Old Exam Papers Dec. 2010 - Video

;fn z ? a ? R esa f zbg fo'ysf"kd rFkk ,dekuh;
Qyu gks rks 0 ? r ? R ds fy, fl) dhft;s fd
f a
1
?
? r
z0
2?
'bg bg
P e d i ? ? ? ?
tgk¡ P ?bg] f a re i?
?
c h dk okLrfod Hkkx gSA
7. (a) Expand e z and sin z in a Taylor's series about
z ? 0.
Qyuksa e z rFkk sin z dk z ? 0 ds lkehI; esa Vsyj
Js.kh esa izlkj dhft;sA
(b) If f zbg has an isolated singularity at z ? a and is
bounded in some deleted neighbourhood of a, then a
is removable singularity. Prove it.
fl) dhft;s fd ;fn z ? a Qyu f zbg dh fo;qDr
fofp=rk gks rFkk z ? a ds fu"dkf"kr izfros'k esa f zbg
ifjc) gks rks a ,d vius; fofp=rk gSA
8. (a) Prove that a polynomial of degree n has a pole of
order n at infinity.
fl) dhft;s fd n dksfV ds cgqin dk vuUr fcUnq
n-dksfV dk vuUrd gSA
600 6 MT-08
Papers (A) (December) 2010
(360)
(b) Prove that the series
b g b g b g...... converges for
2 3
z 1? z ? z 1 ? z ? z 1?
z ?
z ? 1. Also find its sum.
fl) dhft;s fd Js.kh
2 3
zb1 ? zg? z b1 ? zg? z b1 ? zg......vfHklkjh
?
gS
tcfd z ? 1, bldk ;ksx Hkh izkIr dhft;sA
3. (a) For what values of z do the function w defined by
the following equations ceases to be analytic :
(i)
z
w ?
z
? 1
1?
(ii) z ? sinhu cosv ? icoshu sin v, w ? u ? iv
fuEu lehdj.kksa }kjk ifjHkkf"kr Qyu w, z ds fdu ekuksa
ds fy, fo'ysf"kd ugha gS %
(i)
z
w ?
z
? 1
1?
(ii) z ? sinhu cosv ? icoshu sin v, w ? u ? iv
(b) If w ? f ( z ) represents a conformal transformation
of a domain D in the z plane into a domain D' of the
w-plane then f (z) is an analytic function of z in D.
MT-08 3 PTO