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Let zo E C, R> 0, and f : D(2o, R) → C be a holomorphic function whose only root is 2o. (a) Prove that there exists n > 1, such that f(") (2o) # 0. Consider the function g : D(z0, R) \ {2o} → C given by g(z) = 1/f(2). (b) Let m = min {n: f(") (2o) +0} > > 0. Show that zo is a pole of g(z) of degree at most m (i.e., that (z – z0)™g(z) can be extended to a holomorphic function on an open disk centered around zo).

Let zo E C, R> 0, and f : D(2o, R) → C be a holomorphic function whose only root is 2o. (a) Prove that there exists n > 1, such that f(") (2o) # 0. Consider the function g : D(z0, R) \ {2o} → C given by g(z) = 1/f(2). (b) Let m = min {n: f(") (2o) +0} > > 0. Show that zo is a pole of g(z) of degree at most m (i.e., that (z – z0)™g(z) can be extended to a holomorphic function on an open disk centered around zo).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let zo E C, R> 0, and f : D(2o, R) → C be a holomorphic function whose only root is 2o. (a) Prove that there exists n > 1, such that f(") (2o) # 0. Consider the function g : D(z0, R) \ {2o} → C given by g(z) = 1/f(2). (b) Let m = min {n: f(") (2o) +0} > > 0. Show that zo is a pole of g(z) of degree at most m (i.e., that (z – z0)™g(z) can be extended to a holomorphic function on an open disk centered around zo).
Transcribed Image Text:Let zo E C, R> 0, and f : D(2o, R) → C be a holomorphic function whose only root is 2o. (a) Prove that there exists n > 1, such that f(") (2o) # 0. Consider the function g : D(z0, R) \ {2o} → C given by g(z) = 1/f(2). (b) Let m = min {n: f(") (2o) +0} > > 0. Show that zo is a pole of g(z) of degree at most m (i.e., that (z – z0)™g(z) can be extended to a holomorphic function on an open disk centered around zo).
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