Analysis WebNotes
arrow_back
Unfortunately, the story doesn't end there. The definition of complex logarithms is more complicated and ambiguous than that. The reason for this is that there isn't one "right" polar form. The numbers r and t in the polar form are uniquely specified, once we agree that r is positive and t lies between 0 and 2Pi.

Now, there isn't really any reason why you'd want r to be negative, but why should you restrict t that way? Why not restrict t to lie between -Pi and Pi instead? Or indeed in any range of total length 2Pi? It turns out that you could prove a version of Proposition 7.20 for any interval of the form [a, a+2Pi) and then the definition of the complex logarithm would have different imaginary values.

There isn't a simple way round this. The strange behavior of the complex logarithm (the fact that it has infinitely many legitimate "branches") is tied up in deep results in complex analysis.

We shan't be very concerned with complex logarithms in this course. The definition we gave above is a very common one to use, and you should start off by just thinking of that as being the "main" definition. Just bear in mind that there can be others. You might also like to know that:

• All branches of the complex logarithm agree on the positive real numbers.
• In fact, any two branches of the complex logarithm must always differ by an integer multiple of 2Pi