Why can a Complex Logarithm have infinitely many values?What does this mean, that "Due to...

Averi Mitchell

Averi Mitchell

Answered

2022-06-25

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"?
ln z = ln ( cos x + i sin x ) = ?

Answer & Explanation

Elianna Douglas

Elianna Douglas

Expert

2022-06-26Added 23 answers

The complex logarithm is the inverse of the complex exponential. By Euler's formula,
e i y = cos y + i sin y .
Hence for any integer n,
e x + ( y + 2 π n ) i = e x ( cos y + i sin ( y + 2 π n ) ) = e x ( cos y + i sin y ) = e x + i y .
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).
Feinsn

Feinsn

Expert

2022-06-27Added 8 answers

The exponential function f ( x ) = e x 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 log ( y )) such that e log ( y ) = y
But the exponential function f ( z ) = e z = n 0 z n n ! considered as a function on the complex numbers, is not one-to-one. There is in particular the wonderful identity due to Euler,
e i x = cos ( x ) + i sin ( x ) ,
which shows for example that there are infinitely many solutions to the equation e z = 1 namely z = 2 π k for k = , 3 , 2 , 1 , 0 , 1 , 2 , 3 So the complex exponential function is not one-to-one.
More fundamentally, while we could just randomly choose a solution w to the equation e w = z for each z, and call those choices collectively "the logarithm of z there is no way to choose so that the result will give a continuous logarithm function.

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