Old Exam Papers June 2012 (Set 2)

iznf'kZr dhft;s fd Qyu f tks fd
bg bg
4
z
f z e , z 0; f 0 0
}kjk ifjHkkf"kr gS og z = 0 ij fo'ysf"kd ugha gS tcfd
dks'kh&jheku lehdj.k bl fcUnq ij lUrq"V gSA
5. (a) Find the radii of convergence of the power series
F
2
1
1 HG I KJ n
n
z .
n
F
HG I KJ ?kkr Js.kh 1 n
dhft;sA
2
n
1
n
z dk vfHklj.k f=T;k Kkr
(b) If f zbg is continuous on a contour C of length l
and f bg z M for every point z on C, then
zf bg z dz Ml .
C
;fn l yEckbZ okys daVwj C ij Qyu f zbg larr gks
zbg .
rFkk f bg z M z C rks f z dz Ml
600 4 MT-08
C
lhek dh fof/k ls fuEu Qyu dh lEiw.kZ lfEeJ ry ij
fofp=rk,¡ Kkr dhft;s %
F
HG I KJ sinz
1
fbg z exp
2
zcz 9h
z 1
(b) Find out the zeros and discuss the nature of
singularities of f z
bg
z 2 1
z z 1
2 sin .
z
Qyu f bg z
z z
2 1
sin 2 ds 'kwU; eku Kkr dhft;s
1
,oa Qyu dh fofp=rkvksa dh izÑfr dk foospu dhft;sA
9. (a) Prove that by contour integration that :
z
2
log 1
log2
0
2
1
x c hdx
x
ifjjs[kk lekdy }kjk fl) dhft;s fd %
z
2
log 1
log2
0
2
1
x c hdx
x
(b) Find the residue of the following functions at pole :
4
z
2 2
z a
4
c h
MT-08 7 PTO