UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION

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School Of Distce Eductio Questio Bk UNIVERSITY OF ALIUT SHOOL OF DISTANE EDUATION B.Sc MATHEMATIS (ORE OURSE SIXTH SEMESTER ( Admissio OMPLEX ANALYSIS Module- I ( A lytic fuctio with costt modulus is : A ero B costt idetity mp D Noe of these. ( Rel prt of f ( log A log( x y B log( x y log( x iy D Noe of these ( If is positive iteger, the ( i ( i is equl to : A si B cos si D Noe of these. (4 Vlue of ( i ( i equls: A i B i D Noe of these. (5 Rel prt of f ( A x xy B x xy x x y D Noe of these (6 If f ( ( x y xy i( bx y xy is lytic, the vlue of d b is : A -, B, -, D Noe of these (7 If G is ope set i complex ple d f : G is differetible, the o G, f A Alytic B ot lytic Discotiuous D bouded (8 Vlue of i 4 A i 4 B 4 4 D Noe of these e (9 For y complex umber, if: A Re B Re Im D Noe of these omplex Alysis Pge e

School Of Distce Eductio i ( For y complex umber, e equls: A e i B e e D Noe of these ( For y rel umber, i i e e i i i e e A cot B t cos ec D Noe of these ( If S d T re domis i the complex ple, which of the followig eed NOT be true? A S T is domi if S T B S T is ope set S T is ope set D S T is domi ( The derivtive of f ( t i A 99 is : B 4 99 4 D Noe of these (4 If f ( is rel vlued lytic fuctio i domi D, the : A f ( is costt B f ( is ideticlly ero f ( hs modulus D Noe of these (5 If v is hrmoic cojugte for u, the hrmoic cojugte for v A u B -u i u D Noe of these (6 If f( is lytic d o ero i domi D, the i D, l f ( A Alytic B costt ero D hrmoic (7 The fuctio i e hs period: A i B D (8 For rel umbers x d y, si( x iy equls : A Sixoshy i cos xsihy B osxoshy isixsihy Sixoshy i cos xsihy D osxoshy isixsihy (9 The fuctio Log is lytic t: A ll poits i the complex ple B ll o ero complex umbers ll complex umbers except tht o the o positive rel xis D ll complex umbers except tht o the o positive imgiry xis ( Let G be regio ot cotiig. Which of the followig fuctios is NOT hrmoic i G? A x y B x y x y D log( x y. ( Which of the followig is ot hrmoic? A u x( y B u xy y y u x y x y y D Noe of these 99 omplex Alysis Pge

School Of Distce Eductio ( If v is the imgiry prt of lytic fuctio f, lytic fuctio with rel prt v is give by: A f B f if D if ( Vlue of the limit lim equls: A B - D Not exists (4 Rel prt of the fuctio f ( equls: A xy B x y x y D Noe of these (5 For y complex umber, exp( i equls: A exp( B exp( D Noe of these exp( Module II ( A rc ( t ; t b is simple if : A ( t is cotiuous B ( t is oe to oe fuctio ( t is such tht ( = ( b D Noe of these ( Which of the followig is ot simply coected regio? A circulr disk B hlf ples ulus regio D prllel strip ( Which of the followig subset of is simply coected regio? A ; B ; 4 ; ; D (4 Let be y circle eclosig the origi d orieted couter clockwise. The the vlue of the itegrl cos d is : A i B - i D udefied (5 The itegrl Si d ( where the curve is tke ti-clockwise, equls : A -πi. B πi.. D 4πi. (6 The vlue of the itegrl d (, where is - = is : A B i i D Noe of these (7 The oly bouded etire fuctios re: A Rel vlued fuctios B hrmoic fuctios ostt fuctios D Expoetil fuctio omplex Alysis Pge

School Of Distce Eductio (8 Suppose f ( is lytic iside d o uit circle. If f (, with. The upper boud for f ( is : A! B! D Noe of these (9 The itegrl d roud closed curve cotiig - but ot, hs the vlue : A i B i D Noe of these ( The vlue of the itegrl 5 e d, where is = is : A i B i 5 i D Noe of these ( Vlue of the itegrl e i t dt A i B i D Noe of these ( Vlue of d where is ; A i B 6 i D i ( If is y o ero iteger, the e i d equls : A B D Noe of these (4 The vlue of the itegrl d evluted d the semi circle ; rg strtig t i B i A - D Noe of these (5 overse of uchy s itegrl theorem is kow s: A Liouville s theorem B Gourst s theorem Morer s theorem D Euler s theorem. (6 The vlue of the itegrl cos d is : i A B / D -/ (7 The prmetric equtio of the semi circle of rdius with ceter t, lyig i the upper hlf ple from the poit to - A t e t i t ( ; t B ( t e ; t i t i t ( t e ; t D ( t e ; t. e (8 The itegrl d 9 hs o ero vlue if is : A B D 4 omplex Alysis Pge 4

School Of Distce Eductio (9 If f is cotiuous i domi D d if f ( d for every simple closed positively orieted cotour i D, the: A f is costt i D B f is lytic i D f is rel vlued i D D f is purely imgiry i D ( If f is lytic withi d o simple closed, positively orieted cotour d if is poit f ( iterior to, the ( d equls : c i A f (! i f (! B f ( i i D f ( (! Module III ( A Mcluri series is Tylor series with cetre A = B = = D Noe of these ( Let f be lytic fuctio d let f ( ( A f ( (! ( be its Tylor series i some disc. The: B f ( f ( (! D f ( The rdius of covergece of the power series of the fuctio f( = A B¼ ¾ D (4 The coefficiet of / i the Luret series expsio of (5 If f ( f ( A B ½ D 4 is etire, the A f ( (! ( ( ( hs rdius of covergece: B e D Noe of these ( (! bout = /4 is : i the regio < < is : (6 A power series ( lwys coverges for: A t lest oe poit. B ll complex umbers t ll which re either rel or purely imgiry D t ll with R for some R >. (7 If f ( dmits Luret series expsio j j ( j i ulus regio. The j is give by: f ( j! f ( A d j i B d i j i f ( d j j! D Noe of these omplex Alysis Pge 5

School Of Distce Eductio (8 If f ( ( is represeted s the Luret series, the is removble sigulrity of f ( if: A, for > B, for <, for D, for (9 A fuctio f ( give by power series is lytic t : A Every poit of its domi B every poit iside its circle of covergece every poit o its circle of covergece D every poit i the complex ple ( The Tylor series expsio of f ( e i the regio A B!!! D Noe of these ( The sigulr poits of the fuctio f ( 4 re: A d 4 B d 4 4 d 4 D d ( The sigulr poits of the fuctio ( e f ( tht lies iside i re : A d i B d i i d i D d ( The costt term i the Luret series expsio of e f ( i the regio < < is : A B D Noe of these (4 The Luret series expsio of f ( e i the regio A B!! D Noe of these! (5 The power series b b b... coverges: A iside of some circle R B o the circle o some circle R D outside of some circle R omplex Alysis Pge 6

School Of Distce Eductio Module- IV ( The residue of t A B - D. Si x ( Vlue of dx x. A is : B D ( If f ( hs ero of order m t d g ( hs pole of order t d m, the the product f ( g ( hs t : A A essetil sigulrity B pole of order m - A removble sigulrity D pole of order m -. (4 If f ( hs pole of order m t / f (, the g (, t f ( hs: A simple pole B pole of order m pole of order m + D pole of order m -. (5 If f ( hs pole of order m t, the t, hs : f ( A A removble sigulrity B essetil sigulrity A pole of order m D oe of these (6 Zeros of Si ( re : A ; Z B ; Z ; Z D. (7 For t f (, is : A Essetil sigulrity B simple pole Removble sigulrity D double pole (8 If f ( hs pole of order m t, the f ( hs pole of order.. t : A m B m m D m + (9 Which of the followig fuctio hs simple ero t d essetil sigulrity? A e B e ( e D ( e ( If f ( ( is represeted s the Luret series, the is removble sigulrity of f ( if: A, for > B, for <, for D, for ( The fuctio Si f ( ( - t = hs : A pole of order 4 B pole of order pole of order D essetil sigulrity omplex Alysis Pge 7

School Of Distce Eductio ( If rtiol fuctio hs pole of order m t, the its derivtive hs pole of order ------ t A m - B m + m D m ( Residue of 4 9 f ( t A 64 (4 The fuctio 6 5 i f ( t i hs: i i B D Noe of these A Regulr poit B Simple pole double pole D removble sigulrity (5 Sigulrities of rtiol fuctio re: A poles B essetil o isolted D removble (6 If e f (, the t, f( hs: e A Isolted sigulrity B o-isolted sigulrity Pole D ero (7 If fuctio f( hs isolted sigulrity t =, the it is removble sigulrity if : A lim( f ( B lim( f ( lim f ( D lim f ( si (8 The sigulrity of the fuctio (9 Residue of t = is : A Essetil sigulrity B simple pole Removble sigulrity D double pole m A ( ( m! ( m!(! ( m! ( m!(!, m d re turl umbers t = is : B D ( m! m!! ( m! m!! ( At =, the residue of f ( ( A - B D ( The residue of f ( ( t is : A - 5 B D Noe of these ( The fuctio f( = / t (, t = hs : A isolted sigulrity B o- isolted sigulrity simple pole D Noe of these omplex Alysis Pge 8

School Of Distce Eductio ( The ero of first order is kow s A omplex ero B Simple ero Sigulrity D Noe of these (4 The poles of the fuctio f( = si / cos re t A (+ π\, y iteger B π \ ; y iteger π, y iteger D Noe of these (5 For the fuctio is : f ( e, the poit A removble sigulrity B simple pole essetil sigulrity D Noe of these SHOOL OF DISTANE EDUATION - UNIVERSITY OF ALIUT B.Sc DEGREE PROGRAMME MATHEMATIS (ORE OURSE SIXTH SEMESTER - MM6B - OMPLEX ANALYSIS Aswer Key Module I. B. A. B 4. 5. A 6. A 7. A 8. 9. B.. B. D. B 4. A 5. B 6. D 7. B 8. A 9.. B. B. D. D 4. 5. A Module II. B.. D 4. B 5. A 6. A 7. 8. A 9. A.. A.. A 4. 5. 6. D 7. A 8. D 9. B. A Module III. B.. 4. A 5. 6. A 7. A 8. B 9. B. A. B. B. B 4. 5. D Module IV.. A. 4. A 5. A 6. B 7. 8. 9. A. B.. B. D 4. B 5. A 6. B 7. B 8. 9.. D. D. B. B 4. A 5. omplex Alysis Pge 9

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