1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

3.2 • THE CAUCHY-RIEMANN EQUATIONS 101

y v

I.OJ t~~---······················•Z)

4.06

....
....

4 .04

1.005. ~ zplane w plane
w = z2
~

2.98 3. 02 3.04

Figure 3.1 The im~e of a small square. with vertex Zo = 2 + i, using w = z^2.

We know that f is differentiable, so the limit of the difference quotient
[(z]:~•o) exists no matter bow we approach zo = 2 + i. Thus we can approximate
f' (2 + i) by using horizontal or vertical increments in z:

I

I (2 '):::::: f (2.01 + i) - I (2 + i) = 0.0401+0.02i = 4.01 2·

+ i (2.01 + i) - (2 + i) 0.01 + i

and

f

I ( 2 .) - f (2 + 1.0li) -f (2 + i) - -0.0201 + 0.04i - 4 2 oi ·

+i - - - +. t.

(2 + 1.0li) - (2 + i) O.Oli

These comput.ations lead to the idea of taking limits along the horizontal
and vertical directions. When we do so, we get

/

'(2 .) 1. /(2+h+i)-f(2+i) 1· 4h+ h^2 +i2h 4 2·

+ i = 1m = 1m = + i

n- o h 1i- o h

and

!

'(2 ')- Ii f(2+ i +ih)-f(2+i) 1· -2h-h^2 +i4h 4 2·

+ i - m - 1m - + i.

h- o ih 1i- o ih

We now generalize this idea by taking limits of an arbitrary differentiable
complex function and obtain an important result.