Value of 2 − 1 − i in the complexI am trying to find the value of 2 − 1 − i I rewrite it like this, 2 − 1 − i = e ln ⁡ ( 2 ) ( − 1 − i ) = 1 e ln ⁡ 2 e i ln ⁡ 2 = 1 / 2Since e i ln ⁡ 2 = e R e ( i ln ⁡ 2 ) = e 0 = 1This looks way nicer than it should be, I think, can anyone tell me where I go wrong, and maybe a good way to do this problem?Thank you for any input.

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Value of       2          −      1      −      i       in the complexI am trying to find the value of       2          −      1      −      i      I rewrite it like this,       2          −      1      −      i        =      e          ln      ⁡      (      2      )      (      −      1      −      i      )        =            1                        e                      ln            ⁡            2                                    e                      i            ln            ⁡            2                                =  1      /    2Since       e          i      ln      ⁡      2        =      e          R      e      (      i      ln      ⁡      2      )        =      e    0    =  1This looks way nicer than it should be, I think, can anyone tell me where I go wrong, and maybe a good way to do this problem?Thank you for any input.

Tianna Deleon · Accepted Answer

2          −      1      −      i        =      2          −      1        ⋅      2          −      i        =      1    2    ⋅  (      e          Log      2            )          −      i        =            e              i        (        −        Log        2        )              2  where             Log      e        z    =          log      e        ⁡    z    +    2    n    π    i  Now, use Euler Formula

landdenaw · Accepted Answer

This is a multi-valued function. For each   k  ∈      Z  , you can write      2          −      1      −      i        =      e          (      −      1      −      i      )      (      ln      ⁡      2      +      2      π      k      i      )        =      e          −      ln      ⁡      2      +      2      π      k      +      i      (      −      ln      ⁡      2      −      2      π      k      )        =                                          1            2                                e                          2              π              k                                cos          ⁡          ln          ⁡          2          −          i                      1            2                                e                          2              π              k                                sin          ⁡          ln          ⁡          2

Value of 2 <mrow class="MJX-TeXAtom-ORD"> − 1 &

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