1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

7 4 CHAPTER 2 • COMPLEX FUNCTIONS

• EXAMPLE 2.17 Show that Jim (z^2 - 2z + 1) = -1.

z~l+i

S olut ion We let

f (z) = z^2 - 2z + 1 = x^2 - y^2 - 2x + 1 + i (2xy - 2y).

Computing the limits for u and v, we obtain

Jim u(x, y)=l-1-2+1=-1 and

(.,, 11)-(l, 1)

Jim v (x, y) = 2 - 2 = 0,

(:t, 11)-(1, 1)

so our previous theorem implies that lim. f ( z) = -1.

z-l+t

Limits of complex functions are formally the same as those of real functions,
and the sum, difference, product, and quotient of functions have limits given by
the sum, difference, product, and quotient of the respective limits. We state this
result as a theorem and leave the proof as an exercise.