`numpy.polynomial.chebyshev:chebint`

source: /numpy/polynomial/chebyshev.py :962

Signature

   def     chebint (    c    ,    m    =  1   ,    k    =  []   ,    lbnd    =  0   ,    scl    =  1   ,    axis    =  0     )

Summary

Integrate a Chebyshev series.

Extended Summary

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).

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.

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 $u = a x + b$ in an integral relative to x. Then $d x = d u / a$ , 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)

✓

C.chebint(c,3)

✗

C.chebint(c, k=3)
C.chebint(c,lbnd=-2)
C.chebint(c,scl=-2)

✓

Aliases

numpy.polynomial.Chebyshev._int
numpy.polynomial.chebyshev.chebint