Midterm Test of Complex Analysis
Lifelong Education College, SJTU
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�!(20%) For the next statements, mark the correct ones with
, and the wrong ones with ×.
(b) w = z is not differential everywhere in C ( )
(c) For any simple closed contour C in C,
(z2 + 2 sin z − 3ez)dz = 0 ( )
(d) For w = f(z) continuous in a domain Ω ⊂ C, then f(z) is analytic in Ω if and only
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
(a) For z = 1−i, its principal argument ( ) and argument ( )
(b) Write w =
in form a+ ib ( )
(c) The derivative of the power function (1+i)z is ( )
(d) The set of points at which w = znz, n ∈ N, differentiable is ( )
(e) The set of points at which Logz analytic is ( )
n!(10%) Write (1− i)5 is rectangular form, and point out its principal argument and argument.
o!(10%) Present all three 3th roots z1/3 of z = − 1√
, and compute the logarithm of the
second one
Ê!(10%) Compute the limits
5z3 − iz + 1
2z3 − z2 − 1
8!(10%) (a) Verify that z2 is not differentiable at any z0 6= 0, and is differential at z0 = 0.
(b) By (a), explain that why z2 is nowhere analytic in C
Ô!(10%) Evaluate the integral ∫
(2z − 3z2)dz
with C the contour z = z(θ) = eiθ, θ ∈
l!(10%) Let z0 = −2 − 2i, and C be a positively oriented regular octagon centered at z0.
Compute the integral ∫
(z − z0)n−1dz, n ∈ Z
[Needs the detail derivation on your conclusion]
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