Question

Give Examples (this is complex analysis):

(a.) First characterize open and closed sets in terms of their boundary points. Then give two examples of sets satisfying the given condition: one set that is bounded (meaning that there is some real number R > 0 such that |z| is greater than or equal to R for every z in S), and one that is not bounded. Give your answer in set builder notation. Finally, choose one of your two examples and prove that is neither open nor closed.

(b.) Give two examples of a function f: C→C that is continuous at z=0 but not differentiable at z=0 using the Cauchy-Riemann equations.

(c.) Find a cube root of -1, other than -1, in two ways: first, by using high school algebra (solve the equation z^3= -1 by factoring the polynomial z^3+1 as z+1 times a quadratic polynomial and then determine the roots of the quadratic polynomial) and second, by using the formula for computing nth roots of a complex number.

Answer 1