Using definition evaluate the integral integratey1z2 where y

Using definition evaluate the integral: integratey(1/z)2, where y:z = pie^i Theta and pi/2 LE Theta LE 3pi/2

Solution

Solution:

Given integral solution

=-1/z

now apply the limits

as Z = eⁱ

upper limit : eⁱ⁽³/2

lowe limit: eⁱ⁽/2)

Now apply upper limit - lower limit to the evaluated interval

=-1/z

=-1/ eⁱ⁽³/2 ) -(-1/ eⁱ⁽/2))

=-1/ eⁱ⁽³/2 +1/ eⁱ⁽/2))

  =1/{(1/eⁱ/2 -1/e^i3/2)}

   ^=1/{(3-1)/e^3i/2}

  ⁼   2/e^3i/2