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.

~ 178.

Properties of the complex integral.

325

along the path of integration lo also converges to zero, we have by
equation 3):
o+ I
4) 7J f(z)dc = f/ (t + 0k) -+- if(t + O'7),
therefore:
z-t- hL
+" t
Lirn (z) d z (t) -- i (t) -- f '().
The definite integral, regarded as a function of its upper limit,
has therefore generally in the points of the path of integration the
derived function f(z). In particular also:
a f -(z) d z af, /(z) d -
f_ - = f _- = i) ),
z x ' - 7? ay / v' /
by which is expressed, in conformity with ~ 84, that the integral is a
function of the complex variable x + iy = z.
When along the path of integration a unique analytic function
Fv(z) is known, whose derivate is equal to f(z), we have likewise:
z
5) ff(z)dz = F(Z) - F(zo).
Z0
For if by the substitution z= g (t)+ iO (t), f(z) passes into fi (t) + it (t),
and F(s) into F (t) + iF2 (t); we have:
dtz
i'(z- ) f(g)= { F '(f + i Fz'(t) },
therefore:
{/i (1) + if, ( ()} { T(t)0 + 2"(}( -)) + '2(t),
consequently:
Z T T
J'I(z) dl ff(t))+if,2(t)1 { '(t)+ i 4(t)}1 ct 125' (1) + iF7(t) } dt
zo to $o
={F, (T) + iF9(T)} - {(to) + iF2(to) _ F(Z)- F(z).
The definition of the definite integral, furthermore, can be
extended as it was before:
When the function f(z) becomes infinite in discrete points c,,
c2, etc., the equation of definition takes the form:
f(z)>l == Lim Cf(zc)dz.
o a o t oo
Here also, as for the real integral, the following criterion holds: