1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

100 CHAPTER 3 • ANALYTIC AND H A RMONIC FUNCTIONS

(e) Identity (3-12).

13. Consider the differentiable function f ( z) = z^3 and t he two points z1 = 1 and

z 2 = i. Show that there does not exist a point c on the li ne y = 1 - x between

1 and i s uch that flz>~2l-/C•tl -~1 = f' (c). This result shows that the mean value

theorem for d erivatives does not extend to complex funct ions.

14. Let f (z) = zt denote the multivalued "nt h root function," where n is a positive
    integer. Use the chain rule to show t hat if g(z) is any branch of t he nth root
    function, then
    g'(z)= _!.g(z)
    n z
    in some suitably chosen domain (which you should specify).


15. Explain why the composition of two entire functions is an entire function.


16. Let f be different iable at zo. Show that there exists a function ri (z) such that

f(z) = f(zo) + !' (zo)(z-zo) + ri(z) (z - zo),

where 11 (z) --> 0 as z--> zo.

3.2 The Cauchy-Riemann Equations

In Section 3.1 we showed that computing the derivative of complex functions
written in a form such as f (z) = z^2 is a rather simple task. But life isn't always
so easy. Many times we encounter complex functions written in the form of
f (z) = f (x + iy) = u (x, y) + iv (x, y). For example, suppose we had
f (z) = f (x + iy) = u (x, y) + iv (x, y) = (x^3 - 3xy^2 ) + i (3x^2 y -y^3 ). (3-13)
Is there some criterion that we can use to determine whether f is differentiable
and, if so, to find the value off' (z)?
The answer to this question is yes, thanks to the independent discovery of
two important equations by the French mathematician Augustin-Louis Cauchy^1
and the German mathematician Georg Friedrich Bernhard Riemann.

First, let's reconsider the derivative off (z) = z^2. As we have stated, the

limit given in Equation (3-1) must not depend on how z approaches zo. We

investigate two such approaches: a horiz-0ntal approach and a vertical approach

to zo. Recall from our graphical analysis of w = z^2 that the image of a square

is a "curvilinear quadrilateral." For convenience, we let t he square have vertices

zo = 2 + i, z 1 = 2.01 + i, z2 = 2 + l.Oli, and z3 = 2.01 + 1.0li. Then the

image points are wo = 3 + 4i, w 1 = 3.0401 + 4.02i, w 2 = 2.9799 + 4 .04i, and

w 3 = 3.02 + 4.0602i, as shown in Figure 3.1.

(^1) A.L. Cauchy (178!}-1857) played a prominent role in the development of complex analysis,
and you will see his name several times t hroughout this text. The last name is not pronounced
as "kaushee." The beginning syllable has a long "o" sound, like the word kosher, but with the
second syllable having a long "e" instead of "er" at the end. Thus, we pronounce Cauchy as
"kOshe."