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