a) Show that if you have a complex number z, then Re(z) = (z+z*)/2. Now show that Im(z) = (z-z*)/(what #goes here? You decide!) b) There is a useful relation for the product of two complex numbers z₁ and 2₂: It's either Re(Z₁Z2) = Re(Z₁) Re(Z2) + Im(Z₁) Im(Z2), or else it's Re(Z₁Z2) = Re(Z₁) Re(Z2) - Im(z₁) Im(Z2) Work it out; you decide which sign is correct. , show that if you have two complex numbers z₁ and z2, then Im(Z₁Z2) = (+ or -1, you decide!) Re(z₁)Im(Z2) (+ or -, you decide!) Im(z₁)Re(Z2) c) If you have a complex number z, we define |z² = z* z. Is there any difference between z² and |z2? How about Re(z²) and Re(z2)? Briefly, explain.
a) Show that if you have a complex number z, then Re(z) = (z+z*)/2. Now show that Im(z) = (z-z*)/(what #goes here? You decide!) b) There is a useful relation for the product of two complex numbers z₁ and 2₂: It's either Re(Z₁Z2) = Re(Z₁) Re(Z2) + Im(Z₁) Im(Z2), or else it's Re(Z₁Z2) = Re(Z₁) Re(Z2) - Im(z₁) Im(Z2) Work it out; you decide which sign is correct. , show that if you have two complex numbers z₁ and z2, then Im(Z₁Z2) = (+ or -1, you decide!) Re(z₁)Im(Z2) (+ or -, you decide!) Im(z₁)Re(Z2) c) If you have a complex number z, we define |z² = z* z. Is there any difference between z² and |z2? How about Re(z²) and Re(z2)? Briefly, explain.
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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Help me with this problem, assuming I have no prior knowledge of
![a) Show that if you have a complex number z, then Re(z) = (z+z*)/2. Now show that
Im(z) = (z-z*)/(what #goes here? You decide!)
b) There is a useful relation for the product of two complex numbers z₁ and 2₂:
It's either Re(Z₁Z2) = Re(Z₁) Re(Z2) + Im(z₁) Im(Z2), or else it's
Re(Z₁Z2) - Re(Z₁) Re(z2) - Im(Z₁) Im(Z₂)
=
Work it out; you decide which sign is correct.
, show that if you have two complex numbers zi and z2, then
Im(Z₁Z₂) = (+ or -1, you decide!) Re(Z₁)Im(Z₂) (+ or -, you decide!) Im(Z₁)Re(Z₂)
c) If you have a complex number z, we define |z² = z* z. Is there any difference between z² and
|z2? How about Re(z²) and Re(z2)? Briefly, explain.
d) Given z = Ae¹e, find expressions for Re(z), Im(z), z*, and [z], all in terms of A and 0.
e) Using Euler's Theorem, find expressions for cose and sine in terms of eie and e-ie. You
will use these expressions many times](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc10fc8f-c13e-4fd9-87dd-492a474a2ca4%2F11cdfd86-2a38-4a64-b607-0aafc840f434%2F7z2cfcj_processed.png&w=3840&q=75)
Transcribed Image Text:a) Show that if you have a complex number z, then Re(z) = (z+z*)/2. Now show that
Im(z) = (z-z*)/(what #goes here? You decide!)
b) There is a useful relation for the product of two complex numbers z₁ and 2₂:
It's either Re(Z₁Z2) = Re(Z₁) Re(Z2) + Im(z₁) Im(Z2), or else it's
Re(Z₁Z2) - Re(Z₁) Re(z2) - Im(Z₁) Im(Z₂)
=
Work it out; you decide which sign is correct.
, show that if you have two complex numbers zi and z2, then
Im(Z₁Z₂) = (+ or -1, you decide!) Re(Z₁)Im(Z₂) (+ or -, you decide!) Im(Z₁)Re(Z₂)
c) If you have a complex number z, we define |z² = z* z. Is there any difference between z² and
|z2? How about Re(z²) and Re(z2)? Briefly, explain.
d) Given z = Ae¹e, find expressions for Re(z), Im(z), z*, and [z], all in terms of A and 0.
e) Using Euler's Theorem, find expressions for cose and sine in terms of eie and e-ie. You
will use these expressions many times
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