Why can a Complex Logarithm have infinitely many values?What does this mean, that "Due to...
Why can a Complex Logarithm have infinitely many values?
What does this mean, that "Due to the periodicity of the trigonometric functions, a complex logarithm can have infinitely many values"?
Answer & Explanation
The complex logarithm is the inverse of the complex exponential. By Euler's formula,
Hence for any integer n,
So the complex exponential takes each of its values infinitely often, from which it follows that the complex logarithm has infinitely many values (for this reason, we much choose a branch cut for the complex logarithm).
The exponential function as a function that takes real numbers to positive real numbers, is a one-to-one and onto function, and therefore has a well-defined inverse function: for each positive y, there is just one number (denoted ) such that
But the exponential function considered as a function on the complex numbers, is not one-to-one. There is in particular the wonderful identity due to Euler,
which shows for example that there are infinitely many solutions to the equation namely for So the complex exponential function is not one-to-one.
More fundamentally, while we could just randomly choose a solution to the equation for each z, and call those choices collectively "the logarithm of there is no way to choose so that the result will give a continuous logarithm function.