## Mathematics for Engineers and Scientists by Alan Jeffrey

By Alan Jeffrey

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1 Enumerate the elements in the following sets in which I signifies the set of natural positive and negative integers including zero: (a) S={njnel, 5

COSO, y = r v'SlilO Fig. 18 Polar coordinates (r, 0). 6 POLAR COORDINATES I 39 replace 0 by 0 ± 2nn, with n = 0, 1, 2, .... Thus the polar angle 0 is determined up to a multiple of 2n for every point in the plane apart from the origin at which it is not defined. Because of the necessity to use polar coordinates in problems involving differentiation, the angle 0 will always be measured in radians and, for convenience, the angle 0 will be chosen to lie in the interval 0 :5 0 < ZJC. Now let the reference line be identified with the x-axis of a cartesian coordinate system in the same plane with its origin also located at 0.

1 = 2[2"u 0 + (2"- 1)] + 1 = 2"+ 1 u0 +(2"+ 1 -1) showing that S(n)~S(n + 1). The result is thus true for all n. To conclude this section, having introduced the notion of a difference equation let us take the concept a little further so that it can be used in more general circumstances. 3) where a and b are real constants and u. _ 2 , u. _ 1 , u. are three consecutive members of a sequence of numbers. 3) provides an algorithm by which any other member of the sequence may be computed. 4) where A and A.