Function polyint

numpy/lib/_polynomial_impl.py:270–370

Return an antiderivative (indefinite integral) of a polynomial. .. note:: This forms part of the old polynomial API. Since version 1.4, the new polynomial API defined in `numpy.polynomial` is preferred. A summary of the differences can be found in the :doc:`tran

(p, m=1, k=None)

@array_function_dispatch(_polyint_dispatcher)
def polyint(p, m=1, k=None):
    """
    Return an antiderivative (indefinite integral) of a polynomial.

    .. note::
       This forms part of the old polynomial API. Since version 1.4, the
       new polynomial API defined in `numpy.polynomial` is preferred.
       A summary of the differences can be found in the
       :doc:`transition guide </reference/routines.polynomials>`.

    The returned order `m` antiderivative `P` of polynomial `p` satisfies
    :math:`\\frac{d^m}{dx^m}P(x) = p(x)` and is defined up to `m - 1`
    integration constants `k`. The constants determine the low-order
    polynomial part

    .. math:: \\frac{k_{m-1}}{0!} x^0 + \\ldots + \\frac{k_0}{(m-1)!}x^{m-1}

    of `P` so that :math:`P^{(j)}(0) = k_{m-j-1}`.

    Parameters
    ----------
    p : array_like or poly1d
        Polynomial to integrate.
        A sequence is interpreted as polynomial coefficients, see `poly1d`.
    m : int, optional
        Order of the antiderivative. (Default: 1)
    k : list of `m` scalars or scalar, optional
        Integration constants. They are given in the order of integration:
        those corresponding to highest-order terms come first.

        If ``None`` (default), all constants are assumed to be zero.
        If `m = 1`, a single scalar can be given instead of a list.

    See Also
    --------
    polyder : derivative of a polynomial
    poly1d.integ : equivalent method

    Examples
    --------

    The defining property of the antiderivative:

    >>> import numpy as np

    >>> p = np.poly1d([1,1,1])
    >>> P = np.polyint(p)
    >>> P
     poly1d([ 0.33333333,  0.5       ,  1.        ,  0.        ]) # may vary
    >>> np.polyder(P) == p
    True

    The integration constants default to zero, but can be specified:

    >>> P = np.polyint(p, 3)
    >>> P(0)
    0.0
    >>> np.polyder(P)(0)