Begin2.DVI

C =C 1 ∪C 2 ∪···∪ Cm, then the line integral can be broken up and written as a

summation of the line integrals over each section of the curve which is smooth and

one would express this by writing

∫

C

f(z)dz =

∫

C 1

f(z)dz +

∫

C 2

f(z)dz +···+

∫

Cm

f(z)dz (12.188)

Indefinite integration

If F(z)is a function of a complex variable such that

dF (z)

dz =F

′(z) = f(z)

then F(z) is called an anti-derivative of f(z)or an indefinite integral of f(z). The

indefinite integral is denoted using the notation

∫

f(z)dz =F(z) + cwhere cis a con-

stant and F′(z) = f(z)Note that the addition of a constant of integration is included

because the derivative of a constant is zero. Consequently, any two functions which

differ by a constant will have the same derivatives.

Example 12-18. (Indefinite integration)

Let F(z) = 3 sin z+z^3 + 5 z^2 −zwith dF

dz

=F′(z) = 3 cosz+ 3 z^2 + 10z− 1 , then one

can write

∫

(3 cos z+ 3 z^2 + 10z−1)dz = 3 sin z+z^3 + 5 z^2 −z+cwhere cis an arbitrary

constant of integration

The table 12.1 gives a short table of indefinite integrals associated with selected

functions of a complex variable. Note that the results are identical with those derived

in a standard calculus course.