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 ) = ?

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Why can a Complex Logarithm have infinitely many values?What does this mean, that &quot;Due to the periodicity of the trigonometric functions, a complex logarithm can have infinitely many values&quot;?  ln  ⁡  z  =  ln  ⁡  (  cos  ⁡  x  +  i  sin  ⁡  x  )  =  ?

Elianna Douglas · Accepted Answer

The complex logarithm is the inverse of the complex exponential. By Euler&#039;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 · Accepted Answer

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 &quot;the logarithm of   z there is no way to choose so that the result will give a continuous logarithm function.

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

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