11. Solution4. 1 i Wehave t e it cost isint,0 t ,sothepathisthe ockwiseto 1. SeeFigure0. 1 i. ii Here t x t iy t ,x t 1 2cost,y t 1 2sint,0 t 2. Hence x 1 2 y 1 2 22,i. e. i i. See Figure0. 1 ii. iii Here t x t iy t wherex t t,y t cosht,.

 

12. Solution5. 1 i Sincesin2z 1 cos2z 2andcosz 1Xn 0 1 nz2n 2n ! 0. 1 wehavethatsin2z 1 cos2z2 1Xn 1 1 n 14n2 2n !z2n : 0. 2 0. 1 isR 1 0. 2 isalsoR 1. R limn!1jan 1jjanj limn!14n 12 2 n 1 !2 2n !4n limn!142n 2n 1 0 cientsin 0. 2. ii Here11.

 

9. Solution2. 1 i Thissetisopen. LetD fz2CjIm z 0g. Letz02 D. Wehave tond 0suchthatN z0 D. x0 iy0andlet y0 2 0. Supposethatz x iy2N z0. Thenjy y0j pjx x0j2 jy y0j2 jz z0j y02: Hence.

 

13. Solution6. 1Letf z P1m f aroundz 0. i 3,weshouldlook atpowersof1 z:1z 3 1 z1 3 z 1z 1 3z 32z2 1Xm 13 m 1 zm:Hencecm 3 m 1 form0andcm 0form 0. ii Here11 z 1 z z2 1. Hence1z 1 z 1z 1 z z2 isvalidfor0 jzj 1. Hencecm 1ifm 1,cm.

 

14. Solution7. 1 i Thefunctionf z 1 z 1 z2 isdi. 0;. 0; 1. ThenbyLemma7. 3. 1 i wehaveRes f;0 limz!0z1z 1 z2 limz!011 z2 1;Res f;1 limz!1 z 1 1z 1 z2 limz!1 1z 1 z 12;Res f; 1 limz! 1 z 1 1z 1 z2 limz! 11z 1 z 12 : ii Letf z tanz sinz.

 

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subiecu Math- ru M. A/M. Sc:. 2““- -A/10RollNo. SigofDy. Superintendent. University of__ _1rgoQh_a_ M. A/M. Se Part-1 / Composite. 2““ -A/2010 Math- III Complex Analysis Differential.

 

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