Q2P

Page 61

Show from the power series (8.1) that ddzez=ez.

Q2P

Page 59

Find the disk of convergence for each of the following complex power series.

z-z22+z33-z44+

Q2P

Page 77

In each of the following problems, z represents the displacement of a particle from the origin. Find (as functions of t) its speed and the magnitude of its acceleration, and describe the motion.

z=-5eiωt,ω=const

Q30MP

Page 81

Write the series forex(1+i). Write1+iin theformre and so obtain (easily) the powers of (1+i). Thus show, for example, that theexcosxseries has nox2term, nox6term, etc., and a similar result for theexsinxseries. Find (easily) a formula for the general term for each series.

Q30P

Page 64

Use Problems 27 and 28 to find the following absolute values. If you understand Problems 27 and 28 and equation (5.1), you should be able to do these in your head.

|e3-i|

Q30P

Page 71

Find each of the following in the x+iy form and check your answers by computer.

tanh(3πi4)

Q30P

Page 67

Show that the sum of the three cube roots of 8is zero.

Q30P

Page 53

Find the absolute value of each of the following using the discussion above. Try to do simple problems like these in your head-it saves time.

3ii-3.

Q31MP

Page 81

Show that if a sequence of complex numbers tends to zero, then the sequence of absolute values tends to zero too, and vice versa. Hintan+ibn0means an0andbn0.

Q31P

Page 53

Find the absolute value of each of the following using the discussion above. Try to do simple problems like these in your head-it saves time.

5-2i5+2i.

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