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By George Polya, Gordon Latta

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Sin p sin y - cos P sin y sin e>: - cos y sin or. sin p and find an analogous expression for sin (or. + P + y). 6 LOGARITHMS We now extend to the complex domain the concept of natural logarithms, which we call shortly logarithms. We say that w is the logarithm of z, and write (I) w = log z if and only if (2) Thus, we have defined the logarithmic function for complex values of the variable. By our definition , the two relations (I) and (2) are fully equivalent; both have exactly the same meaning.

1) . (2) SID Z 2! 4! 3 z6 6! = Z - - + - + -- + · · · 3! 5! 7! z z1 It follows immediately that (3) cos ( - z) = cos z sin ( -z) = -sin z since the series (I) contains only even powers of z and (2) only odd powers. 2 (5). We cannot help observing a certain regularity if we write 38 Chapter Two the three series as follows : zs z' z' z2 + ez = 1 + -z + +-+-+··· 1 ! 2 ! 3 ! 4! 5 ! z' z2 cos z = 1 - + -··· 2! 4! z z3 z 6 sm z = - - - + - + · · . 1! 3! 5! We may observe that the signs in the last two series recur periodically with the period 4.

On the left-hand side we take logarithms of arbitrary complex numbers, using the new, unrestricted meaning of the term. On the right-hand side we take the logarithm of the positive number r = l z l , using the old meaning of the term, which is restricted to real and uniquely determined logarithms of positive numbers. It would be natural to expect that the new definition of logarithm, when applied to positive numbers, should give the same result as the old definition ; then the above distinction would be unnecessary.

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