K.S. Narain , ICTP

We keep continuing with the rest of summaries. Simply connected or not simply connected region and Stokes theorem are given. We define Cauchy theorem is simply connected Region R if f(z) is analytic in R, note that we do not follow its proof from the book. The proof of Stokes theorem in 2 dimension is given . Some application like Total flux of electric field is equal to total charged enclosed the sphere is known. We mention Cauchy integral representation for an analytic function. Proof of this statement is directly given in the class and we give some example. By taking modulus of an analytic function we prove this does not make a local maximum. We have seen that Cauchy theorem says if a function is analytic in any region then the integral over any closed contour vanishes. The Converse of the Cauchy theorem, Cauchy Criteria for Convergence,Geometric series are said. If we know that there is a continuous function whose integral over any closed contour in some region vanishes then that function is analytic in that region. Then we showed that if f is continuous for contour C (C is a closed contour in some region R) then f(z) is analytic and in fact we proved it. We have proved Cauchy Criteria for Convergence and talk about uniform Convergence and gave some examples. Necessary conditions for convergent series are provided . We give some examples of convergency like geometric series and we also proved it.