Python Program to Integrate a Chebyshev Series and Set the Integration Constant

In this article, we will discuss how to integrate a Chebyshev Series and set the integration constant

To perform Chebyshev integration, NumPy provides a function called chebyshev.chebint which can be used to integrate Chebyshev series. 

Syntax: chebyshev.chebint(c, m=1, k=[], lbnd=0, scl=1, axis=0)

Parameters:

• c - Array of Chebyshev series coefficients.
• m - (integer) Order of integration, must be positive
• k -  Integration constant. 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
• lbnd - The lower bound of the integral. (Default: 0) 
• scl - Following each integration the result is multiplied by scl before the integration constant is added. (Default: 1)
• axis - Axis over which the integral is taken.

Example 1:

In the first example. let us consider a 1D array with a first-order integration and 3 as an integration constant. Import the necessary packages as shown and pass the appropriate parameters as shown below.

import numpy as np
from numpy.polynomial import chebyshev

# co.efficient array
c = np.array([11, 12, 13, 14, 15])

print(f'The shape of the array is {c.shape}')
print(f'The dimension of the array is {c.ndim}D')
print(f'The datatype of the array is {c.dtype}')

res = chebyshev.chebint(c, m=1, k=3)

# integrated chebyshev series 
# with integration constant of 1
print(f'Resultant series ---> {res}')

Output:

Example 2:

In the second example. let us consider a 2D array with a first-order integration and 5 as an integration constant. Import the necessary packages as shown and pass the appropriate parameters as shown below.

import numpy as np
from numpy.polynomial import chebyshev

# co.efficient array
c = np.array([[11, 12, 13, 14, 15], [3, 4, 5, 6, 7]])

print(f'The shape of the array is {c.shape}')
print(f'The dimension of the array is {c.ndim}D')
print(f'The datatype of the array is {c.dtype}')

res = chebyshev.chebint(c, m=1, k=5)

# integrated chebyshev series
# with integration constant of 5
print(f'Resultant series ---> {res}')

Output:

Example 3:

In the third example. let us consider a 3D array with a fifth-order integration and 7 as an integration constant. Import the necessary packages as shown and pass the appropriate parameters as shown below.

import numpy as np
from numpy.polynomial import chebyshev

# co.efficient array
c = np.array([[[11, 12, 13, 14, 15],
               [3, 4, 5, 6, 7],
               [21, 22, 23, 24, 25]]])

print(f'The shape of the array is {c.shape}')
print(f'The dimension of the array is {c.ndim}D')
print(f'The datatype of the array is {c.dtype}')

res = chebyshev.chebint(c, m=5, k=7)

# integrated chebyshev series
# with integration constant of 7
print(f'Resultant series ---> {res}')

Output: