推 arsenefrog : 收電機系精華 05/02 19:59
課程名稱︰工程數學-複變
課程性質︰必修
課程教師︰陳士元
開課學院：電機資訊學院
開課系所︰電機工程學系
考試日期（年月日）︰104/4/21
考試時限（分鐘）：110分鐘
試題 :
COMPLEX ANALYSIS
Midterm (2015/4/21, 10:20 AM-12:10 PM)
_
z
1. Prove that f(z) = e is nowhere analytic. (8%)
2. Find the imgae of the line y = a (a is a real-valued constant) in the
w-plane under the mapping f(z) = sin z . (8%)
-1 2
3. Find the derivative of tanh (z + 4√2 z + 8) at z = -√2 . (8%)
4. Find all values of the given quatity.
-1
(1) cos (coshπ) [Hint: use the formula cosz = cosx coshy - i sinx sinhy]
(10%)
-1
(2) ln(tan (0)) (10%)
(1-i)^2
(3) (1-i) (8%)
5. Evaluate the given integrals along the indicated contour C. (10% each)
1
(1) ∮ ───── dz , where C: |z| = 2 .
C z^4 - 1
2 2
cosh z 2 y
(2) ∮ ──────── dz , where C: 100x + ─── = 1
C (z - 3πi)^2 100
2π ircosθ
(3) ∫ e cos(irsinθ - nθ)dθ, where n is a positive integer and r
0
is a posistive real number. [Hint: form a corresponding comples contour
integral]
1
6. Find the Maclaurin series representation of f(z) = ──────── . (8%)
(z - i)(z - 2)
7. What is the radius of convergence R of the power series expansion of
z^3 - 1
f(z) = ─────────── centered at the origin. (4%)
z^2 - (1 + 2i)z + 2i
π
8. Find all isolated singular points of f(z) = csc(---) . (6%)
z
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※ 編輯: NTUkobe (140.112.73.49), 05/02/2015 18:02:22