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6. Evaluate Jezdz, where C is the circle |z = 4 taken with counter-clockwise (i.e. anti- clockwise) orientation. Hint: Let C: z(t) = 4 cost+i4 sint for 0≤t≤2. (Ans: 0) 7. Evaluate √ Edz, where C is the circle [2] = 4 taken with counter-clockwise (i.e. anti- clockwise) orientation. (Ans: 32mi) (Ans: 8. Evaluate √(z+1)dz, where C is given by z(t) = cost+isint for 0 ≤t≤π/2. i-2) - 9. Evaluate fezdz, where C is the line segment from i to 1 and 2(t) = + (1 − t)i for (Ans: 1) 10. Evaluate fez²dz, where C is the line segment from 1 to 1+i and z(t) = 1 + it for 0≤t≤1. 11. Evaluate (2-iy²))dz, where C is the upper semicircle C : z(t) 0sts, taken counter-clockwise. 12. Evaluate fc2dz, where C is given by C: z(t)=t+ it² for 0 1), show that and HENCE that (+4) that lim IR = 0. R-+00 dz, where CR is the semicircle below, centre 0, radius R > 2. Show Ri CR -Ri о

6. Evaluate Jezdz, where C is the circle |z = 4 taken with counter-clockwise (i.e. anti- clockwise) orientation. Hint: Let C: z(t) = 4 cost+i4 sint for 0≤t≤2. (Ans: 0) 7. Evaluate √ Edz, where C is the circle [2] = 4 taken with counter-clockwise (i.e. anti- clockwise) orientation. (Ans: 32mi) (Ans: 8. Evaluate √(z+1)dz, where C is given by z(t) = cost+isint for 0 ≤t≤π/2. i-2) - 9. Evaluate fezdz, where C is the line segment from i to 1 and 2(t) = + (1 − t)i for (Ans: 1) 10. Evaluate fez²dz, where C is the line segment from 1 to 1+i and z(t) = 1 + it for 0≤t≤1. 11. Evaluate (2-iy²))dz, where C is the upper semicircle C : z(t) 0sts, taken counter-clockwise. 12. Evaluate fc2dz, where C is given by C: z(t)=t+ it² for 0 1), show that and HENCE that (+4) that lim IR = 0. R-+00 dz, where CR is the semicircle below, centre 0, radius R > 2. Show Ri CR -Ri о

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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6. Evaluate Jezdz, where C is the circle |z = 4 taken with counter-clockwise (i.e. anti- clockwise) orientation. Hint: Let C: z(t) = 4 cost+i4 sint for 0≤t≤2. (Ans: 0) 7. Evaluate √ Edz, where C is the circle [2] = 4 taken with counter-clockwise (i.e. anti- clockwise) orientation. (Ans: 32mi) (Ans: 8. Evaluate √(z+1)dz, where C is given by z(t) = cost+isint for 0 ≤t≤π/2. i-2) - 9. Evaluate fezdz, where C is the line segment from i to 1 and 2(t) = + (1 − t)i for (Ans: 1) 10. Evaluate fez²dz, where C is the line segment from 1 to 1+i and z(t) = 1 + it for 0≤t≤1. 11. Evaluate (2-iy²))dz, where C is the upper semicircle C : z(t) 0sts, taken counter-clockwise. 12. Evaluate fc2dz, where C is given by C: z(t)=t+ it² for 0 <t≤1. 8/15+15/6) (Ans: -1+21/3) cost+i sint for (Ans: 2/3+14/3) (Ans: 13. Evaluate fez -12dz, where C is the upper half of the circle || = 1 taken with the counter-clockwise orientation. (Ans: -4-mi) 14. Evaluate (1/2)dz, where C is the circle |=|=2 taken with clockwise orientation. Hint: C: z(t)=2 cost-12 sint for 0 St≤2. (Ans: -2πi) 15. Evaluate c(1/2)dz, where C is the circle |2|= 2 taken with clockwise orientation. (Ans: 0) 16. Evaluate fee'dz, where C is the straight line segment joining 1 to 1+i. (Ans: -2e) 17. Show that cos zdz = sin(1 + i) where C is the polygonal path from 0 to 1+i that consists of segments from 0 to 1 and 1 to 1+i. = 18. Show that fee'dze+-1 where C is the straight line segment joining 0 to 1+i. 19. Evaluate √cze dz where C is the square with vertices 0, 1, 1+i and i taken with the counter-clockwise orientation. 20. If IR= dz z²(z+1)' CR (a) IR 2 R(R-1) (b) lim IR = 0. R-400 21. If IR = z-5 where CR is the circle |2|= R (R> 1), show that and HENCE that (+4) that lim IR = 0. R-+00 dz, where CR is the semicircle below, centre 0, radius R > 2. Show Ri CR -Ri о
Transcribed Image Text:6. Evaluate Jezdz, where C is the circle |z = 4 taken with counter-clockwise (i.e. anti- clockwise) orientation. Hint: Let C: z(t) = 4 cost+i4 sint for 0≤t≤2. (Ans: 0) 7. Evaluate √ Edz, where C is the circle [2] = 4 taken with counter-clockwise (i.e. anti- clockwise) orientation. (Ans: 32mi) (Ans: 8. Evaluate √(z+1)dz, where C is given by z(t) = cost+isint for 0 ≤t≤π/2. i-2) - 9. Evaluate fezdz, where C is the line segment from i to 1 and 2(t) = + (1 − t)i for (Ans: 1) 10. Evaluate fez²dz, where C is the line segment from 1 to 1+i and z(t) = 1 + it for 0≤t≤1. 11. Evaluate (2-iy²))dz, where C is the upper semicircle C : z(t) 0sts, taken counter-clockwise. 12. Evaluate fc2dz, where C is given by C: z(t)=t+ it² for 0 <t≤1. 8/15+15/6) (Ans: -1+21/3) cost+i sint for (Ans: 2/3+14/3) (Ans: 13. Evaluate fez -12dz, where C is the upper half of the circle || = 1 taken with the counter-clockwise orientation. (Ans: -4-mi) 14. Evaluate (1/2)dz, where C is the circle |=|=2 taken with clockwise orientation. Hint: C: z(t)=2 cost-12 sint for 0 St≤2. (Ans: -2πi) 15. Evaluate c(1/2)dz, where C is the circle |2|= 2 taken with clockwise orientation. (Ans: 0) 16. Evaluate fee'dz, where C is the straight line segment joining 1 to 1+i. (Ans: -2e) 17. Show that cos zdz = sin(1 + i) where C is the polygonal path from 0 to 1+i that consists of segments from 0 to 1 and 1 to 1+i. = 18. Show that fee'dze+-1 where C is the straight line segment joining 0 to 1+i. 19. Evaluate √cze dz where C is the square with vertices 0, 1, 1+i and i taken with the counter-clockwise orientation. 20. If IR= dz z²(z+1)' CR (a) IR 2 R(R-1) (b) lim IR = 0. R-400 21. If IR = z-5 where CR is the circle |2|= R (R> 1), show that and HENCE that (+4) that lim IR = 0. R-+00 dz, where CR is the semicircle below, centre 0, radius R > 2. Show Ri CR -Ri о
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