[OBJECTIVE]
Subject: Complex Analysis-II
Time Allowed: 15 Minutes
Maximum Marks: 10
NOTE: Attempt this Paper on this Question Sheet only. Please encircle the correct option. Division of marks is given in front of each question. This Paper will be collected back after expiry of time limit mentioned above.
Part-I Encircle the right answer, cutting and overwriting are not allowed. (10)
1. An elliptic function with no poles in a cell is a ___________.
a) Entire function
b) Constant function
c) Periodic function
d) None of these
2. For f(z) =sinz/z, z=0 is _________________.
a) Isolated singularity
b) Non isolated singularity
c) Essential singularity
d) Removable singularity
3. If the principal part in the Laurent series contains an infinitely many nonzero terms, then the singularity is called a _______________.
a) Isolated singularity
b) Non isolated singularity
c) Essential singularity
d) Removable singularity
4. Res (e-2/z2,0) = ____________.
a) 2
b) 3
c) 1
d) 0
5. Sum of Residues of f(z) = 1/(z-1)2(z-3) is ____________.
a) 0
b) -1
c) 7
d) -7
6. The number of poles of an elliptic function in any cell is _____________.
a) Finite
b) Infinite
c) Zero
d) None of these
7. An elliptic function of order less than two is __________.
a) Entire function
b) Constant function
c) Periodic function
d) None of these
8. f(z)= (2z+5)/(z-1)(z+5)(z-2)4 has a zero at z=2 of order ________________.
a) 1
b) 4
c) 2
d) 3
9. The principal part of f(z) = sinz/z2 is ______.
a) 1/z
b) -1/z
c) 1/z2
d) 0
[SUBJECTIVE]
Subject: Complex Analysis-II
Time Allowed: 2 Hours 45 Minutes
Maximum Marks: 50
NOTE: ATTEMPT THIS (SUBJECTIVE) ON THE SEPARATE ANSWER SHEET PROVIDED.
Part-II Give short notes on following, each question carries equal marks. (20)
Q#1: Determine whether z=0 is an essential singularity of f(z) = e2+1/z
Q#2: Evaluate O/e3/z where the contour C is the circle |z|=1.
Q#3: Prove that an elliptic function with no poles in a cell is a constant function.
Q#4: Find the order of the poles of f(z) = tanz.
Part-III Give detailed answers, each question carries equal marks. (30)
Q#1: Prove that the sum of residues at the poles in a cell of an elliptic function is zero.
Q#2: State and Prove Weierstrass’s Factorization theorem.
