M-iii

1 . C i r c l e of c o nve r g e n c e of ∞ n = 0 a n z n i s ( a) | z | = | a n | - 1 n (b) |z | = Lim n→∞ |an | 1 n (c) |z | = Lim n→∞ |an | -1 n (d) |z | = |an | 1 n 2 . I f Tay l or ‘ s s e r i e s e x p an s i o n o f f (z ) = ∞ n = 0 a n ( z - z 0 ) n t h e n a n = . ( a) f ( n - 1 ) ( z 0 ) ( n - 1 ) ! ( b ) f ( n + 1 ) ( z 0 ) ( n + 1 ) ! ( c ) f ( n) (z0 ) n! (d) f ( n) (z -z0 ) n! 3 . T h e t ot al nu mb e r o f d i s t i n c t L a u r e nt s e r i e s e x p an s i on s of 7 z 2 + 9 z - 1 8 z 3 - 9 z a b ou t i t s s i n g u l ar p oi nt s a r e ( a) 2 ( b ) 8 ( c ) 5 ( d ) 3 4 . T h e L au r e nt s e r i e s e x p a n s i on of 1 ( z + 1 ) ( z + 2 ) a b ou t z = - 1 , va l i d i n 0 < | z + 1 | < 2 i s ( a) ∞ n = 0 ( - z - 1 ) n ( b ) 1 z + 1 + ∞ n = 0 ( z + 1) n ( c ) 1 z + 1 - ∞ n = 0 ( - z - 1 ) n ( d ) - ∞ n = 0 ( - z - 1 ) n 5 . z = 0 f or e 1 z i s . ( a) R e m ovab l e s i n gu l a r i ty ( b ) E s s e nti a l s i n gu l a ri ty ( c ) s i m p l e p ol e ( d ) b r an ch p oi nt 6 . R e s i d u e of z 2 - 2 z ( z + 1 ) 2 ( z 2 + 4 ) a t z = - 1 i s . ( a) 1 ( b ) 3 / 5 ( c ) -14/25 (d) -1/2 7 . T h e r e s i d u e o f 1 - e 2 z z 4 a t z = 0 i s . . ( a) - 2 / 3 ( b ) - 2 ( c ) - 4 / 3 (d) -3/4 8 . ∞ - ∞ x 2 d x ( a 1 + x 2 ) ( 4 + x 2 ) = ( a) π ( b ) 2 π ( c ) π / 3 ( d ) π 2 9 . 2 π 0 d θ ( √ 2 - c o s θ ) = ( a) π ( b ) π 2 ( c ) π √ 2 ( d ) 2 π 1 0. To c al c ul a t e ∞ 0 x p - 1 1 + x d x we u s e C z p-1 1+z dz wh e r e C i s t h e b o u n d a ry of . . ( a) 0 ← ε < < R → ∞ E x c l u d i n g - ve I m ag i n a ry ax i s ( b ) 0 ← ε < | z | < R → ∞ E x c l u d i n g - ve R e al ax i s ( c ) 0 ← ε < | z | < R → ∞ R E x c l u d i n g + ve Re al ax i s ( d ) 0 ← ε < | z | < R → ∞ E x c l u d i n g + ve I m a g i n ar y a x i s 1 1. i f f ( z ) = S i n π z t h en 2π i (c) 14π i (d) -14π i

|z |=π f

( z ) f ( z ) d z = ( a) 7 ( b ) 1

1 2. R e ga r d i n g th e T a n z = a z w i t h a > 0 r o ot s of w h i ch o f t h e f o l l ow i n g i s f al s e ( a) O n l y two p u r e l y i m a g i na r y r o ot s i f 0 < a < 1 ( b ) N o C o m p l e x r o ot s i f a ≥ 1 ( c ) i n fi n i t e r e a l r o ot s ( d ) a l l i m a gi n a ry ro otsifa≥ 1 1 3. W h i ch o f th e f o l l ow i n g i s t h e M a x i mu m M o d u l u s T h e or e m ( a) I f a f u n c t i o n f ( z ) i s an a l y t i c i n & o n S i m p l e C l os e d C u r ve C , a n d i f f (z ) = 0 i n s i d e | f ( z ) | a tt a i n s i t s m i n . va l u e o n C ( b ) I f a N o n C on s ta nt f u n c ti o n f ( z ) i s a n al y t i c i n & on S i m p l e C l o s e d C u rve C , t h e

n M a x i mu m Va l u e o f | f ( z ) | o c c u r s o n C ( c ) I f a N o n C on s t a nt f u n c t i o n f ( z ) i s an a l y ti c i n & on S i m p l e C l o s e d C u r ve C , t h e n M ax i mu m Val u e o f f ( z ) o c c u r s o n C ( d ) I f a f u n c t i o n f ( z ) i s an a l y t i c i n & o n S i m p l e C l os e d C u r ve C , t h e n M a x i mu m Va l u e o f | f ( z ) | o c c u r s on C 1 4. T h e nu mb e r of r o ot s of z 3 e 1 - z = 1 h a s e xa c t l y r o ot s i n s i d e | z | = 1 ( a) 1 ( b ) 3 ( c ) 2 ( d ) ∞ 1 5. L i ov i l l e ‘ s t h e or e m c an b e p rove d u s i n g ( a) Fu n d a m e nt al th e or e m of al g e b r a ( b ) R ou ch e ‘ s T h e o r e m ( c ) C a u chy‘ s i n e q u al i ty ( d ) A r gu m e nt pr i n c i p l e 1 6. T h e m ag n i t u d e o f a n g l e o f r o ta t i on i nvo l ve d i n f ( z ) = ( 1- i ) z - ( 4 +2 i ) ( a) 7 π 4 ( b ) 5 π 4 ( c ) π 4 ( d ) 3 π 4 1 7. I f u +i v = w = f (z )= e z t h e n u , v r e s p e c ti ve l y a r e ( a) S i ny ; C o s y ( b ) e x S i ny ; e x C o s y ( c ) e x C o s y ; e x S i ny ( d ) C o s y ; S i ny 1 8. I f u +i v = S i n z th e n u , v ar e . . . . . . . . . . . . . . . . . ( a) S i n x C o s hy ; C os xS i n hy ( b ) S i ny C os h x; C o s y S i n h x ( c ) S i ny C o s hy ;C os x S i n h x ( d ) S i n x C o s h x ; C o s y S i n hy 1 9. T h e m ap w = z + a 2 z t ake s | z | = R i n z - p l a ne t o . . . . . . . . i n w - p l an e ( a) E l l i p s e s w i t h d i ff e r e nt f o c c i ( b ) C o n f o c a l e l l i p s e s w i t h focciat ±a (c) Confocalellipseswithfocciat ±2a (d) Linesp ar a l l e l t o X a x i s 2 0. w = e i θ 0 z - z 0 z - ¯ z 0 m a p s u p p e r h al f p l an e [ y > 0 ] c o nta i n i n g z 0 t o . . . . . . . . . . ( a) | w | < 1 ( b ) | w | > 1 ( c ) | w | ≤ 1 ( d ) | w | ≥ 1 ANS :CCBCBCCCDCCDBCCACACA