Problem 1.2 (Complex Conjugation) Consider a complex number w = x +jy, having real partRe(w) = x and imaginary part Im (w) = y, with its complex conjugate defined as w* = x - jy.(a) Derive complex conjugation in polar form.1(b)Prove that (i) Re(w) = 1/2 (w +w*) and (ii) Im (w) = , (w – w*)-(c) Prove that |w? = ww* = |w*P, where |w| denotes the magnitude of w.(d) Determine (i) Re(e" ) and (ii) Re(e"*).(e) Determine (i) Im(e" ) and (ii) Im(e"* ).(f) Sketch the complex plane (i) the solution to |w - j2| = 2 and (ii) the solution to |w* – j2| = 2.Which number(s) are common to both solutions?

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Problem 1.2 (Complex Conjugation) Consider a complex number w = x +jy, having real part Re(w) = x and imaginary part Im (w) = y, with its complex conjugate defined as w* = x - jy. (a) Derive complex conjugation in polar form. (b) Prove that (i) Re(w)= 1/2 (w + w*) and (ii) Im (w) : (w – w*) 2j (c) Prove that |wP = ww* = |w*}, where |w| denotes the magnitude of w. (d) Determine (i) Re(e" ) and (ii) Re(e"*). (e) Determine (i) Im(e" ) and (ii) Im(e"* ). (f) Sketch the complex plane (i) the solution to |w – j2|=2 and (ii) the solution to |w* – j2| = 2. Which number(s) are common to both solutions?

Problem 1.2 (Complex Conjugation) Consider a complex number w = x +jy, having real part Re(w) = x and imaginary part Im (w) = y, with its complex conjugate defined as w* = x - jy. (a) Derive complex conjugation in polar form. (b) Prove that (i) Re(w)= 1/2 (w + w) and (ii) Im (w) : (w – w) 2j (c) Prove that |wP = ww* = |w}, where |w| denotes the magnitude of w. (d) Determine (i) Re(e" ) and (ii) Re(e"). (e) Determine (i) Im(e" ) and (ii) Im(e"* ). (f) Sketch the complex plane (i) the solution to |w – j2|=2 and (ii) the solution to |w* – j2| = 2. Which number(s) are common to both solutions?

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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