Integrate a Legendre series.
Returns a Legendre series that is the Legendre series cs, integrated m times from lbnd to x. 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 cs is a sequence of coefficients, from lowest order Legendre series “term” to highest, e.g., [1,2,3] represents the series P_0(x) + 2P_1(x) + 3P_2(x).
Parameters : | cs : array_like
m : int, optional
k : {[], list, scalar}, optional
lbnd : scalar, optional
scl : scalar, optional
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Returns : | S : ndarray
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Raises : | ValueError :
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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 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 “re-projected” onto the C-series basis set. Thus, typically, the result of this function is “un-intuitive,” albeit correct; see Examples section below.
Examples
>>> from numpy.polynomial import legendre as L
>>> cs = (1,2,3)
>>> L.legint(cs)
array([ 0.33333333, 0.4 , 0.66666667, 0.6 ])
>>> L.legint(cs,3)
array([ 1.66666667e-02, -1.78571429e-02, 4.76190476e-02,
-1.73472348e-18, 1.90476190e-02, 9.52380952e-03])
>>> L.legint(cs, k=3)
array([ 3.33333333, 0.4 , 0.66666667, 0.6 ])
>>> L.legint(cs, lbnd=-2)
array([ 7.33333333, 0.4 , 0.66666667, 0.6 ])
>>> L.legint(cs, scl=2)
array([ 0.66666667, 0.8 , 1.33333333, 1.2 ])