6
Diffe1rentiation
6.1 THE CONCEPT OF THE DERIVATIVE.
MONOGENIC AND ANALYTIC FUNCTIONSDefinition 6.1 Let w = f(z) be a single-valued complex function defined
in some open set G of the complex plane, and let z 0 be a point of G. If
l~zl is small enough, the point zo + ~z will belong to the same component
of G to which z 0 belongs. By the derivative off at z 0 , denoted f1(z 0 ), it
is understood the limit
!
,( ) _
1. f(zo + ~z) - J(zo)
Zo :- Im A
Az-->0 ~z
(6.1-1)provided that this limit exists (as a finite complex number). By the defi-
nition of limit the indicated limit then has a unique value for all methods
of approach of ~z to zero. Alternafive notations for the derivative of f at
Zo are Df(zo) and (dw/dz)z=zo·If the derivative of f exists at points of a certain subset G1 C G, its
value depends on the point z chosen in G 1 , and there results a single-
valued function, denoted f1 ( z ), which is called the derived function off on
G1. This function is then defined by the formula
J1(z) = lim f(z + ~z) - f(z),
Az ...... o ~z
(6.1-2)it being understood that G 1 is a subset of G for which the limit in (6.1-2)
exists.
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