## 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)/z^{2} 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 λ^{3}+2λ^{2}+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)(2x^{2} + axy + by^{2}) 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 = (x^{2} - yz) + (y^{2} - zx)↑ + (z^{2} - 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 x^{2} + y^{2} + z^{2} = 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)