.52)

where we have used Cauchy’s integral formula (24.48) for the derivatives of

f(z). Cancelling the factors of 2πi, we thus establish the result (24.50) with

an=f(n)(z 0 )/n!.

Show that iff(z)andg(z)are analytic in some regionR, andf(z)=g(z)within some

subregionSofR,thenf(z)=g(z)throughoutR.

It is simpler to consider the (analytic) functionh(z)=f(z)−g(z), and to show that because
h(z)=0inSit follows thath(z) = 0 throughoutR.
If we choose a pointz=z 0 inS, then we can expandh(z) in a Taylor series aboutz 0 ,
h(z)=h(z 0 )+h′(z 0 )(z−z 0 )+^12 h′′(z 0 )(z−z 0 )^2 +···,

which will converge inside some circleCthat extends at least as far as the nearest part of
the boundary ofR,sinceh(z)isanalyticinR. But sincez 0 lies inS, we have

h(z 0 )=h′(z 0 )=h′′(z 0 )=···=0,

and soh(z)=0insideC. We may now expand about a new point, which can lie anywhere
withinC, and repeat the process. By continuing this procedure we may show thath(z)=0
throughoutR.
This result is called theidentity theoremand, in fact, the equality off(z)andg(z)
throughoutRfollows from their equality along any curve of non-zero length inR,oreven
at a countably infinite number of points inR.

So far we have assumed thatf(z) is analytic inside and on the (circular)

contourC.If,however,f(z) has a singularity insideCat the pointz=z 0 ,thenit

cannot be expanded in a Taylor series. Nevertheless, suppose thatf(z) has a pole