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On the real axis The function  along the real axis.  is an oscillating periodic function with period  . The absolute value and the argument of  along the real axis. The left graphic shows  and the right graphic shows  . The argument is piecewise constant. On the real axis at infinity The function  along the real axis. At  , the function f=sin(1/x) has an essential singularity and oscillates infinitely often. The absolute value and the argument of  along the real axis. The left graphic shows  and the right graphic shows  . The argument is piecewise constant. On the imaginary axes The real part and the imaginary part of  along the imaginary axis. The left graphic shows  and the right graphic shows  . Along the imaginary axis,  is purely imaginary and its absolute value is exponentially increasing. The absolute value and the argument of  along the imaginary axis. The left graphic shows  and the right graphic shows  . Because  is purely imaginary on the imaginary axis, the argument is piecewise constant. On the imaginary axis at infinity The function  along the imaginary axis. The left graphic shows  and the right graphic shows  . At  , the function  has an essential singularity. The absolute value and the argument of  along the imaginary axis. The left graphic shows  and the right graphic shows  . On the unit circle The real part and the imaginary part on the unit circle. The left graphic shows  and the right graphic shows  . The absolute value and the argument on the unit circle. The left graphic shows  and the right graphic shows  .
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