课程名称︰工程数学-复变
课程性质︰必修
课程教师︰陈士元
开课学院：电机资讯学院
开课系所︰电机工程学系
考试日期（年月日）︰104/4/21
考试时限（分钟）：110分钟
试题 :
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(