#### Previous topic

scipy.signal.unique_roots

#### Next topic

scipy.signal.residuez

# scipy.signal.residue

`scipy.signal.``residue`(b, a, tol=0.001, rtype='avg')

  Compute partial-fraction expansion of b(s) / a(s).

  If `M = len(b)` and `N = len(a)`, then the partial-fraction expansion H(s) is defined as:

              b(s)     b[0] s**(M-1) + b[1] s**(M-2) + ... + b[M-1]
      H(s) = ------ = ----------------------------------------------
              a(s)     a[0] s**(N-1) + a[1] s**(N-2) + ... + a[N-1]

               r[0]       r[1]             r[-1]
           = -------- + -------- + ... + --------- + k(s)
             (s-p[0])   (s-p[1])         (s-p[-1])

  If there are any repeated roots (closer together than tol), then H(s) has terms like:

        r[i]      r[i+1]              r[i+n-1]
      -------- + ----------- + ... + -----------
      (s-p[i])  (s-p[i])**2          (s-p[i])**n

  Returns:

  r : ndarray

  > Residues.

  p : ndarray

  > Poles.

  k : ndarray

  > Coefficients of the direct polynomial term.

  See also

  `invres`, `numpy.poly`, `unique_roots`