[SOLVED] Math3979 Midterm Test P0

Midterm Test of Complex Analysis
√
!(20%) For the next statements, mark the correct ones with , and the wrong ones with ×.

(b) w = z is not differential everywhere in C ( )
Z
(c) For any simple closed contour C in C, (z2 + 2sinz − 3ez)dz = 0 ( )
C
(d) For w = f(z) continuous in a domain Ω ⊂C, then f(z) is analytic in Ω if and only if RC f(z)dz = 0 with C any closed contour interior to Ω ( )
(e) If f is analytic in a domain Ω, and f ≡ 0 on the curve S ⊂ Ω, then f ≡ 0 in Ω
( )
!(20%) Putting your answers in the paces Assignment Project Exam Help
(a) For z = 1−i, its principal argument ( ) and argument ( )
(c) The derivative of the power function (1+i)z is ( )
(d) The set of points at which w = znz, n ∈N, differentiable is ( )
2 n!(10%) Write (1−i)5 is rectangular form, and point out its principal argument and argument.
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o!(10%) Present all three 3th roots , and compute the logarithm of the
second one
3
˚!(10%) Compute the limits
,
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(b) By (a), explain that why z2 is nowhere analytic in C
4
!(10%) Evaluate the integral
with C the contour
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[Needs the detail derivation on your conclusion]