Prove for any real number k, prove that the exponential function e z is a bijection (z is a complex number) from the strip k &lt; I m ( z ) ≤ k + 2 π to the complex plane minus the point 0, C − { 0 }

Question

Prove for any real number k, prove that the exponential function       e    z   is a bijection (z is a complex number) from the strip   k  &amp;lt;  I  m  (  z  )  ≤  k  +  2  π to the complex plane minus the point 0,       C    −  {  0  }

Gabriella Hensley · Accepted Answer

Hint To solve       e          x      +      i      y        =  ω, write   ω in trigonometric form and solve.

KesseTher12 · Accepted Answer

A hint: Don&#039;t solve equations, but investigate what the exponential function  z  =  x  +  i  y     ↦         e    z    =      e    x    ⋅      e          i      y      does to horizontal lines      g    v    :    y  :=  v     (  =      c    o    n    s    t    .    )     ,    −  ∞  &amp;lt;  x  &amp;lt;  ∞     ,and to vertical lines      h    u    :    x  :=  u     (  =      c    o    n    s    t    .    )  ,    −  ∞  &amp;lt;  y  &amp;lt;  ∞     .

Prove for any real number k, prove that the exponential function e^z is a bijection (z is a complex number) from the strip k<Im(z)=<k+2p to the complex plane minus the point 0, C−{0}.

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