(b) For any complex number c and any real number R > 0, let D(c, R) be the closed disk of radius R centred at c. Suppose that (hn(z))n=1 is a sequence of functions that are each analytic on the closed disk D(0, R). Suppose further that g(z) is an analytic function on the closed disk D(0, R) such that g(z) – h,(z) is a non-constant function in the interior of D(0, R) and Ig(z) – h,(z)| = 1/n whenever |z| = R. (i) Show that for all positive integers n, there exists some wn E D(0, R) such that g(Wn) = h, (wn). (ii) If there is some k such that |hx(z)| < 1/k for all z E D(0,R), show that there exists some v E D (0, R) such that g(v) = 0. If |hn(z)| < 1/n for all n and for all z E D(0, R), show that there is a subsequence of (wn)n=1 that converges to a root of g(z). (iii)
(b) For any complex number c and any real number R > 0, let D(c, R) be the closed disk of radius R centred at c. Suppose that (hn(z))n=1 is a sequence of functions that are each analytic on the closed disk D(0, R). Suppose further that g(z) is an analytic function on the closed disk D(0, R) such that g(z) – h,(z) is a non-constant function in the interior of D(0, R) and Ig(z) – h,(z)| = 1/n whenever |z| = R. (i) Show that for all positive integers n, there exists some wn E D(0, R) such that g(Wn) = h, (wn). (ii) If there is some k such that |hx(z)| < 1/k for all z E D(0,R), show that there exists some v E D (0, R) such that g(v) = 0. If |hn(z)| < 1/n for all n and for all z E D(0, R), show that there is a subsequence of (wn)n=1 that converges to a root of g(z). (iii)
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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Transcribed Image Text:(b)
For any complex number c and any real number R > 0, let D(c, R) be the closed disk
of radius R centred at c. Suppose that (hn(z))n=1 is a sequence of functions that are
each analytic on the closed disk D(0, R). Suppose further that g(z) is an analytic
function on the closed disk D(0, R) such that g(z) – h,(z) is a non-constant function
in the interior of D(0, R) and
Ig(z) – h,(z)| = 1/n
whenever |z| = R.
(i)
Show that for all positive integers n, there exists some wn E D(0, R) such that
g(Wn) = h, (wn).
(ii)
If there is some k such that |hx(z)| < 1/k for all z E D(0,R), show that there
exists some v E D (0, R) such that g(v) = 0.
If |hn(z)| < 1/n for all n and for all z E D(0, R), show that there is a
subsequence of (wn)n=1 that converges to a root of g(z).
(iii)
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