STUPIDSID
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.
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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
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
is irrotational. Also find the corresponding potential function
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3 (a)
Evaluate the following:
using the residue theorem.
using the residue theorem.
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3 (b)
If 
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
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:
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
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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