Question

what's  
dz $cz2+42+13
, Where C is the contour  

Answer 1

Here basically we use the cauchy residue formula for integrals, using the pole.

We have the contour 112-3;}|= 3 Now given integral dz Ic 2²+42 +13 ļ Thus 22+42 +13= (2-(-2+ 3i) (2 -(-2-3i)) function Hence the 2² +42 +13 (2-(-2+3;))(2-6-2-3i)) has two poles at pointe and Z=-2-31 Z = -2+31 of order 1 Now Risice 11 -2 +31 - Jill = 243. and 11-2-3) - Bill = 54 +36 >3 le. 2=-2+3) les 112-3111=3 but in the contour 2=-2-3) does not

Thus the function has only z²+42 +13 one pole al z=-2+b; in the contour 112-3ill=3. As we know. For he heide the The Cauchy Residue theorem: cony contour C, and the function of with pole at Zo, contour. $ }(2) dz= aaj Res (4,20). where C is poortively orvented closed we have curve. (-2-33) dz 2-(-2+31) =raj Res -2+gil * (* 10273123

톱 19. - 기 - -1+3j + +3 ul = - - +3) - (-3) -\ul = (11) -1 !G+z-42 = 2ni lim - + 러2+3 2x; lim (+) -)

Thus we are done.