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 iz^3+z^2-z+i = 0 , then show that |z|=1.

If z_1,z_2 in C , show that (z_1+ z_2)^2= z_1^2+2 z_1 z_2+ z_2^2 .

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 n gt 1 and if z ^(n) = (z +1) ^(n) then :

If w= z/(z-i1/3) and |w|=1,then z lies on:

Complex numbers z_(1),z_(2),z_(3) are the vertices of A,B,C respectively of an equilteral triangle. Show that z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1).

If z_(1),z_(2)andz_(3) are the vertices of an equilasteral triangle with z_(0) as its circumcentre , then changing origin to z^(0) ,show that z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=0, where z_(1),z_(2),z_(3), are new complex numbers of the vertices.

Define addition and multiplication of two complex numbers z_1 and z_2 . Hence show that : I_m (z_1 z_2)=R_e (z_1)I_m (z_2)- I_m (z_1)R_e(z_2) .

Define addition and multiplication of two complex numbers z_1 and z_2 . Hence show that : R_e (z_1 z_2)=R_e (z_1)R_e (z_2)- I_m (z_1)I_m(z_2) .

If z_(1),z_(2),z_(3) andz_(4) are the roots of the equation z^(4)=1, the value of sum_(i=1)^(4)z_i^(3) is