Page 402 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 16] FUNCTIONS OF A COMPLEX VARIABLE 393

LIMITS AND CONTINUITY
Definitions of limits and continuity for functions of a complex variable are analogous to those for a
real variable. Thus, f ðzÞ is said to have the limit l as z approaches z 0 if, given any > 0, there exists a
> 0 such that j f ðzÞ lj < whenever 0 < jz z 0 j < .
Similarly, f ðzÞ is said to be continuous at z 0 if, given any > 0, there exists a > 0 such that
j f ðzÞ f ðz 0 Þj < whenever jz z 0 j < . Alternatively, f ðzÞ is continuous at z 0 if lim f ðzÞ¼ f ðz 0 Þ.
z!z 0
Note:While these definitions have the same appearance as in the real variable setting, remember that
jz z 0 j < means
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
ðx x 0 Þ þð y y 0 Þ < :
jðx x 0 jþ ið y y 0 Þj ¼
Thus there are two degrees of freedom as ðx; yÞ!ðx 0 ; y 0 Þ:

DERIVATIVES
If f ðzÞ is single-valued in some region of the z plane the derivative of f ðzÞ, denoted by f ðzÞ,is defined
0
as
lim ðf ðz þ zÞ f ðzÞ
z!0 z ð1Þ
provided the limit exists independent of the manner in which z ! 0. If the limit (1) exists for z ¼ z 0 ,
then f ðzÞ is called analytic at z 0 .If the limit exists for all z in a region r, then f ðzÞ is called analytic in r.
In order to be analytic, f ðzÞ must be single-valued and continuous. The converse, however, is not
necessarily true.
We define elementary functions of a complex variable by a natural extension of the corresponding
functions of a real variable. Where series expansions for real functions f ðxÞ exists, we can use as
definition the series with x replaced by z. The convergence of such complex series has already been
considered in Chapter 11.
z 2 z 3 z 3 z 5 z 7 z 2 z 4 z 6
EXAMPLE 1. x þ .
2! þ 3! þ ; sin z ¼ z 3! þ 5! 7! þ ; cos z ¼ 1 2! þ 4! 6!
We define e ¼ 1 þ z þ
x
x
From these we can show that e ¼ e xþiy ¼ e ðcos y þ i sin yÞ,aswell as numerous other relations.
Rules for differentiating functions of a complex variable are much the same as for those of real
d n n 1 d
variables. Thus, ðz Þ¼ nz ; ðsin zÞ¼ cos z, and so on.
dz dz
CAUCHY-RIEMANN EQUATIONS
Anecessary condition that w ¼ f ðzÞ¼ uðx; yÞþ ivðx; yÞ be analytic in a region r is that u and v
satisfy the Cauchy-Riemann equations
@u @v @u @v
;
@x ¼ @y @y ¼ @x ð2Þ
(see Problem 16.7). If the partial derivatives in (2) are continuous in r, the equations are sufficient
conditions that f ðzÞ be analytic in r.
If the second derivatives of u and v with respect to x and y exist and are continuous, we find by
differentiating (2) that
2
2
2
2
@ u @ u ¼ 0; @ v @ v
@x 2 þ @y 2 @x 2 þ @y 2 ¼ 0 ð3Þ
Thus, the real and imaginary parts satisfy Laplace’s equation in two dimensions. Functions satisfying
Laplace’s equation are called harmonic functions.