Give Examples (this is complex analysis):
(a.) First characterize open and closed sets in terms of theirboundary points. Then give two examples of sets satisfying thegiven condition: one set that is bounded (meaning that there issome real number R > 0 such that |z| is greater than or equal toR for every z in S), and one that is not bounded. Give your answerin set builder notation. Finally, choose one of your two examplesand prove that is neither open nor closed.
(b.) Give two examples of a function f: C?C that is continuousat z=0 but not differentiable at z=0 using the Cauchy-Riemannequations.
(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 byfactoring the polynomial z^3+1 as z+1 times a quadratic polynomialand then determine the roots of the quadratic polynomial) andsecond, by using the formula for computing nth roots of a complexnumber.
