Department of Mathematics
MAT–3003: Complex Variables and Partial Differential Equations
Digital Assignment–I
Slot: A2+TA2+TAA2
Faculty: Dr. R. Nageshwar Rao
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1. Show that u(x, y) is harmonic in some domain and find a harmonic conjugate v(x, y) when
(a) u(x, y) = y 3 − 3x 2 y
(b) u(x, y) = 2x(1 − y)
2. The velocity potential for a planar flow of an incompressible and irrotational fluid is given by
x
φ(x, y) = 2
. Find the complex velocity potential and the velocity field F of the flow.
x + y2
3. Suppose f (z) = φ + iψ is an analytic function.
(a) Show that φ and ψ satisfy Laplace’s equations.
(b) Discuss any possible physical significance for the functions φ and ψ.
(c) If ψ = x 2 − y 2 + 2 y, find φ(x, y).
4. Find the image of the region bounded by the lines x − y < 2 and x + y > 2 under the mapping
w = 1/z. Sketch the regions in the z− plane and their images in the w−plane.
5. Determine the image of |z − 2| = 2 under the transformation w = z 2 and sketch the graph of
the region in z− plane and its image in the w− plane.
6. Consider the points ∞, 1, 0 in z−plane is mapped onto the points 0, 1, ∞ under the bilinear
transformation f (z).
(a) Find the bilinear transformation .
(b) Determine the fixed points of the transformation.
(c) Show that the imaginary axis is mapped onto itself.
7. Find the Bilinear transformation that maps the points ∞, i, 0 in the z−plane to −1, −i, 1 in the
w−plane. Find the invariant points and also find the image of the circle |z| < 1 in the w−plane.
8. Expand the following functions in Taylor series with the center z0 as specified. Also mention
the region of validity in each cases.
1
(a)
about origin.
2
(z + 1)(z − 1)
1
(b)
about z = −1.
(z − 3)(z + 2)2
1
(c)
about z = −i.
(z + 1)2
(d) cos z about z = π/2.
3−i
(e) f (z) =
about z0 = 4 − 2i.
1−i+z
9. Expand the following functions in Laurent series valid in the regions specified.
1
(a) f (z) = 2
about z = 0 in 1 < |z| < 2.
z − 3z + 2
1
(b) f (z) =
about z = 1 in all possible regions.
(z + 1)(z + 2)2
10. Write a short note on (i) Zeros (ii) Singularities (iii) Poles of Analytic Functions. Give suitable
examples in each case.
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