ExampleEn=1 on D = (z : [2] <1) converges, is uniform on the setsD,={z:z≤r] for0≤r<1 andhenceconvergesuniformlyonallclosed disks in D.Thus we can conclude that is analytic on D and that the derivativeiszn-1,whichalsoconvergesonD.Remark: Applying Theorem 4.1.8, we see that the k-th derivatives f(k)converge uniformly to f(k) on closed disks in A. We do have pointwise but not uniform conergence on D:Convergenceisuniformonlyoneachclosed subdiskinDFCV&ITAugust 19, 201918/50Uni. of Sci&Tech)
Seqences of complex numbers Example P∞ n=1 z n n on D = {z : |z| < 1} converges, is uniform on the sets Dr = {z : |z| ≤ r} for 0 ≤ r < 1 and hence converges uniformly on all closed disks in D. Thus we can conclude that P z n n is analytic on D and that the derivative is Pz n−1 , which also converges on D. Remark: 1 Applying Theorem 4.1.8, we see that the k-th derivatives f (k) n converge uniformly to f (k) on closed disks in A. 2 We do have pointwise but not uniform conergence on D; Convergence is uniform only on each closed subdisk in D. Ming Li (Changsha Uni. of Sci & Tech) FCV & IT August 19, 2019 18 / 50
niimherExample(4.1.3)Show that f(2) = En=1 is analytic on D={z:|z|<1].Writeaseries for f'(z)Solution.Let Mn =,clearly.Mn converges and<=Mfor all z ED.(1) By the Weierstrass M test, m= converges uniformly on D.Therefore, the series converges on closed disks in D(2) By Theorem 4.1.8, the sum n=1 is an analytic function on D.1Furthermorenzn-1>n-f(2) =n2nn=1n=1FCV&IT19/50aUni.ofSci&Tech)August 19,2019MineLilch
Seqences of complex numbers Example (4.1.3) Show that f(z) = P∞ n=1 z n n2 is analytic on D = {z : |z| < 1}. Write a series for f 0 (z). Solution. Let Mn = 1 n2 , clearly, PMn converges and z 2 n2 < 1 n2 = Mn for all z ∈ D. (1) By the Weierstrass M test, P∞ n=1 z n n2 converges uniformly on D. Therefore, the series converges on closed disks in D. (2) By Theorem 4.1.8, the sum P∞ n=1 z n n2 is an analytic function on D. Furthermore f 0 (z) = X∞ n=1 nzn−1 n2 = X∞ n=1 z n−1 n Ming Li (Changsha Uni. of Sci & Tech) FCV & IT August 19, 2019 19 / 50