CSIR NET MATHEMATICS| Complex Analysis | How to use Cauchy’s Integral Formula? (June 2018)

In this video, we are going to discuss the problem of complex analysis from the CSI Net June 2018. Example, the value of integral over the closed contour 1 minus z equal to 1 e power z divided by z square minus 1 dz is the first option 0, second option pi i e, third option pi i e minus pi i e power minus 1 and the fourth option e plus e power minus 1. We are given the Integration over the closed contour modulus 1 minus z is equal to 1 e power z divided by z square minus 1 dz. Factorize the denominator z square minus 1 is equal to z minus 1 z plus 1. Now, integrand e power z divided by z minus 1 z plus 1. So, we have two singularities. One is z is equal to 1 and second one is z is equal to minus 1. So, we are getting the closed contour modulus 1 minus z z is equal to 1. So which represent this circle with the center 1, 0 and the radius 1. Now this circle contain positive 1 but not contain the negative 1. Therefore from the two singularities z is equal to 1 and z is equal to minus 1. Only z is equal to 1 is inside the closed contour and minus 1 outside the closed contour. Therefore z is equal to 1 can be considered as a singularity for this function, so this integrand can be written in the form of e power z divided by z plus 1, this whole divided by z minus 1, so numerator will be the analytic function for this closed contour and z minus 1 will become a singularity. Now using the Cauchy integral formula, if f is analytic then Integration over the closed contour fz divided by z minus 1 dz is equal to 2 pi i f of a. Here f of z is equal to the numerator e power z divided by z plus 1 which is analytic on the closed contour modulus 1 minus z equal to 1 and a is 1. So the integral value is 2 pi i into f of a. So a is equal to 1. 2 pi i f of 1, so 2 pi i, if you put z is equal to 1, e power 1 divided by 1 plus 1, so it is e by 2, and 2 to cancel, the final answer is pi i e, which is the second option, so this example is the answer, second option, pi i e. Thanks for watching, for more videos, like, subscribe, press the bell icon of Next Gen Maths.