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.

~ 84. 85.

Differential quotients.

143

du. dv
df(z) d -+ idv _ dx + dx
dz dx +- idy dy
dx
since f'(z) - + -i V, we obtain the equation:
du dv (au av\( i dy)
d x -t dx ' ( + +T i d)'
from which and 6) we find for the total differential quotients of te and v:
du a av dy 8u u dy
dx ax a x 'dxax a -y 'ay dx'
dv av a u dy _ v av dy
dz ax + ax dax a x + y dx
Conversely, if the Theorem of the Total Differential be supposed to
hold for u and v, equations 6) are sufficient conditions that the combination u + iv = w be expressible by arithmetical operations on the
single variable S = x + iy. For then, replacing x by z - iy in the
combination u + iv, this must become altogether independent of the
variable y, i. e. its partial differential quotient with respect to y must
vanish. (~ 26. 1.)
Denoting the result of substituting x = - iy in u + iv by
(i) - i(v), we find
a (m) am au a (v\) __ v av
(y -- ~ (- i) + y -- (_ i) +
ay a sx oy y Ax ay
when z - iy is substituted for x in the derived functions on the right.
Accordingly, combining these, we have:
a u) - ( acv= ) (am av (-a av\
ay + aiy \ ay axJ ax ay~
In consequence of equations 6) the expressions in the brackets vanish;
they are therefore sufficient conditions that w should depend on z only.
85. The property of an analytic function of a complex variable, that
its first derived f'(z) is independent of the ratio dy, is important in
the geometrical transformation upon plane A of plane B that represents the values of the function w = f(z). If we consider in plane A
a triangle PP'P", whose vertices belong to the values a, s + Az,
z +- Az', to these correspond in plane B three points QQ'Q", whose
values we may denote by w, wz Aw, w + Aw'. Transposing the
system of coordinates in each plane so as to make the points P and Q
the origins of the systems, and putting:
A - Ar. eA'P, e Az ' - A r'. eiP' At e w A -. A ei', A w g' e
the quantities introduced are in each plane the polar coordinates of
the other two vertices of. the triangle in regard to the origin. But
by the analytic property, the quotients:
/Aw = ei(P-9) and l w' =. e('-')
Az Ar Az' Ar'