B. Tech Degree IV Semester Examination, April 2009 IT/CS/EC/CE/ME/SE/EB/EFEE/FT 401 ENGINEERING MATHEMATICS III CUSAT

B. Tech Degree IV Semester Examination, April 2009

IT/CS/EC/CE/ME/SE/EB/EFEE/FT

401 ENGINEERING MATHEMATICS III

(2006 Scheme)

Time : 3 Hours Maximum Marks : 100

PART - A

(Answer ALL questions)
(8 x = 40)
I. (a) • Show that the function f (z) = sin z is analytic.
(b) What do you mean by conjugate harmonic function? Verify whether the function ex sin y is harmonic.
(c) State and prove Canchy's integral formula.
(d) Define the residue of a function at an isolated singularity and determine the poles
z2 2z
(.z +1)2 (z2 +1)
(e) Form the 'partial differential equation by eliminating the arbitrary constants in ‘2
z=(x—a) -1Jy—b‘2) +1.
(0 Solve p2 + q2 = x2 + y2.
(g) Derive one dimensional heat equation.
au au
(h) Solve the equation —= 4—, u (0, y) = 8
ax ay
PART — B
(4 x 15 = 60)
II. (a) Show that the function f (z) = .\11xyl is not regular at the origin, eventhough Canchy
Riemann equations are satisfied at the origin. (5)
(b) Find the analytic function w = u + iv, given that v e2x (X cos 2y y sin 2y). (5)
(c) Find the equation of the orthogonal trajectories of the family of curves given by
3x2y + 2x2 — y3 — 2y2 = a, where a is an arbitrary constant. (5)
OR
III. (a) What do you mean by conformal mapping? Also discuss about
(i) Translation
(ii) Magnification and rotation (5)
(b) Discuss the transformation about w = sin z. (5)

(a) Verify Canchy's theorem for the integral of 23 taken over the boundary of the
rectangle with vertices —1,1,1 +1,-1 + 1.
z2 —
(b) Find the Lausent's series expansion of about z = 0 in the region
z +5z+6
< Izi < 3.
(c) Using Residue theorem, evaluate is 3z2 4-
2
 dz where C :12 — 21 = 2.
OR
dx
V. (a) Evaluate using contour integration where a > b > 0.
(x(x2 + b
(b) Evaluate '1°s/sin mx
tine using contour integration where m > O.
X
VI. Solve (i) p(1+0=qz
(ii) z2 (p2x2 q2 =1
(iii) (x2 + _2
y + yz) p + (x2 + y2 — xz
q=z(x+y)
IV. Solve — OR
(i) (IY —3D Di + 4.1TY z ex+2Y
(ii) (D2 —3DDI +13t2 z = sin x cos y
(iii) (D3 —7 D.D12 — 6 = x2 y + sin (x + 2y)
VIII. (a) A string is stretched and fastened to two points £ apart. Motion is started by displacing
the string in the form y = a sin (,rx 1 £) from which it is released at time t = O. Show that the displacement of any point at a distance x from one end at time t is given
by y (x, t) = a sin (rx I Ocos(zct I t) (8)
(b) Obtain D'Alembert's solution of the wave equation by the method of separation of
variables. (7)
OR
Ix. (a) A string is stretched and fastened to two points x = 0 and x = apart. Motion is
started by displacing the string into the form y = k (Ix x2 ) from which it is
released at time t = O. Find the displacement of any point on the string at a distance
of x from one end at time t.