Section-A 1) z Suppose w= . Without doingany calculations , explain why |w|=1 z 2) W can ^ be said about complex number z if z=z ? If ( z )2=( z )2 3) Student A states that , even thoughshe cant find it thetext , she thinks that arg ( z )=arg ( z ) . For example , she says ,if z =1+ i, then z =1iarg ( z )= arg ( z ) = . 4 4 Student B disagreesbecause he feels that he has a counter example :if z=iz =i ; we can take arg (i 4) Defineremovable singuarities pole . 5) Evaluate (6 z +4 )dz where C isthe straight line 1+i 2+3 i . 6) For any complex number , z 0, evaluate z 0 . c Section-B 1) ez U se e =e cos y+i e sin y prove that z =e z e z . e 2) If |sin z| 1 , thenwhat can you say about z ? Justify your answer . z x x Section-C 1) Show that the function 1 u= log ( x 2 + y 2 ) isharmonic determine its conjugate . 2 2) z1 ( z +1 )2 ( z2 ) dz where C isthe |zi|=2using Cauchys Residuetheorem . E valuate c 3) Obtain Laurents series expansion of f ( z )= 4) Evaluate ,using contour integration 2 0 THE END 1 1