Complex Analysis Answers

show that Lagz can have real value if and only if < is positive?

if x is the set of complex numbers whose real and imaginary parts and rational,what is Int x,x,8x?

Suppose p is a polynomial with n distinct real roots. Show that p' has
at least (n-1) distinct real roots.

True or False with explanation:-
1.The mean value theorem of a real-valued function of a real variable does not hold in general for complex-valued functions.
2.A real-valued function u in domain D cannot be analytic in D unless it is a constant function.
3.There exists a non-constant real-valued function u in open set D which is analytic in D.
4.A real-valued function u(x,y) is harmonic in D iff u(x,-y) is harmonic in D.
5.If f:D→D’ is analytic and u: D'→R is harmonic then the composition of u & f is harmonic inD.
6.{fn(z)} be a sequence of analytic function in a domain D such that fn→f uniformly inD. Then fn’→f' uniformly in D .

Find all the 5 roots of the equation x^5 + 243 = 0 .
[ Give your answers in the form of r(cosθ + isinθ). ]