By Smirnov V.I.
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Extra info for A course of higher mathematics, vol. 2
24(a). 48. Find a point (x0 , y0 , u0 ) on the unit sphere that corresponds to the complex number 2 + 5i. 24(b). 49. Describe the set of points on the unit sphere that correspond to each of the following sets in the complex plane. (a) the numbers on unit circle |z| = 1 (b) the numbers within the open disk |z| < 1 (c) the numbers that are exterior to unit circle, that is, |z| > 1 50. 24(b) in terms of the coordinates of the point (a, b, 0) in the complex plane. Use these formulas to verify your answer to Problem 48.
10) Because of (10), the results in (7) are valid. Moreover, note that Euler’s formula (4) is a special case of (9) when z is a pure imaginary number, that is, with x = 0 and y replaced by θ. Euler’s formula provides a notational convenience for several concepts considered earlier in this chapter. The polar form of a complex number z, z = r(cos θ + i sin θ), can now be written compactly as z = reiθ . (11) This convenient form is called the exponential form of a complex number z. √ For example, i = eπi/2 and 1 + i = 2eπi/4 .
25. 1+ 27. 1 2 29. √ 3i 9 10 + 12 i √ √ π π 2 cos + i 2 sin 8 8 12 26. (2 − 2i)5 √ √ 28. (− 2 + 6i)4 √ 2π 2π 30. 3 cos + i sin 9 9 6 In Problems 31 and 32, write the given complex number in polar form and in then in the form a + ib. π π π 12 π 5 31. cos + i sin 2 cos + i sin 9 9 6 6 3π 3π + i sin 8 8 π π 2 cos + i sin 16 16 3 8 cos 32. 10 33. Use de Moivre’s formula (10) with n = 2 to ﬁnd trigonometric identities for cos 2θ and sin 2θ. 34. Use de Moivre’s formula (10) with n = 3 to ﬁnd trigonometric identities for cos 3θ and sin 3θ.
A course of higher mathematics, vol. 2 by Smirnov V.I.