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MODEL QUESTION PAPER
FOUR YEAR B.TECH DEGREE END EXAMINATION
THIRD SEMESTER EXAMINATION
COMPLEX VARIABLES AND SPECIAL FUNCTIONS(CV&SF)
(SCHEME – 2013)
(Common To ECE & EEE branches)
Time: 3 Hours Max. Marks: 70
Note: 1) Question No. 1 is compulsory & it must be answered first in sequence at one place only
2) Answer any four from the remaining questions.
1. (a). Functions which satisfy Laplace’s equation in a region R are called -------------- in R
10X1M
(1) Harmonic (2) Analytic (3) Non -harmonic (4) None
(b). If the mapping w=f(z) is conformal then the function is --------------------
(c). Poles of 12 z
z are given by
(1) z=1 (2) z=-1 (3) z= 1 (4) z= i
(d). State the Cauchy’s residue theorem
(e). ( xe )=------------ taking h=1
(f). If )(1 xxJdx
d= ------------------------
(g). Write the Jacobi series
(h). Rodrigue’s formula for )(xPn is -------------------
(i).The value of )1(nP is
(1) 0 (2) l (3) 2 (4) None
(j).The second order Runge-Kutta formula is ----------------------
2. (a).Derive Cauchy-Riemann equations in Cartesian form . (7)
(b). Find the bilinear transformation which maps the points ( ,i,0) into the points (-1,-i,1) (8)
3. (a).. Expand f(z) = )3)(4(
1
zz
z in Taylor’s series about the point z = 2 (7)
(b). Evaluate by using contour integration
2
0cos2
d (8)
4. (a). Using Newton’s forward interpolation formula find y at x =8 from the following table: (7)
x 1.1 1.3 1.5 1.7 1.9
y 0.21 0.69 1.25 1.89 2.61
(b). Using Lagrange’s interpolation formula find y when x =10 from the following table: (8)
x 5 6 9 11
y 12 13 14 16
5. (a) Show that )()( 1 xJxxJxdx
dn
n
n
n
(7)
(b). Prove that )(
)1
(
2 xJte n
n
nt
tx
(8)
6. (a). State and prove Rodrigue’s formula. (7)
(b) Prove that
dxxx )(P)(P n
1
1
m nmif ,0 (8)
7. (a). Using Taylor’s series method solve 2yxdx
dy , y (0) = 1 at x = 0.1, 0.2 . (7)
(b).Find the correlation coefficient from the following table : (8)
x 10 14 18 22 26 30
y 18 12 24 6 30 36