On the real axis The function along the real axis. is real‐valued outside the interval . The function along the real axis. The left graphic shows and the right graphic shows . In the interval the function is complex valued and in the interval it is purely imaginary. At , the function has a logarithmic singularity. The absolute value and the argument of along the real axis. The left graphic shows and the right graphic shows . For and , the argument is piecewise constant. On the real axis at infinity The function along the real axis. The left graphic shows and the right graphic shows . The absolute value and the argument of along the imaginary axis. The left graphic shows and the right graphic shows . For and ,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 the imaginary part is an asymptotically decreasing function. At the point , the function has a logarithmic singularity. The absolute value and the argument of along the imaginary axis. The left graphic shows and the right graphic shows . On the imaginary axis at infinity The function along the imaginary axis. The left graphic shows and the right graphic shows . 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 of on the unit circle. The left graphic shows and the right graphic shows . The absolute value and the argument of on the unit circle. The left graphic shows and the right graphic shows .
