Exponential of z ¯ I am currently reading the book Complex Variables by Stephen Fisher, there is one paragraph that was written like this: Establishing the following relation, and they write e x p ( z ¯ ) = e x p ( z ) ¯ the bar on the right is long and span throughout the whole 3 letters and z, what does this mean? And what do they mean by establishing the relation, do I show that they equal each other? There are no further explanations, they just said left for readers.

Question

Exponential of             z      ¯      I am currently reading the book Complex Variables by Stephen Fisher, there is one paragraph that was written like this: Establishing the following relation, and they write  e  x  p  (            z      ¯        )  =            e      x      p      (      z      )        ¯  the bar on the right is long and span throughout the whole 3 letters and z, what does this mean? And what do they mean by establishing the relation, do I show that they equal each other? There are no further explanations, they just said left for readers.

Riya Richard · Accepted Answer

The bar is the complex conjugate:             x      +      i      y        ¯    =  x  −  i  y. Your book is saying that for   z  ∈      C  , then the exponential of z-conjugate is the conjugate of the exponential of z.Recall Euler&#039;s formula:       e          i      y        =  cos  ⁡  (  y  )  +  i  sin  ⁡  (  y  ). Now, consider       e          x      −      i      y        =      e    x    (  cos  ⁡  (  −  y  )  +  i  sin  ⁡  (  −  y  )  ). Sine is an odd function and cosine is an even function, so this is equivalent to       e    x    (  cos  ⁡  (  y  )  −  i  sin  ⁡  (  y  )  ), which is the complex conjugate of       e    x    (  cos  ⁡  (  y  )  +  i  sin  ⁡  (  y  )  )  =      e          x      +      i      y      , Q.E.D.

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