S.E. Sem. III [EXTC] Applied Mathematics - III

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1 S.E. Sem. III [EXTC] Applied Mhemic - III Time : 3 Hr.] Prelim Pper Soluio [Mrk : 8 Q.() Fid Lplce rform of e 3 co. [5] A.: L{co }, L{ co } d ( ) d () L{ co } y F.S.T. 3 ( 3) Le co 3 Q.() Prove h : f ( + y + z)i + ( 3y z)j + ( + cy + z)k i oleoidl d deermie he co,, c if f i irroiol. A.: f 3 + f i oleoidl. give f i j k y z yz 3yz cyz (c + )i ( )j + ( ) k i + j + k c,, Q.(c) Show h f(z) ih z i lyic. Hece fid i derivive. [5] A.: f(z) ih z ih( + iy) ih coh iy + coh ih iy u + iv ih coy + i coh iy u ih coy u coh coy () u y ih iy () v coh iy v ih iy (3) v y coh coy () From (), () d (), (3) u v y v u y f(z) ih z i lyic. Now F(z) u iv z,y coh coy i ih iy z,y Vidylkr f(z) coh z [5]

2 Vidylkr : S.E. Mh III Q.(d) Prove h () co d A.: we kow h () d d () d () d i d d i co d () co Q.() Show h he fucio rform he righ lie c i he z-ple z io circle i w-ple. Fid i cere d rdiu. A.: c () Give z + iy z u iv + iy u iv u u v u v u v uig hi i () u c u + v u u v c cere z,, c rd c Q.() A.: Show h e iu dud u iu iu e dud L u du u Now iu iu L du L u u d, Vidylkr [5]

3 Q.(c) Oi fourier erie for f() Hece deduce h A.: f() i eve fucio. f() co() () d co()d ( ) uig hi i () f() ( ) co() for deducio pu d oe h f() ( ) Q.3() Evlue y Gree Theorem e iyd e coydy A.: C where C i recgle wih verice (, ), (, ), (, ), (, ) Pd Qdy Q Pyddy, P e i y, Q e co y Prelim Pper Soluio i () co () e coye coy ddy e coydyd e iy d e Vidylkr (e ) (,/) (,/) (,) (,) 3

4 Vidylkr : S.E. Mh III Q.3() Prove h () () () A.: We kow h + pu, + y recurrece formul, lo equig + pu, uig hi i () dy Q.3(c) Solve he differeil equio y yd d A.: () rform give y() L y() L y() L y()d L{i} 5y() y() y() y() y() y() c d y() ( ),, 3, c, d 3 y() () kig ivere Lplce, y() 3 e e i i uig Lplce Vidylkr

5 Q.() A.: Prelim Pper Soluio Fid orhogol rjecory o he fmily of curve e co y + y co i X Y ple. If f(z) u + iv lyic he v i orhogol o u Le u e co y + y dv v d + v y d y u y d + u d y [ u v y, v u y ] dv e iy e coy ydy which i ec D.E. iegrig e iy d e coy y dyc v yco freefrom y v e iy c m Q.() Show h co 8 i(m), If < < m m A.: Hlf rge ie erie f() i() coi()d co ( ) co( ) i( ) i() d ( ) ( ) ( ) Now icod i d co co ( ) i() pu m co (m) m ( ) i(m) m m co 8 m i (m) m Vidylkr m () 5

6 Vidylkr : S.E. Mh III Q.(c) Fid Bilier rformio which mp he poi, i, oo he poi i,, i. Hece fid fied poi d imge of z <. A.: Coider B. I. z cz d () give z, i () i cd ic + id + () give z i, () i icd i (3) give z, i () i c d ic id + () () + () ic ic i c (5) () () id d i (6) uig (3), (5), (6), i () zi i z zi iz (7) i For fied poi w z, (7) z z iz z + iz i z z + (i + ) z i i ii z o fid imge of z < : (7) i + z iz ( + )z i i ii(u iv) z uiv z < iiuv uiv i iu + v + u + iv v ( u) < (u) v v + u + u < + u + u + v < u < u Vidylkr Q.5() Ue Soke Theorem o evlue f dr c Where fyi zj k d c i oudry of he urfce + y z, z >. A.: f i j k y z y z f i j k, k i + j + k d d dy +y 6

7 Prelim Pper Soluio c f d f dr () d dy Are of circle Q.5() If f() c () + c () + c 3 3 (), where c, c, c 3 re co d,, 3 re orhoorml fucio, o he e (, ). Show h [f()] d c c c 3 A.: Give, 3 re orhoorml o (, ) Q.5(c) d d d d d 3 3 d Now f c + c + c 3 3 qurig 3 f c c c c c c c c c fd 3 3 uig equio (), () we ge c d c d c d c c dc c dc c d f() d c c c Fid ivere Lplce Trform of (i) log (ii) e A.: (i) Le f() log log f() log ( ) log f() kig L we ge L f () co Vidylkr f() co (co ) f() () () 7

8 Vidylkr : S.E. Mh III (ii) Le f() 3 L F() L 3 Q.6() A.: Q.6() e 3 F() e L i 3 3 we kow h, L {e f()} f( ) H( ) e ( ) 3 L e i () H() 3 Oi comple form of fourier erie for F() e, i (, ) where i o ieger. Comple form of fourier erie i give y f() c i e () c i e e d i e i (i) ( ) e e ( ) (i) c ( ) ih( ) uig hi i () e (i) ( ) ih( )e i + j + k f i irroiol. 8 i ( i) (i)( i) i i e e e e Prove h f (ye y co z)i + (e y co z)j + (e y i z)k i iroiol. Alo fid clr poeil d work doe i movig pricle from (,, ) o (,, ) i j k Vidylkr A.: f y z y ye coz y e coz y e iz y y y y y y y y e i z e iz i ye i z ye i z j e cozye coze cozy e coz k

9 Prelim Pper Soluio y y y Sclr poeil work doe (,, ) (,,) work doe e ye coz d e co z dy e i z dz c e y co z + c y ( d,, ) e coz (,,) Q.6(c) Fid imgiry pr of lyic fucio whoe rel pr i e ( co y y i y), Alo verify h v i hrmoic fucio. A.: Give u e ( co y y i y) f(z) u + iv lyic u v y, v u y } () dv v d + v y dy uy d + u dy [ ()] dv e ( iy iy ycoy) d +{e (co y) + e ( co y y i y)dy} which i ec. y iegrio, v e (iy iy ycoy) d dyc yco freefrom e e iy iy ycoy ( iy) c e v e (iy ycoy) iy iy ycoy iy v e (iy yco y) e iy v e iy ycoy e iy e iy v e iy ye co y+ e iy () v y e ( co y + co y y i y) v yy e ( i y i y i y y co y) v yy e i y e i y y e co y (3) ddig (), (3) V + V yy v i hrmoic. Vidylkr 9

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