On the real axis The function along the real axis. is real‐valued in the interval . At , the function has a logarithmic singularity. The function along the real axis. The left graphic shows and the right graphic shows . In the interval the function is purely imaginary and the real part vanishes identically. 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 , 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, the real part of is a decreasing function and the imaginary part is piecewise constant. 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 .
