Old Exam Papers Dec. 2010 - Video

(b) Show that the necessary condition that a function
bg b g b g
f z ? u x, y ? iv x, y be analytic in a domain
D is that u and v satisfy Cauchy-Riemann equation
in D.
bg b g b g
iznf'kZr dhft;s fd Qyu f z ? u x, y ? iv x, y
ds fdlh izkUr D eas fo'ysf"kd gksus ds fy, vko';d
izfrcU/k ;g gS fd u rFkk v izkUr D esa dks'kh&jheku
lehdj.k lUrq"V djrs gksaA
2. (a) If f ( z ) is an analytic function of z, prove that :
(i)
(ii)
F
2 2
? ?
2 2
1
? f z 2 f z
2 2 HG ? x ? y
I KJ Re bg ? bg
F
2 2
? ?
2 2
1
? f z 4 f z
2 2 HG ? x ? y
I KJ bg ? bg
;fn f ( z ) , z dk ,d fo'ysf"kd Qyu gS rks fl)
dhft;s fd %
(i)
(ii)
F
2 2
? ?
2 2
1
? f z 2 f z
2 2 HG ? x ? y
I KJ Re bg ? bg
F
2 2
? ?
2 2
1
? f z 4 f z
2 2 HG ? x ? y
I KJ bg ? bg
600 2 MT-08
?1
2 (b) Prove that the function z fbg? z e has no singularity.
Papers (A) (December) 2010
(359)
fl) dhft;s fd Qyu f z e z
ugha gSaA
bg?
?1 2 dh dksbZ fofp=rk,¡
9. (a) Use the method of contour integration to prove that
a cos?
the integral
d
a cos?
?
?
?z is equals to :
? ?
R
S|
T|
2?a 1 ?
U
V|
W|
a
a ? 1
2 ca ? 1h
,
ifjjs[kk lekdy fof/k dk mi;ksx dj] fl) dhft;s
fd %
R
T|
?
2 ?z S| 1
?
a cos?
d a
a cos?
? ? ? ?
?
U
V|
W|
a
, a ? 1
2 ca ? 1h
(b) Suppose f zbg and g zbg are analytic inside and
on a simple closed contour c with g bg z ? f bg z
on c, then show that f zbg and fbg? z g( z ) have
same number of zeros inside c.
MT-08 7 PTO