Complex analysis Demonstration 5 26. 10. 2004
1. Iff(z) =u+iv is entire (kokonainen), and ifu=x3−3xy2, ndv from the Cauchy- Riemann equations, and expressf(z)as a polynomial inz, which is unique up to a pure imaginary constant.
2. Show thatf(z) = (1 +z2)/(z2−1)is analytic at ∞. 3. Test P
αn for convergence where αn is (a) n!
nn (b) n3(n+ 1)n
(3n)n (use √n
n →1 as n → ∞).
4. Find the radius of convergence of Xzk k2 . 5. Find the radius of convergence of X ³n+ 1
n
´n2 zn. 6. Find the radius of convergence of X 1
npzn and X
npzn, p >0.
