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1. Specify the order of the zero z = 0 of the following functions: a) f(2) = z²(e* – 1), b) f(2) = ein = – etan z

1. Specify the order of the zero z = 0 of the following functions: a) f(2) = z²(e* – 1), b) f(2) = ein = – etan z

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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H) ZEROS AND UNIQUENESS (SESSION 9) 1. Specify the order of the zero z = 0 of the following functions: a) f(2) = 2°(e² – 1), b) f(2) = e*in = - etan : 2. Find the zeros and orders of zeros of the following functions 2? +1 22 -1' 1 b) f(2) = ÷+3 Log z f) f(2) = c) f(2) = 2² sin z, d) f(2) = cos z – 1, e) f(2) = sinh² z+cosh² z, a) f(2) = 3. Show that sin? z + cos² z = 1, z € C, assuming the corresponding identity for z e R and using the uniqueness principle. 4. Show that if f and g are an alytic on a domain D and f(2)g(2) = 0 for all z e D, then either f or g must be identically zero in D. 5. Is there any fun ction f, analytic in |z| < 1, su ch that 1 and f() k = 1,2, 3, ...? 2k 2k 2k
Transcribed Image Text:H) ZEROS AND UNIQUENESS (SESSION 9) 1. Specify the order of the zero z = 0 of the following functions: a) f(2) = 2°(e² – 1), b) f(2) = e*in = - etan : 2. Find the zeros and orders of zeros of the following functions 2? +1 22 -1' 1 b) f(2) = ÷+3 Log z f) f(2) = c) f(2) = 2² sin z, d) f(2) = cos z – 1, e) f(2) = sinh² z+cosh² z, a) f(2) = 3. Show that sin? z + cos² z = 1, z € C, assuming the corresponding identity for z e R and using the uniqueness principle. 4. Show that if f and g are an alytic on a domain D and f(2)g(2) = 0 for all z e D, then either f or g must be identically zero in D. 5. Is there any fun ction f, analytic in |z| < 1, su ch that 1 and f() k = 1,2, 3, ...? 2k 2k 2k
H) ZEROS AND UNIQUENESS 1. a) 3, b) 3. a) z = ti, order 1, b) z = e(#/4+k#/2), k = 0, 1,2,3, order 1, c) z = 0, order 3; 2 = kr, k e Z \ {0}, order 1, d) z = 2kr, k e Z, order 2, e) z = i(z/4+ ka /2), k e Z, order 1, f) z = 1, order 1. 2. 5. No. It is impossible by the uniquene ss principle. z +1 f(2) = 22 + 1' 6.
Transcribed Image Text:H) ZEROS AND UNIQUENESS 1. a) 3, b) 3. a) z = ti, order 1, b) z = e(#/4+k#/2), k = 0, 1,2,3, order 1, c) z = 0, order 3; 2 = kr, k e Z \ {0}, order 1, d) z = 2kr, k e Z, order 2, e) z = i(z/4+ ka /2), k e Z, order 1, f) z = 1, order 1. 2. 5. No. It is impossible by the uniquene ss principle. z +1 f(2) = 22 + 1' 6.
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