Real and Complex Analysis (Rudin)

ELEMENTARY PROPERTIES OF HOLOMORPHIC FUNCTIONS 201
points Sj' IX = So < Sl < ... < Sn = p, and the restriction of Y to each interval
[Sj-l' Sj] has a continuous derivative on [Sj-l, Sj]; however, at the points
Sl' ... , Sn-l the left- and right-hand derivatives of Y may differ.
A closed path is a closed curve which is also a path.
Now suppose Y is a path, and f is a continuous function on y*. The
integral off over y is defined as an integral over the parameter interval [IX, P]
ofy:
r
if(Z) dz = f(y(t))y'(t) dt. (1)
Let qJ be a continuously differentiable one-to-one mapping of an interval
[lXI' PI] onto [IX, P], such that qJ(lXl) = IX, qJ(Pl) = p, and put Yl = Y 0 qJ. Then
Yl is a path with parameter interval [1X1o PI]; the integral off over Yl is
fJI J.fJI J.fJ