EXPONENTIAL FUNCTION OF A COMPLEX NUMBER

  

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The function  of a real variable x, is called the exponential function of a real variable x. is For a complex variable z=x+iy, the exponential function of Z,   written as expz, defined by 

                         exp z = exp(x+iy)=  (cos y+i sin y). 

This definition agrees with the real exponential function when z is purely real. When z is purely real, y = 0 and 

                           exp z = exp(x+0)= (cos0+isin 0) =

. When z is purely imaginary, x =0 and 

                           exp = exp(0+iy)=   (cos y+isin y) = cos y +isin y. 

Since  >0 for all real x,  (cos y +isin y) represents a complex number in polar form,   being the modulus and y being an amplitude of expz. Since   >0 for any real number x, the expz is a non-zero complex number for any complex number z.

Let u+iv be a non-zero complex number and let it's polar re-representation be r(cosθ+isin θ). Since r is positive, log r is real and r can be expressed as   Therefore 

                              u + iv =  (cosθ+i sin θ) = exp (logr+iθ) 

Thus when u +iv is a given non-zero complex number, there exists a complex number, Z= logr+iθ such that exp z =u+iv.    This means that the range of the exponential function of z is the entire complex plane excluding the origin.