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.

~ 177. 178.

Definition of the complex integral.

discrete set of points, a progressive differential quotient that is itself
generally a continuous function and is identical with the regressive
differential quotient; or in geometric terms: the curve has no angles
except at discrete points. The intervals can therefore (~ 100) be assumed
so small, that generally for each value of t:
(tk + 1) -- )(t) +
tk+1 - tk
where 6 signifies an arbitrarily small quantity. Accordingly each of
the four sums takes a form such as:
k=n-.1 ke-n-1
_f, (tk) '(tk) (tk+I -- tk) +- 6 f1 (tk)(tk+ -l tk),
kA= k=0
that passes over into the definite integral:
T
f, (t) '(t) dt
to
as the value of n increases arbitrarily. Therefore:
z T
I. Lim f(,k)(Zk+- k) j f(s)d==j {f(t)(t) - f9(t), t)}t
Zo to
[s T
+ I {f2 ({)'( + fl(t)(t) (} i-t if (t) + if2(t) } { '(t) + i'(t) }d
to to
is a determinate finite quantity.
For instance, if the integral is to be formed in a straight path from
the point z0 = xo + iyo to the point Z =- X + iY, we have:
X = X0 +- (X - o)t, y =- o + (Y - yo)t,
therefore:
Z t=-1
A( d= {- X-Xo +i (Y-yo ) ff(X(+ i+yo+ —t(-xo)+it(Y -yo))ct.
zo tO
When we require to integrate along the are of a circle of radius r
whose centre is the point xo + iyo, we have by the equations:
x = xo +rcost, y ==yo - rsint dx = - rsintdt, dy = rcostdt;
Z T T
Jf(Z)dZ Af(xfx + iyo -+- r eit)ir eit = irf f(x + iyo + r e)eit d eIt,
so to to
where to and T denote the values of t belonging to the initial and
terminal points of the arc.
178. It results from Equation I. that we also have for the
complex integral the following theorems:
21*