Integrate a Chebyshev series over specific axis in Python

To integrate a Chebyshev series over a specific axis in Python, use the numpy.polynomial.chebyshev.chebint() method. This function returns the Chebyshev series coefficients integrated m times from a lower bound along the specified axis.

Syntax

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

Parameters

The chebint() method accepts the following parameters ?

• c ? Array of Chebyshev series coefficients. For multidimensional arrays, different axes correspond to different variables
• m ? Order of integration (must be positive). Default is 1
• k ? Integration constants. If empty list (default), all constants are set to zero
• lbnd ? Lower bound of integration. Default is 0
• scl ? Scale factor applied after each integration. Default is 1
• axis ? Axis over which integration is performed. Default is 0

Example

Let's integrate a 2D Chebyshev series coefficient array along different axes ?

import numpy as np
from numpy.polynomial import chebyshev as C

# Create a multidimensional array of Chebyshev series coefficients
c = np.arange(4).reshape(2,2)

# Display the coefficient array
print("Our coefficient Array...\n", c)

# Check the dimensions and properties
print("\nDimensions of our Array...\n", c.ndim)
print("\nDatatype of our Array object...\n", c.dtype)
print("\nShape of our Array object...\n", c.shape)

# Integrate along axis 1
print("\nResult (axis=1)...\n", C.chebint(c, axis=1))

Our coefficient Array...
 [[0 1]
 [2 3]]

Dimensions of our Array...
2

Datatype of our Array object...
int64

Shape of our Array object...
(2, 2)

Result (axis=1)...
 [[0.25 0.   0.25]
 [0.75 2.   0.75]]

Integration Along Different Axes

You can also integrate along axis 0 to see the difference ?

import numpy as np
from numpy.polynomial import chebyshev as C

c = np.arange(4).reshape(2,2)
print("Original array:\n", c)

# Integrate along axis 0 (default)
result_axis0 = C.chebint(c, axis=0)
print("\nIntegration along axis 0:\n", result_axis0)

# Integrate along axis 1
result_axis1 = C.chebint(c, axis=1)
print("\nIntegration along axis 1:\n", result_axis1)

Original array:
 [[0 1]
 [2 3]]

Integration along axis 0:
 [[1.   1.5 ]
 [0.   0.  ]
 [0.25 0.375]]

Integration along axis 1:
 [[0.25 0.   0.25]
 [0.75 2.   0.75]]

How It Works

When integrating a Chebyshev series, the function:

• Increases the degree of the polynomial by 1 along the specified axis
• Applies the integration formula specific to Chebyshev polynomials
• Multiplies by the scale factor scl after each integration
• Adds integration constants k as needed

Conclusion

The chebint() method provides efficient integration of Chebyshev series along any axis. Use the axis parameter to control which dimension is integrated, allowing flexible manipulation of multidimensional polynomial data.