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.

142

Functions of a complex variable.

Bk. II. ch. II.

equal and moreover are continuous functions of the complex variable
x -+ iy, we have also:
df (z) af O af
For, we have:
f(x-+-iy+A x +i Ay) —f(x+i y) f(x-i y+A X+ iAy) —f(x +iy+ iAy) Ax
Ax + iAy Ax Ax+iAy
+ f(x +i y+iAy)- f(x-+iy). ay.
iAy Ax + iAy
Now Ay can be chosen so small that:
f (x + iy + iAy) - f(x +- iy) ' f(x +iy),
iAy i ay
where ' denotes a quantity arbitrarily small in amount. Further we
can choose the value of Ax so that:
f(x +iy x + iA/y) - f(x +iy+ iAy) _ f(x +-iy +OAx + iAy)
Ax ax
f -- (x --- i y _
From this it follows: values can be assigned to Ax and Ay such that
for them and for all smaller values the above quotient of differences
I Of
shall differ from the value - f at most by the quantity
+ da
_Ax;TiAy
Ax + iAy + A+x +- iAy
the modulus of 6 being arbitrarily small. Since the amount of the
quotients by which 6 is multiplied cannot increase beyond all limits,
what we have stated is proved.
The real and imaginary constituents u +- iv into which the complex
function f(z) resolves, are functions of the two real variables x and y.
But as the equation of condition 4) must be fulfilled, they are functions
of two variables of a special kind: the functions u and v cannot be
independent. In fact from:
f(z)= — u + iv; we have: '- a +. Y + X -.a
- / - i ( x x' +yx - - -y -yi r
then by equation 4) we find:
5)(at 3)+ ay = 0.
By separating the real and the imaginary, this equation resolves into:
6) au _ V a _ av
Dax - y' a y — x
The two constituents of an analytic function are therefore generally
continuous functions with determinate differential quotients. For these
functions the Theorem of the Total Differential holds. For, if we write: