23 trace the variation of z when z starting at the

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23. Trace the variation of z when Z , starting at the point exp π , moves round the larger circle in the positive direction to the point - exp π , along the barrier, round the smaller circle in the negative direction, back along the barrier, and round the remainder of the larger circle to its original position. 24. Suppose each plane to be divided up into an infinite series of rings by circles of radii . . . , e - (2 n +1) π , . . . , e - π , e π , e 3 π , . . . , e (2 n +1) π , . . . . Show how to make any ring in one plane correspond to any ring in the other, by taking suitable values of the powers in the equations z = Z i , Z = z - i .
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[X : 237] THE GENERAL THEORY OF THE LOGARITHMIC, 540 25. If z = Z i , any value of the power being taken, and Z moves along an equiangular spiral whose pole is the origin in its plane, then z moves along an equiangular spiral whose pole is the origin in its plane. 26. How does Z = z ai , where a is real, behave as z approaches the origin along the real axis? [ Z moves round and round a circle whose centre is the origin (the unit circle if z ai has its principal value), and the real and imaginary parts of Z both oscillate finitely.] 27. Discuss the same question for Z = z a + bi , where a and b are any real numbers. 28. Show that the region of convergence of a series of the type -∞ a n z nai , where a is real, is an angle, i.e. a region bounded by inequalities of the type θ 0 < am z < θ 1 [The angle may reduce to a line, or cover the whole plane.] 29. Level Curves. If f ( z ) is a function of the complex variable z , we call the curves for which | f ( z ) | is constant the level curves of f ( z ). Sketch the forms of the level curves of z - a ( concentric circles ) , ( z - a )( z - b ) ( Cartesian ovals ) , ( z - a ) / ( z - b ) ( coaxal circles ) , exp z ( straight lines ) . 30. Sketch the forms of the level curves of ( z - a )( z - b )( z - c ), (1+ z 3+ z 2 ) /z . [Some of the level curves of the latter function are drawn in Fig. 59 , the curves marked i vii corresponding to the values . 10 , 2 - 3 = . 27 , . 40 , 1 . 00 , 2 . 00 , 2 + 3 = 3 . 73 , 4 . 53 of | f ( z ) | . The reader will probably find but little difficulty in arriving at a general idea of the forms of the level curves of any given rational function; but to enter into details would carry us into the general theory of functions of a complex variable.] 31. Sketch the forms of the level curves of (i) z exp z , (ii) sin z . [See Fig. 60 , which represents the level curves of sin z . The curves marked i viii correspond to k = . 35, . 50, . 71, 1 . 00, 1 . 41, 2 . 00, 2 . 83, 4 . 00.] 32. Sketch the forms of the level curves of exp z - c , where c is a real constant. [ Fig. 61 shows the level curves of | exp z - 1 | , the curves i vii corresponding to the values of k given by log k = - 1 . 00, - . 20, - . 05, 0 . 00, . 05, . 20, 1 . 00.]
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[X : 237] EXPONENTIAL, AND CIRCULAR FUNCTIONS 541 - 6 - 5 - 4 - 3 - 2 0 1 2 3 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 - 1 I I II III IV V VI VII VII Fig. 59. 33. The level curves of sin z - c , where c is a positive constant, are sketched in Figs. 62, 63. [The nature of the curves differs according as to whether c < 1 or c > 1. In Fig. 62 we have taken c = . 5, and the curves i viii correspond to
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[X : 237] THE GENERAL THEORY OF THE LOGARITHMIC, 542 O - π 2 π 2 I II III IV V VI VII VIII Fig. 60. 0
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