課程名稱︰Complex Analysis
課程性質︰必修
課程教師︰王暉
開課系所︰電機系
考試時間︰2006/4/18 10:20~12:10
試題 :
1. True or false (If it is false, please explain the reasons briefly) (20%)
(a) Ln z is analytic for |z|＞0 and its deivative is 1/z.
(b) A function f is analytic in a simple connected domain D and C is any
contour in D. Then ∫f(z) is independent of the path C.
c
(c) A function f is analytic at point z_0 if and only if f is differentiable
at z_0 and every point in every neighborhood of z_0.
(d) The only bounded entire function is zero.
(e) If f is analytic in simply connected domain D. Then f possesses
derivatives of all orders at every point z in D, and they are all
analytic in D.
2. Suppose the function f(z) = u(r,θ)+iυ(r,θ) is analytic at point z whose
polar coordinates are (r,θ).
Please prove (15%) (偏微分)
du 1 dυ
(a) the Cauchy-Riemann equations in the polar coordinate is ── = ─ ──
dr r dθ
dυ 1 du
and ── = - ─ ── , and
dr r dθ
du dυ
(b) the derivative of f at (r,θ) is f'(z) = e ^(-iθ)(── + i ──)
dr dr
3. Suppose the function f(z) = u(x,y) + iv(x,y) is analytic in domain D. Then
the real and imaginary parts of f can be used to define two families of
curves , u(x,y) = c1 and v(x,y) = c2, in D, where c1 and c2 are arbitrary
real constants. Please prove that these two families of curves are
orthogonal. (20%)
4. Please find all values of the given quantity: (20%)
-1
(a) sinh i
(b) cosh(1+iπ/6)
(c) ln(-2+2i)
(d) (1+i)^(1-i)
5. Please find the values (15%)
e^z _
(a) ∮( ── - 3z) dz, where |z| = 1
c z+3
i
(b) ∫e^z cosz dz
π
(c) ∮Ln(z+10)dz, where |z| = 2
c
6. Please find the values (10%)
z^2
(a) ∮─── dz, where |z-i| = 2
c z^2+4
1
(b) ∮───── dz, where |z-2| = 5
c z^2 (z-1)^2
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※ 編輯: supersatan 來自: 220.139.90.41 (04/18 19:17)