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- 2. Evaluate the following. Express all answers in rectangular form. Solutions for complex number transformation are not required in these items. c. (√(2+i)²arrow_forwardLet a = V3 – i and b = –1+ i be two complex numbers. (9a) Compute the value of b18 Z = a7 (9b) Find all solutions w to the equation w³ 462. (Note: the final answers to 9a and 9b should be given in the standard form x + iy).arrow_forwardSuppose z0 is any constant complex number on the interior of any closed simple curve. of the contour C. Show that for a positive integer n :arrow_forward
- For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly find æ, Y, r, 0 in your head for these simple problems. Then plot the number and label it in five ways as in Figure 3.3. Also plot the complex conjugate of the number. 1. 1+i 2. i - 1 3. 1- iv3 4. -v3+i 5. 2i 6. -4i 7. -1 8. 3 9. 2i – 2 10. 2 – 2i 11. 2 ( cos +i sin 2т 37 12. 4 i sin 13. + i sin COS Cos 2 2 14. 2 (cos + 2 ( cos 15. COS IT i sin 7 16. 5(cos 0 + i sin 0) Vze-iz/4 17. 18. 3e/2 19. 5(cos 20° + i sin 20°) 20. 7(cos 110° – i sin 110°)arrow_forwardI. X D f(z) is equal to the above Let z be a non-zero complex number. Then Log(z²) = 2Logz if -TTSArg(z)s0 if OsArg(z)e for all complex numbers z. Then f is an exponentia' ction such a function does not existarrow_forwardDraw f (E) in complex numbers system if f (z) = z +7+ izī and E = {z E C:1 < |z| < 3}. %3Darrow_forward
- Find all the roots of f (z) = (-5+ j4) z² + (3 – j5) z + 4 + j3 For all computed DECIMAL values that are NOT the final answer, round off decimal values to 4 decimal places. Then, round off the final answer to 2 decimal places. For this question, whenever permitted by the calculation, such as rectangular to polar conversion or operations of complex numbers, it is recommended you use a calculator. But, as discussed in class, you cannot use a calculator to compute directly the square root of a complex number since it will display an error. O -7.48 - j4.75, -6.71 - j5.87 O -0.31 - jo.58 , 1.16 + j0.27 none of the choices O -053 - jo.82, 1.12 + jo.56 -0.64 - jo.17, 1.00 + jo.46arrow_forwardGiven the complex numbers z, = 2+ j0.51, zz = -0.4- j0.8, and z3 = 3+j3 find; |21Z2 (Z – 23) + Im(zz + z1)| In decimal form: z = x + jy Re(z)+ )Im(z) %3Darrow_forwardA set S is said to be convex if each pair of points P and Q in S can be joined by a line segment PQ such that every point on the line segment also lies in S. 4. Determine which of the sets S in the complex plane defined by the following conditions are convex. (a) -2+il<3 (b) yarrow_forwardLet x₁, * are the roots of the quadratic equation x² + ax+b=0, where a, & are complex numbers and y₁, y are the roots of the quadratic equation y³ + [a]y+ [4] = 0. If=p=1, then prove that yllyl=1.arrow_forwardConsider the set C of complex numbers. Let (c², d) be a function where c² represents an element from the set of complex numbers and d(x, y) is defined as: - d(x, y) = √(L₁ – M₁)² + (L2 − M2)² where x = (L₁, L2₂) and y = (M₁, M₂), with L1, L2, M₁, M2 belonging to the set of complex numbers. Prove that (C², d) forms a metric space, where:arrow_forwardConsider the complex-valued function f (x + iy) = x² + ay² – 2xy + i(bx – y² + 2xy). Find values of a and b Correct answers are integers. No spaces, no punctuation except for minus signs if necessary. so that f (x +iy) will be analytic. a= type your answer... and b = type your answer...arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage