1 (a)
Show that there does not exist any analytic function f(z) = u + iv such that:
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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 M
1 (c)
Show that:
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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.
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2 (a)
Show that the bilinear transformation
maps |z| ≤ 1 onto |w| ≤ 3
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2 (b)
Show that the matrix is diagonalisable
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2 (c)
Show that
is irrotational. Also find the corresponding potential function
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3 (a)
Evaluate the following:
using the residue theorem.
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3 (b)
If ![](data:image/png;base64,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)
show that
show that
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3 (c)
Verify Green's theorem for
over the region bounded by 1 ≤ x ≤ 2 and 1 ≤ y ≤ 3
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4 (a)
Show that:
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4 (b)
Evaluate the following:
![](data:image/png;base64,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)
over the region bounded by y = 0, y = 2x, x + y = 3
over the region bounded by y = 0, y = 2x, x + y = 3
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4 (c)
Show that A is diagonalisable if and only if A is derogatory
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5 (a)
Show that the Eigen values are unit of modulus and check if the eigen vectors are orthogonal
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5 (b)
Find a and b such that u = (5x + 3y)(2x2 + axy + by2) is a harmonic function.
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5 (c)
Find the analytic function f(z) whose real part is
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6 (a)
Evaluate ∫C Zdz over the upper half of C: |z|=2, traversed in the anti-clockwise direction.
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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
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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
(i) |z| < 1; (ii) |z| > 3; (iii) 0 < |z+1| < 2
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7 (a)
Verify Stoke's Theorem for
where S is the upper hemisphere x2 + y2 + z2 = 1, z > 0
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7 (b)
Diagonalise the quadratic form Q = 2xy + 2xz - 2yz using an orthogonal transformation.
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7 (c)
Prove the following equation:
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