By George Polya, Gordon Latta
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20 Describe the region formed by those points z for which the term z"fn ! is greater in absolute value than the other terms of the infinite series z z2 z3 DoZ + 1 +- +- +- + ··· 3! 2! 1! 21 Let A and B denote complex constants, and z a complex variable. ,t: 0, and show that A + A + Bz + Bz = 0 is the equation of an (arbitrary) straight line. 22 Let A, B, and C denote complex constants, and z a complex variable. 4 + Bz + Bz + ( C + C) z1. = 0 is the equation of an (arbitrary) circle. , determine the three vertices of a triangle, and w, wit w2 the corresponding vertices of another triangle.
For instance, they lead quickly to various expressions for 1 /z, the reciprocal of z: Let us write 1 Z X -iy 1 - = -2 = 2 = - (cos 6 - i sin 6 ) z lz1 x + y2 r z1 = x1 + i y1 = r1 (cos 6 1 + i sin 6J where x1 , y 1 0 rh 61 are real and r1 positive, as the notation suggests. Then we can express the quotient z1/z as follows : r1 z1 z1z = 2 = - [cos (6 1 - 6 ) + ism (6 1 - 6 )] r z lzl If we wish to write the conjugate of a composite expression we use longer horizontal lines. For instance, if we suppose, as usual that x and y are real, z2 = (x + iy)2 = x2 - y2 - 2ixy .
If n is an integer. sin 2n7T = 0 cos 2n7T = 1 Consequences of Euler's Theorem 41 and therefore e12 11 " = cos 2mr + ; sin 2mr = 1 (3) Hence (4) This last equation is apparently more general than (2) which is contained i n it as a special case for n = 1 . But we may easily derive (4) from ( 2) by repeated application. For instance, We obtain similarly that and so on. (c) Since cos we have 11' = -1 sin 11' =0 ei" = cos 11' + i sin 11' = - 1 We may remember this result in the form ei " + 1 = 0 (5) Somebody called this the "most beautiful" formula since, he said, it combines the five "most important" numbers, 0, 1 , i, 11', and e.
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