Applied Mathematics 4 - Dec 2013

Electronics & Telecomm. (Semester 4)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks. 1 (a) Show that there does not exist any analytic function f(z) = u + iv such that:
(5 marks) 1 (b) Find the poles of f(z) = (sec z)/z² which lie inside the circle C: |z| = 2. Also find the residues of f(z) at these poles.(5 marks) 1 (c) Show that:
(5 marks) 1 (d) A is a 3×3 matrix whose characteristic polynomial is λ³+2λ²+3λ+4. Find the sum of the eigen values of A^-1.(5 marks) 2 (a) Show that the bilinear transformation
maps |z| ≤ 1 onto |w| ≤ 3(7 marks) 2 (b) Show that the matrix is diagonalisable
(6 marks) 2 (c) Show that
is irrotational. Also find the corresponding potential function(8 marks) 3 (a) Evaluate the following:
using the residue theorem.(6 marks) 3 (b) If
show that
(6 marks) 3 (c) Verify Green's theorem for
over the region bounded by 1 ≤ x ≤ 2 and 1 ≤ y ≤ 3(6 marks) 4 (a) Show that:
(6 marks) 4 (b) Evaluate the following:

over the region bounded by y = 0, y = 2x, x + y = 3(6 marks) 4 (c) Show that A is diagonalisable if and only if A is derogatory
(6 marks) 5 (a) Show that the Eigen values are unit of modulus and check if the eigen vectors are orthogonal
(6 marks) 5 (b) Find a and b such that u = (5x + 3y)(2x² + axy + by²) is a harmonic function.(6 marks) 5 (c) Find the analytic function f(z) whose real part is
(8 marks) 6 (a) Evaluate ∫_C Zdz over the upper half of C: |z|=2, traversed in the anti-clockwise direction.(6 marks) 6 (b) Verify the Gauss divergence theorem F = (x² - yz) + (y² - zx)↑ + (z² - xy)↑ Over the surface S: 0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z ≤ c(6 marks) 6 (c) Find the Laurent Series expansion of f(z)=1/(z+1)(z+3) in
(i) |z| < 1; (ii) |z| > 3; (iii) 0 < |z+1| < 2(8 marks) 7 (a) Verify Stoke's Theorem for
where S is the upper hemisphere x² + y² + z² = 1, z > 0(6 marks) 7 (b) Diagonalise the quadratic form Q = 2xy + 2xz - 2yz using an orthogonal transformation. (6 marks) 7 (c) Prove the following equation:
(8 marks)