Begin2.DVI

illustrated in the figure 12-16.

(b) If one had used the binomial expansion on the last term in the representation (i)

for f(z), one obtains a series which converges for |z− 1 |> 2. After multiplication by the

first term, the resulting series would converge in the annular region r 1 <|z− 1 |< r 2

where r 1 = 2 and r 2 =lim r→∞ r. This is not the annular region which isolates the

singular point at z= 1 because the singular point z= 3 is also inside this region.

Figure 12-16. Annular region of convergence.

In general, the radius of convergence for the power series with positive powers

or non-principal part of the Laurent series, is represented by the distance from the

center of the series to the nearest other singular point of the function being studied.

The correct Laurent series expansion, which isolates the singular point at z= 1,

shows that f(z)has a simple pole at z= 1 and the residue of f(z)at z= 1 has the

value − 1 / 2.

The above examples represent a small fraction of the many concepts presented

in the study of functions of a complex variable.