Unformatted text preview: Complex Analysis (Mid—term Exanm April 18$ 2006 [10:20am — 12zl0am)
1. True or false (lfit is false: please explain the reasons brieﬂy) (20%)
(a) Ln 3 is analytic for Jeri :- 0 and its derivative is 1!: . (b) A function f is analytic in a simple connected domain D and C is any contour in D. Then If(s)d: is independent of the path C.
C (c) A function f is analytic at point an if and only iff is differentiable at an and every point in
every neighborhood of Zn- (d) The only bounded entire function is zero (e) If f is analytic in a simply connected domain D. Then f possesses derivatives of all orders at
every point a in D: and. they are all analytic in D. I
2.: Suppose the function f(:} : n(r,,a)+ ems) is analytic at point :2: whose polar coordinates are (:36).
Please prove (15%) r1 1": "
W=mﬂ and wai=——}rEf—5and (a) the Cauchy-Riemann equations in the polar coordinate is T
or r of} Sr r 39 (b) the derivative of fat (ea) is f’(s)=e‘l9(%1i+i;ﬂ).
I" I' 3. Suppose the function f(3) 2 am y) + iv(xj y) is analytic in domain D. Then the real and
imaginary parts off can be used to deﬁne two families of curves, Mr: 32) r: c] and 1/10.; 3:) 2 c2 3
in D; where c; and c; are arbitrary real constants. Please prove that these two families of curves
are orthogonal. (20%) I I i 4. Please find all values of the given quantity: (20%)
(a) sinh‘l i (b) cosh(l+%i) f' (c) h1(—2+2i')
5. Please ﬁnd the values (15%) (a) (hi-3 —3s)ae, where |e| =1
C. 1::- (b) Pile: cos ads
(C) Cf Lama-10M; where lEl = 2
1‘: 6. Please ﬁnd the values (10%) s i
(a) (grim dz, where ls—r‘f :2
'T—"mti-‘J wher 3—2 :5
(b) gs'gel)‘ Ei ' l ...
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- Spring '07