Butterworth filters are widely used as easy-to-handle filters because they are flat (maximum flatness) with no ripple in the passband and have characteristics that gradually attenuate with respect to frequency.
The transfer function is below. It is possible to express from 1st order to nth order.
Here, a symbol like π is a multiplication, and indicates the multiplication of functions as follows.
Specific calculation examples of multiplication are as follows.
The calculation process is explained here, and the results are as follows.