If `I_(n)=int z^(n)e^(1//z)dz`, then show that `(n+1)! I_(n)=I_(0)+e^(1//z)(1!z^(2)+2!z^(3)+...+n!z^(n+1))`.

If I_(1)=int_(0)^(x) e^("zx ")e^(-z^(2))dz and I_(2)=int_(0)^(x) e^(-z^(2)//4)dz , them

If |z_(1)|= |z_(2)|= ….= |z_(n)|=1 , prove that |z_(1) + z_(2) + …+ z_(n)|= |(1)/(z_(1)) + (1)/(z_(2)) + …(1)/(z_(n))|

If z_(0)=(1-i)/2 , then the value of the product (1+z_(0))(1+z_(0)^(2))(1+z_(0)^(2^(2))(1+z_(0)^(2^(3)))…..(1+z_(0)^(2^(n))) must be

if z_(1) = 3i and z_(2) =1 + 2i , then find z_(1)z_(2) -z_(1)

if z_(1),z_(2),z_(3),…..z_(n) are complex numbers such that |z_(1)|=|z_(2)| =….=|z_(n)| = |1/z_(1) +1/z_(2) + 1/z_(3) +….+1/z_(n)| =1 Then show that |z_(1) +z_(2) +z_(3) +……+z_(n)|=1

Let 1, z_(1),z_(2),z_(3),…., z_(n-1) be the nth roots of unity. Then prove that (1-z_(1))(1 - z_(2)) …. (1-z_(n-1))= n . Also,deduce that sin .(pi)/(n) sin.(2pi)/(pi)sin.(3pi)/(n)...sin.((n-1)pi)/(n) = (pi)/(2^(n-1))

Let 1, z_(1),z_(2),z_(3),…., z_(n-1) be the nth roots of unity. Then prove that (1-z_(1))(1 - z_(2)) …. (1-z_(n-1))= n . Also,deduce that sin .(pi)/(n) sin.(2pi)/(pi)sin.(3pi)/(n)...sin.((n-1)pi)/(n) = (pi)/(2^(n-1))

If z_(1),z_(2),z_(3),…,z_(n-1) are the roots of the equation z^(n-1)+z^(n-2)+z^(n-3)+…+z+1=0 , where n in N, n gt 2 and omega is the cube root of unity, then

If z_(1),z_(2),z_(3)………….z_(n) are in G.P with first term as unity such that z_(1)+z_(2)+z_(3)+…+z_(n)=0 . Now if z_(1),z_(2),z_(3)……..z_(n) represents the vertices of n -polygon, then the distance between incentre and circumcentre of the polygon is

If z + 1/z = 2costheta, prove that |(z^(2n)-1)//(z^(2n)+1)|=|tanntheta|