# numpy.polynomial.chebyshev.chebint

polynomial.chebyshev.chebint(c, m=1, k=[], lbnd=0, scl=1, axis=0)

  Integrate a Chebyshev series.

  Returns the Chebyshev series coefficients c integrated m times from lbnd along axis. At each iteration the resulting series is multiplied by scl and an integration constant, k, is added. The scaling factor is for use in a linear change of variable. (“Buyer beware”: note that, depending on what one is doing, one may want scl to be the reciprocal of what one might expect; for more information, see the Notes section below.) The argument c is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series `T_0 + 2*T_1 + 3*T_2` while [[1,2],[1,2]] represents `1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + 2*T_0(x)*T_1(y) + 2*T_1(x)*T_1(y)` if axis=0 is `x` and axis=1 is `y`.

  Parameters:
    c array_like

      Array of Chebyshev series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index.

    m int, optional

      Order of integration, must be positive. (Default: 1)

    k{[], list, scalar}, optional

      Integration constant(s). The value of the first integral at zero is the first value in the list, the value of the second integral at zero is the second value, etc. If `k == []` (the default), all constants are set to zero. If `m == 1`, a single scalar can be given instead of a list.

    lbnd scalar, optional

      The lower bound of the integral. (Default: 0)

    scl scalar, optional

      Following each integration the result is multiplied by scl before the integration constant is added. (Default: 1)

    axis int, optional

      Axis over which the integral is taken. (Default: 0).

      New in version 1.7.0.

  Returns:
    S ndarray

      C-series coefficients of the integral.

  Raises:
    ValueError

      If `m < 1`, `len(k) > m`, `np.ndim(lbnd) != 0`, or `np.ndim(scl) != 0`.

  See also

  `chebder`

  Notes

  Note that the result of each integration is multiplied by scl. Why is this important to note? Say one is making a linear change of variable \(u = ax + b\) in an integral relative to x. Then \(dx = du/a\), so one will need to set scl equal to \(1/a\)- perhaps not what one would have first thought.

  Also note that, in general, the result of integrating a C-series needs to be “reprojected” onto the C-series basis set. Thus, typically, the result of this function is “unintuitive,” albeit correct; see Examples section below.

  Examples

      >>> from numpy.polynomial import chebyshev as C
      >>> c = (1,2,3)
      >>> C.chebint(c)
      array([ 0.5, -0.5,  0.5,  0.5])
      >>> C.chebint(c,3)
      array([ 0.03125   , -0.1875    ,  0.04166667, -0.05208333,  0.01041667, # may vary
          0.00625   ])
      >>> C.chebint(c, k=3)
      array([ 3.5, -0.5,  0.5,  0.5])
      >>> C.chebint(c,lbnd=-2)
      array([ 8.5, -0.5,  0.5,  0.5])
      >>> C.chebint(c,scl=-2)
      array([-1.,  1., -1., -1.])