Ch.5:Series Representations for Analytic Functions Ch.5:Series Representations for Analytic Functions L5.3 Power Series L5.3 Power Series Uniform Convergence Uniform Convergence(Cont'd) Knowing that the uniform limit of a sequence of continuous Uniform convergence is a powerful feature of a sequence,as functions is continuous,we can integrate this limit.In fact the the next three results show integral of the limit is the limit of integrals The first says that the uniform limit of continuous functions is Theorem itself continuous Let fn be a sequence of functions continuous on a set TCC Lemma containing the contour T,and suppose that fn converges uniformly Let fn be a sequence of functions continuous on a set T CC and to f on T.Then the sequencefn(z)dz converges tof(z)dz converging uniformly to f on T.Then f is also continuous on T Combining these results with Morera's theorem (page 210). we can prove the following theorem in the next slide 白·0+之。，急，是2风C Ch.5:Series Representations for Analyti Functions Ch.5:Scrics Representations for Analytic Functions L5.3 Power Series 5.3 Power Series Uniform Convergence(Cont'd) Uniform Convergence(Cont'd) Theorem Let fn be a sequence of functions analytic in a simple connected domain D and converging uniformly to f in D.Then f is analytic But any point within the circle of convergence lies inside every in D such a subdisk,so we can state the following Since the partial sums of a power series are analytic functions Theorem (indeed,polynomials)and since they converge uniformly in A power series sums to a function that is analytic at every point any closed subdisk interior to the circle of convergence,we inside its circle of convergence know that the limit function is analytic inside every such subdiskCh.5: Series Representations for Analytic Functions 5.3 Power Series Uniform Convergence Uniform convergence is a powerful feature of a sequence, as the next three results show The first says that the uniform limit of continuous functions is itself continuous Lemma Let fn be a sequence of functions continuous on a set T ⊂ C and converging uniformly to f on T. Then f is also continuous on T Ch.5: Series Representations for Analytic Functions 5.3 Power Series Uniform Convergence (Cont’d) Knowing that the uniform limit of a sequence of continuous functions is continuous, we can integrate this limit. In fact the integral of the limit is the limit of integrals Theorem Let fn be a sequence of functions continuous on a set T ⊂ C containing the contour Γ, and suppose that fn converges uniformly to f on T. Then the sequence Γ fn(z)dz converges to Γ f(z)dz Combining these results with Morera’s theorem (page 210), we can prove the following theorem in the next slide Ch.5: Series Representations for Analytic Functions 5.3 Power Series Uniform Convergence (Cont’d) Theorem Let fn be a sequence of functions analytic in a simple connected domain D and converging uniformly to f in D. Then f is analytic in D Since the partial sums of a power series are analytic functions (indeed, polynomials) and since they converge uniformly in any closed subdisk interior to the circle of convergence, we know that the limit function is analytic inside every such subdisk Ch.5: Series Representations for Analytic Functions 5.3 Power Series Uniform Convergence (Cont’d) But any point within the circle of convergence lies inside every such a subdisk, so we can state the following Theorem A power series sums to a function that is analytic at every point inside its circle of convergence