May 2024
Intermediate to advanced
481 pages
17h 15m
English
Content preview from Complex Analysis
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“ca: ch_04” — 2012/3/27 — 21:16 — page 115 — #4
Complex Integration
|
115
Solution: We have,
π
0
te
−it
dt =
π
0
t( cos t − i sin t)dt
=
π
0
t cos tdt− i
π
0
t sin tdt
=
(
t sin t + cos t
)
π
0
− i
(
−t cos t + sin t
)
π
0
=−2 − iπ
4.4 CONTOURS
Unlike real functions, the complex-valued functions of a complex variable are integrated along curves
in the complex plane, instead of intervals of real line. Some adequate classes of curves are given
below.
4.4.1 Path
A path or an arc C in a complex plane is defined as set of points z = x + iy if x = x
(
t
)
, y = y
(
t
)
;
a ≤ t ≤ b, where x
(
t
)
and y
(
t
)
are continuous functions of real parameter t.
The points of C can be described by the equation ...
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