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Remembering that any complex number can be written in the form re" by (9.4), we get Section 9 Euler's Formula 63 2 e2 = ri r2 e4(02+@2), e 21 22 (9.6) In words, to multiply two complex numbers, we multiply their absolute values and add their angles. To divide two complex numbers, we divide the absolute values and subtract the angles Example. Evaluate (1 + i)?/(1 - i). From Figure 5.1 we have 1 2et/4. We plot 1 - i in Figure 9.5 and find r2, 0=/4 (or +7/4), so 1-i= /2e-i#/4. Then (VEetr/a)2 V2e-i7/4=J%-in/s = /2e3i#/4 2 eir/2 (1+i)2 1-1 Figure 9.5 From Figure 9.6, we find =-1, y = 1, so (1i)2 1-1 =riy-1+i We could use degrees in this problem. By (9.6), we find that the angle of (1 i)2/(1-i) is 2(45°) - (-45°) 135° as in Figure 9.6 Figure 9.6 PROBLEMS, SECTION 9 Express the following complex numbers in the r + iy form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others-try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers. 3. 93ri/2 e-2i -4mi - 2. ei/2 1. ei/4 (a/3)(344mi 4. 6. 5. 7. 3e2(1+i 2esri/6 9. 2e-i/2 /4 4e-Sin/3 10. 11. 12. (i) 1-i (1+ W) 15. (1 (1 i)* 14. 13. ( (-)(1+v) 17. 16 () 19. (1-) 21 20.

Remembering that any complex number can be written in the form re" by (9.4), we get Section 9 Euler's Formula 63 2 e2 = ri r2 e4(02+@2), e 21 22 (9.6) In words, to multiply two complex numbers, we multiply their absolute values and add their angles. To divide two complex numbers, we divide the absolute values and subtract the angles Example. Evaluate (1 + i)?/(1 - i). From Figure 5.1 we have 1 2et/4. We plot 1 - i in Figure 9.5 and find r2, 0=/4 (or +7/4), so 1-i= /2e-i#/4. Then (VEetr/a)2 V2e-i7/4=J%-in/s = /2e3i#/4 2 eir/2 (1+i)2 1-1 Figure 9.5 From Figure 9.6, we find =-1, y = 1, so (1i)2 1-1 =riy-1+i We could use degrees in this problem. By (9.6), we find that the angle of (1 i)2/(1-i) is 2(45°) - (-45°) 135° as in Figure 9.6 Figure 9.6 PROBLEMS, SECTION 9 Express the following complex numbers in the r + iy form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others-try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers. 3. 93ri/2 e-2i -4mi - 2. ei/2 1. ei/4 (a/3)(344mi 4. 6. 5. 7. 3e2(1+i 2esri/6 9. 2e-i/2 /4 4e-Sin/3 10. 11. 12. (i) 1-i (1+ W) 15. (1 (1 i)* 14. 13. ( (-)(1+v) 17. 16 () 19. (1-) 21 20.

Advanced Engineering Mathematics
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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Problems,Section 9 number 3,4,6. Please answer the questions

Remembering that any complex number can be written in the form re" by (9.4), we get Section 9 Euler's Formula 63 2 e2 = ri r2 e4(02+@2), e 21 22 (9.6) In words, to multiply two complex numbers, we multiply their absolute values and add their angles. To divide two complex numbers, we divide the absolute values and subtract the angles Example. Evaluate (1 + i)?/(1 - i). From Figure 5.1 we have 1 2et/4. We plot 1 - i in Figure 9.5 and find r2, 0=/4 (or +7/4), so 1-i= /2e-i#/4. Then (VEetr/a)2 V2e-i7/4=J%-in/s = /2e3i#/4 2 eir/2 (1+i)2 1-1 Figure 9.5 From Figure 9.6, we find =-1, y = 1, so (1i)2 1-1 =riy-1+i We could use degrees in this problem. By (9.6), we find that the angle of (1 i)2/(1-i) is 2(45°) - (-45°) 135° as in Figure 9.6 Figure 9.6 PROBLEMS, SECTION 9 Express the following complex numbers in the r + iy form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others-try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers. 3. 93ri/2 e-2i -4mi - 2. ei/2 1. ei/4 (a/3)(344mi 4. 6. 5. 7. 3e2(1+i 2esri/6 9. 2e-i/2 /4 4e-Sin/3 10. 11. 12. (i) 1-i (1+ W) 15. (1 (1 i)* 14. 13. ( (-)(1+v) 17. 16 () 19. (1-) 21 20.
Transcribed Image Text:Remembering that any complex number can be written in the form re" by (9.4), we get Section 9 Euler's Formula 63 2 e2 = ri r2 e4(02+@2), e 21 22 (9.6) In words, to multiply two complex numbers, we multiply their absolute values and add their angles. To divide two complex numbers, we divide the absolute values and subtract the angles Example. Evaluate (1 + i)?/(1 - i). From Figure 5.1 we have 1 2et/4. We plot 1 - i in Figure 9.5 and find r2, 0=/4 (or +7/4), so 1-i= /2e-i#/4. Then (VEetr/a)2 V2e-i7/4=J%-in/s = /2e3i#/4 2 eir/2 (1+i)2 1-1 Figure 9.5 From Figure 9.6, we find =-1, y = 1, so (1i)2 1-1 =riy-1+i We could use degrees in this problem. By (9.6), we find that the angle of (1 i)2/(1-i) is 2(45°) - (-45°) 135° as in Figure 9.6 Figure 9.6 PROBLEMS, SECTION 9 Express the following complex numbers in the r + iy form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others-try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers. 3. 93ri/2 e-2i -4mi - 2. ei/2 1. ei/4 (a/3)(344mi 4. 6. 5. 7. 3e2(1+i 2esri/6 9. 2e-i/2 /4 4e-Sin/3 10. 11. 12. (i) 1-i (1+ W) 15. (1 (1 i)* 14. 13. ( (-)(1+v) 17. 16 () 19. (1-) 21 20.
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