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polyint(p, m=1, k=None)
note

This forms part of the old polynomial API. Since version 1.4, the new polynomial API defined in :None:None:`numpy.polynomial` is preferred. A summary of the differences can be found in the :None:doc:`transition guide </reference/routines.polynomials>`.

The returned order m antiderivative :None:None:`P` of polynomial p satisfies $\frac{d^m}{dx^m}P(x) = p(x)$ and is defined up to :None:None:`m - 1` integration constants k. The constants determine the low-order polynomial part

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

of :None:None:`P` so that $P^{(j)}(0) = k_{m-j-1}$ .

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 :None:None:`m = 1`, a single scalar can be given instead of a list.

Return an antiderivative (indefinite integral) of a polynomial.

See Also

poly1d.integ

equivalent method

polyder

derivative of a polynomial

Examples

The defining property of the antiderivative:

>>> 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)
0.0
>>> np.polyder(P, 2)(0)
0.0
>>> P = np.polyint(p, 3, k=[6,5,3])
... P
poly1d([ 0.01666667,  0.04166667,  0.16666667,  3. ,  5. ,  3. ]) # may vary

Note that 3 = 6 / 2!, and that the constants are given in the order of integrations. Constant of the highest-order polynomial term comes first:

>>> np.polyder(P, 2)(0)
6.0
>>> np.polyder(P, 1)(0)
5.0
>>> P(0)
3.0
See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

numpy.polymul numpy.polyder numpy.polydiv numpy.polyadd numpy.polynomial.polynomial.polyder numpy.lib.polynomial.poly1d.integ

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GitHub : /numpy/lib/polynomial.py#267
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