# Complex Variables by George Polya, Gordon Latta

By George Polya, Gordon Latta

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J X2 + y 2 = I Z I (cos 0 + ; sin 0)2 = cos2 0 - sin2 0 + (a) The interior of a circle, center at z = - ! /3 (minor). The midpoint of side b, c is (b + c)f2, and the point i the distance toward a on the line joining (b + c) /2 to a is (a + b + c)f3 ; use symmetry. � l < 1 lab = l al l b l sin ({J - IX} The area of the oriented parallelogram (positive if the shortest rota­ tion from a to b is in the positive sense. ) Chapter One Chapter TWO COMPLEX FUNCTIONS Complex functions of a complex variable can be obtained by extension of the usual analytic expressions to complex numbers, and can be used in problems dealing with maps or two dimensional vector fields.

These vectors may have some physical significance as displace­ ments, velocities, and f01ces, for example. We may represent these vectors by complex numbers a , b, ... z; in order to do so we must choose a system of coordinates. Physical phenomena do not depend, however, on our choice of the coordinate system. Therefore, we should pay particular attention to operati ons wi th vectors whose results are independent of the choi ce of the coordi nate system; we call such operations vectori al operati ons.

Substituting -z for z in (4) and taking (3) in account, we obtain (5) Combining e-iz = cos z - i sin z (4) and (5) by addition and subtraction, we obtain (6) Thus we have obtained expressions for the trigonometric functions cosine and sine in terms of the exponential function. If we do not consider complex values of the variable, we have no such expressions. By (4) and ( 5), or by (6), the study of trigonometric functions is reduced to that of the exponential function. For instance, we can deduce (3) from (6).