Let U c C be an open set. A function f: U → C is analytic if for every 20 € U there is a disk D(20,e) CU and a sequence (Cn) in C such that for all z e D(zo, e), f(2) = Cn(2 – 20). %3D n=0 Prove that if U is open and connected, V CU is open, f:U → C is analytic and satisfies flv = 0, then f = 0. (That is, any analytic function that vanishes on an open set vanishes everywhere. It's not necessary for the homework, but you can make this stronger by replacing V with any subset of U that has an accumulation point in U.)

Let U c C be an open set. A function f: U → C is analytic if for every 20 € U there is a disk D(20,e) CU and a sequence (Cn) in C such that for all z e D(zo, e), f(2) = Cn(2 – 20). %3D n=0 Prove that if U is open and connected, V CU is open, f:U → C is analytic and satisfies flv = 0, then f = 0. (That is, any analytic function that vanishes on an open set vanishes everywhere. It's not necessary for the homework, but you can make this stronger by replacing V with any subset of U that has an accumulation point in U.)

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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Concept explainers

Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0). The multivariable limit of this function as it approaches the points (x0, y0) and is denoted as:

Question

Let U c C be an open set. A function f: U → C is analytic if for
every 20 € U there is a disk D(20,e) CU and a sequence (Cn) in C such that for all
z e D(zo, e),
f(2) = Cn(2 – 20).
%3D
n=0
Prove that if U is open and connected, V CU is open, f:U → C is analytic and
satisfies flv = 0, then f = 0. (That is, any analytic function that vanishes on an open
set vanishes everywhere. It's not necessary for the homework, but you can make this
stronger by replacing V with any subset of U that has an accumulation point in U.)

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