An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 83. 84.

Analytic functions.

141

this quotient tend for Az = 0, constitute the derived of the complex
function.
We shall concern ourselves in the sequel only with functions for
which, except in singular points, this limiting quantity is a function of
S = x + iy exclusively and is thus independent of the value qS or of the
ratio A y: Ax; such functions are called a al y tic functi o n s. For an
analytic function f(z) therefore:
1) Lim f ( + - f ( );
it has a derived function, not only as in real functions identical when
taken progressively and regressively but the same in every direction.
We shall show that every function expressed by an infinite series of
powers is analytic within the convergency of this series, and that
also conversely every function that is analytic within a domain can
within this domain be expressed by infinite series of powers*).
Equation 1) can also be written thus:
2) df (Z) -
2) dr() f'(z) or: cf(z) =- '(). d- = f'(z) (dx + i dy).
This last form is the equation for the Total Differential of the
complex function.
If we make the complex variable z change only by the real part
Ax or by the purely imaginary part iAy, we obtain as limiting
values of the quotient of differences the partial derived functions with
regard to x or to y. But these likewise, in consequence of our hypothesis, satisfy the following equations:
~f(z) =Limi f(z + Ax) - f(Z) f
Ax
3) O Li f (z + iA ) ( f (z).
-Li -i. f __ )
iay inzJy
Therefore the analytic function regarded as a function of the two
variables x and y satisfies the equation:
4) Of- O oraf r f + af;4) a~ - x ~y, i- - - -- i ax ay
If we ask whether these equations are also sufficient conditions,
that there may be at a point one derived function depending only on
z for every direction; the answer is:
Provided there exist in the neighbourhood of the point, definite
values of the partial derived functions - and 1 O that are ways
e_ x i ay always
*) Riemann styled functions on the hypothesis of their analytical property,
simply, functions of a complex variable. Cauchy called functions that are
analytic in a domain without exception, synectic. Briot and Bouquet call such
functions holomorphe.