numpy.polynomial.hermite_e.hermeint
-
numpy.polynomial.hermite_e.hermeint(c, m=1, k=[], lbnd=0, scl=1, axis=0)
[source] -
Integrate a Hermite_e series.
Returns the Hermite_e series coefficients
c
integratedm
times fromlbnd
alongaxis
. At each iteration the resulting series is multiplied byscl
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 wantscl
to be the reciprocal of what one might expect; for more information, see the Notes section below.) The argumentc
is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the seriesH_0 + 2*H_1 + 3*H_2
while [[1,2],[1,2]] represents1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) + 2*H_1(x)*H_1(y)
if axis=0 isx
and axis=1 isy
.Parameters: c : array_like
Array of Hermite_e 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
lbnd
is the first value in the list, the value of the second integral atlbnd
is the second value, etc. Ifk == []
(the default), all constants are set to zero. Ifm == 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
Hermite_e series coefficients of the integral.
Raises: ValueError
If
m < 0
,len(k) > m
,np.isscalar(lbnd) == False
, ornp.isscalar(scl) == False
.See also
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 variablein an integral relative to
x
. Then .. math::dx = du/a
, so one will need to setscl
equal to- 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.hermite_e import hermeint >>> hermeint([1, 2, 3]) # integrate once, value 0 at 0. array([ 1., 1., 1., 1.]) >>> hermeint([1, 2, 3], m=2) # integrate twice, value & deriv 0 at 0 array([-0.25 , 1. , 0.5 , 0.33333333, 0.25 ]) >>> hermeint([1, 2, 3], k=1) # integrate once, value 1 at 0. array([ 2., 1., 1., 1.]) >>> hermeint([1, 2, 3], lbnd=-1) # integrate once, value 0 at -1 array([-1., 1., 1., 1.]) >>> hermeint([1, 2, 3], m=2, k=[1, 2], lbnd=-1) array([ 1.83333333, 0. , 0.5 , 0.33333333, 0.25 ])
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Licensed under the NumPy License.
https://docs.scipy.org/doc/numpy-1.11.0/reference/generated/numpy.polynomial.hermite_e.hermeint.html