complex numbers 6.(a) Show that for all complex numbers z and w|z – w? + ]z + w? = 2|z|² + 2\w]*.(b) Let u, v be complex numbers such that |u| = |v| =1 and |u – v| = 2.Use part (a) to express u in terms of v.

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Answered: 6. (a) Show that for all complex…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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(Review of Complex Numbers) The three most important facts to know about complex numbers are that • 12 = -1 • every complex number can be written in the form a + bi, where a and b are real numbers ⚫elt = cos(t)+isin(t) (this is known as Euler's formula) When manipulating complex numbers, you are free to always treat i as you would with normal algebra (in terms of factoring, rearranging, etc.). If you ever have an i², you can replace it with -1. To divide complex numbers, you can multiply a fraction's numerator and denominator by the complex conjugate of the denominator, and then simplify: a+bi a+bi c-di = c+di cdi cdi You should be able to rearrange it to something of the form A + Bi, where A and B are real numbers. Write each of the following numbers in the form a + bi, where a and b are real numbers. (a) (2+3)(2-3i) 2+3i (b) 2-i
-5 +4i / 2 +5i Complex numbers Simplify the fraction
(Review of Complex Numbers) The three most important facts to know about complex numbers are that • i² = -1 ⚫ every complex number can be written in the form a + bi, where a and b are real numbers ⚫eit = cos(t) + i sin(t) (this is known as Euler's formula) When manipulating complex numbers, you are free to always treat i as you would with normal algebra (in terms of factoring, rearranging, etc.). If you ever have an i², you can replace it with -1. To divide complex numbers, you can multiply a fraction's numerator and denominator by the complex conjugate of the denominator, and then simplify: a+bi = abic di c+ di c+ di c - di =... You should be able to rearrange it to something of the form A + Bi, where A and B are real numbers. Write each of the following numbers in the form a + bi, where a and b are real numbers. (c) iii2-i+1 (d) eлi

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6. (a) Show that for all complex numbers z and w |z - w? + |z + w? = 2|2[² + 2|w/*. %3D (b) Let u, v be complex numbers such that |u| = |v| =1 and |u – v| = 2. Use part (a) to express u in terms of v.