Page 46 - Calculus with Complex Numbers

Ffg ure 4.2

An integrable function is one for which the above limit exists linite. A primitive
for f @) is any F @) such that F/ @) = f (x). Theorem 1 guarantees that the
integral exists for any continuous function. Theorem 2 gives us a practical method
for evaluating integrals. Together with the following combination rules.

*. Linear combination rule

*. Product rule (integration by parts)
b
J
?
/J ? . , - - jb ,-(-).,(-)-s.-.
,(-).,-(-)-s.- - E ,(.-).?(.-)q'-
*. Composite rule (integration by substitution)

b
yt-vl-ï.x - j' y(g(,))g-(,),,,
/ -
J
where g ((y) = a, g @) = b.
For integrals which cannot be evaluated exactly we have the inequalities

b
/- -
flx) dx :% jb I./*(.r)1dx :% M(b - J),

4 .2 C ontou rs
lnstead of intervals we shall integrate complex functions flz) of the complex
variable z along contours. By a contour y we mean a continuous curve in the
complex plane. A parametrisation of y is a representation of y as
?' = 1/ (/): a :i t :i /$ ),