[ACS] For each of the following statements, either show it's true (justifying your answer) or give an (explicit) counterexample. (a) For all complex numbers z, 22 +23 = 2² +2³. (b) For all complex numbers z, Re(z²+z³) = Re(z)² + Re(z)³. (c) For all complex numbers z, | Re(z)| ≤|z|. 1.
[ACS] For each of the following statements, either show it's true (justifying your answer) or give an (explicit) counterexample. (a) For all complex numbers z, 22 +23 = 2² +2³. (b) For all complex numbers z, Re(z²+z³) = Re(z)² + Re(z)³. (c) For all complex numbers z, | Re(z)| ≤|z|. 1.
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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1
![[ACS] For each of the following statements, either show it's true (justifying your answer) or give
an (explicit) counterexample.
(a) For all complex numbers z, z² + 2³ = 2² +2³.
(b) For all complex numbers z, Re(z²+z³) = Re(z)² + Re(z)³.
(c) For all complex numbers z, | Re(z)| ≤|z|.
(d) For all complex numbers z, |z+i| ≥ |z|.
(e) For all real numbers x, x + i] ≥ |x|.
1.
(f) For all complex numbers z, |z + i| ≥ |2| − 1.
(g) For all complex numbers z ‡0, [1/z| = 1/|z|.
(h) For all complex numbers z, w € C: Re(zw) = Re(zw).
(i) For all complex numbers z, w € C: Im(zw) = Im(zw).
(j) For all complex numbers z, if |z| ≥ 10 then || ≤1/9.
>
1+z](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F490cbcd2-ad81-426b-824f-903aced284ba%2F3ff32f9b-da19-41a3-af2e-f68c3d414422%2Famfpki_processed.png&w=3840&q=75)
Transcribed Image Text:[ACS] For each of the following statements, either show it's true (justifying your answer) or give
an (explicit) counterexample.
(a) For all complex numbers z, z² + 2³ = 2² +2³.
(b) For all complex numbers z, Re(z²+z³) = Re(z)² + Re(z)³.
(c) For all complex numbers z, | Re(z)| ≤|z|.
(d) For all complex numbers z, |z+i| ≥ |z|.
(e) For all real numbers x, x + i] ≥ |x|.
1.
(f) For all complex numbers z, |z + i| ≥ |2| − 1.
(g) For all complex numbers z ‡0, [1/z| = 1/|z|.
(h) For all complex numbers z, w € C: Re(zw) = Re(zw).
(i) For all complex numbers z, w € C: Im(zw) = Im(zw).
(j) For all complex numbers z, if |z| ≥ 10 then || ≤1/9.
>
1+z
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