Assignment 2 PHY3604 Semester 1 2013/2014 Problems are taken from Chapter 2 of M.L. Boas’ “Mathematical Methods in the Physical Sciences”. Section 4 For each of the following numbers, visualize where it is in the complex plane. Find x, y, r, for these numbers and plot and label them on the complex plane. Also plot the complex conjugate of the number. 4.  3 i 8. 3 cos 32  i sin 32 13. 18. 20. 3 ei / 2 7  cos 110  i sin 110  Section 5 First simplify each of the following numbers to the x  iy form or to the r ei form. Then plot the number in the complex plane. i 2  2i  1 4. 5. 14. 15. 16. i  3  2 2.8 e  i (1.1) 5  2i 5  2i 1 0.5  cos 40  i sin 40   Find each of the following in rectangular  a  bi  form if z  2  3i ; if z  x  iy . 20. 22. 1 z2 1 z i Find the absolute value of each of the following using the discussion above. 5  2i 31. 5  2i Solve for all possible values of the real numbers x and y in the following equations. 35. x  iy  3i  4 41. 44. 47.  2 x  3 y  5  i  x  2 y  1 2 x  iy  1  i  3  x  iy    1  0 Section 9 Express the following complex numbers in the x  iy form. Try to visualize each complex number, using sketches if necessary. ei  e  i 10. 15. 1  i  20.  2     i 1 2  1  i  4 10 24.  1i 3 i  1  21 38 Find the following absolute values. Results of problems 27 and 28 may be assumed. 31. 5 e2 i /3 34. 4 e2i 1 Section 11 Find each of the following in rectangular form x  iy and check your results by computer. 4. 6. e3ln 2  i cos  i ln 5  8. cos   2i ln 3 In the following integrals express the sines and cosines in exponential form and then integrate to show that: 13. 14.    sin 2x sin 3x dx    sin 4x dx    2 0 2 0 Evaluate  e a  ib  x dx and take real and imaginary parts to show that: 17.  eax cos bx dx  eax  a cos bx  b sin bx  a 2  b2 Section 12 Verify each of the following by using equations (11.4), (12.2) and (12.3). cosh 2 z  sinh 2 z  1 11. 12. cos4 z  sin 4 z  1  12 sin 2 2 z tan x  i tanh y 18. tan z  tan  x  iy   1  i tan x tanh y Find the real part, the imaginary part, and the absolute value of 23. cosh  ix  31. sinh  ln 2  i 3  36. sinh 1  i 2  37. cos  i  Section 14 Evaluate each of the following in x  iy form, and compare with a computer solution.  3. ln i  13. i 2i /  1 i  2i  14. 3  Section 15 Find each of the following in the x  iy form and compare a computer solution. 5.   arccos i 8 1  i  6. tanh 11. sinh 1 i / 2 14. arcsin  3i / 4   