transforms – Function Transforms

Transform module: function parsing and mathematical transforms (Fourier, Laplace, Taylor).

transforms.parse_scalar_function(expression, parameters=None)

Parse a scalar function expression f(x) into a vectorized callable.

The expression should use x as the independent variable. Supports the same math functions as the ODE parser.

Parameters:

• expression (str) – Python expression string (e.g. "sin(x)", "exp(-a*x)").

• parameters (dict[str, float] | None) – Named parameter values (e.g. {"a": 1.0}).

Return type:

Callable[[ndarray], ndarray]

Returns:

A vectorized callable f(x) that accepts a numpy array and returns the evaluated values.

Raises:

EquationParseError – If the expression is invalid.

class transforms.DisplayMode(*values)

Bases: str, Enum

How to display the transform result.

CURVE = 'Curve (f vs x)'

COEFFICIENTS = 'Coefficients (aᵢ vs i)'

class transforms.TransformKind(*values)

Bases: str, Enum

Available transformation types.

ORIGINAL = 'Original (f(x))'

FOURIER = 'Fourier (FFT)'

LAPLACE = 'Laplace (real axis)'

TAYLOR = 'Taylor series'

HILBERT = 'Hilbert (discrete)'

Z_TRANSFORM = 'Z-transform (discrete)'

transforms.apply_transform(func, kind, x_min, x_max, n_points=1024, *, taylor_order=5, taylor_center=None, laplace_s_min=0.1, laplace_s_max=10.0, laplace_n_points=200, fourier_amp_threshold=0.01, z_transform_amp_threshold=0.01, laplace_amp_threshold=0.01)

Apply a transformation to a function and return (x, y) for plotting.

Parameters:

• func (Callable[[ndarray], ndarray]) – Vectorized callable f(x) -> y.

• kind (TransformKind) – Type of transformation.

• x_min (float) – Lower bound of the original domain.

• x_max (float) – Upper bound of the original domain.

• n_points (int) – Number of sample points for Original/Fourier/Taylor.

• taylor_order (int) – Order of Taylor expansion (for Taylor kind).

• taylor_center (float | None) – Center point for Taylor (default: midpoint).

• laplace_s_min (float) – Minimum s for Laplace (real axis).

• laplace_s_max (float) – Maximum s for Laplace.

• laplace_n_points (int) – Number of s values for Laplace.

• fourier_amp_threshold (float) – Relative amplitude threshold (fraction of max) to trim Fourier spectrum at low and high frequencies. Frequencies with |F(ω)| below this fraction of the maximum are excluded from the displayed range. Default 0.01.

• z_transform_amp_threshold (float) – Same as fourier_amp_threshold but for Z-transform magnitude spectrum. Default 0.01.

• laplace_amp_threshold (float) – Same for Laplace: trim s range where |L(s)| is below this fraction of max. Default 0.01.

Return type:

tuple[ndarray, ndarray, str, str]

Returns:

Tuple of (x_axis, y_values, x_label, y_label).

Raises:

ValueError – If kind is not a known transform type.

transforms.compute_function_samples(func, x_min, x_max, n_points=1024)

Sample a function over a range.

Parameters:

• func (Callable[[ndarray], ndarray]) – Vectorized callable f(x) -> y.

• x_min (float) – Lower bound.

• x_max (float) – Upper bound.

• n_points (int) – Number of sample points.

Return type:

tuple[ndarray, ndarray]

Returns:

Tuple of (x, y) arrays.

transforms.get_transform_coefficients(func, kind, x_min, x_max, n_points=512, *, taylor_order=5, taylor_center=None, laplace_s_min=0.1, laplace_s_max=10.0, laplace_n_points=50, fourier_amp_threshold=0.01, z_transform_amp_threshold=0.01, laplace_amp_threshold=0.01, hilbert_amp_threshold=0.01)

Return coefficient representation with physical axis and metadata.

Uses frequency ω (or s for Laplace) as x-axis when meaningful for interpretation.

Parameters:

• func (Callable[[ndarray], ndarray]) – Vectorized callable f(x) -> y.

• kind (TransformKind) – Transform kind.

• x_min (float) – Lower bound.

• x_max (float) – Upper bound.

• n_points (int) – Sample count.

• taylor_order (int) – Taylor order.

• taylor_center (float | None) – Taylor center.

• laplace_n_points (int) – Number of s values for Laplace.

• fourier_amp_threshold (float) – Amplitude threshold for Fourier coefficient trimming.

• z_transform_amp_threshold (float) – Amplitude threshold for Z-transform coefficient trimming.

• laplace_amp_threshold (float) – Amplitude threshold for Laplace coefficient trimming.

• hilbert_amp_threshold (float) – Amplitude threshold for Hilbert coefficient trimming.

• laplace_s_min (float)

• laplace_s_max (float)

Return type:

tuple[ndarray, ndarray, str, str, dict[str, object]]

Returns:

Tuple of (x_axis, coefficients, x_label, y_label, metadata). metadata: dict with domain, n_points, and transform-specific params.

Raises:

ValueError – If kind is not a known transform type.

transforms.function_parser

Safe parsing and evaluation of user-written scalar functions f(x).

transforms.function_parser.parse_scalar_function(expression, parameters=None)

Parse a scalar function expression f(x) into a vectorized callable.

The expression should use x as the independent variable. Supports the same math functions as the ODE parser.

Parameters:

• expression (str) – Python expression string (e.g. "sin(x)", "exp(-a*x)").

• parameters (dict[str, float] | None) – Named parameter values (e.g. {"a": 1.0}).

Return type:

Callable[[ndarray], ndarray]

Returns:

A vectorized callable f(x) that accepts a numpy array and returns the evaluated values.

Raises:

EquationParseError – If the expression is invalid.

transforms.transform_engine

Mathematical transforms: Fourier, Laplace, Taylor series.

class transforms.transform_engine.TransformKind(*values)

Bases: str, Enum

Available transformation types.

ORIGINAL = 'Original (f(x))'

FOURIER = 'Fourier (FFT)'

LAPLACE = 'Laplace (real axis)'

TAYLOR = 'Taylor series'

HILBERT = 'Hilbert (discrete)'

Z_TRANSFORM = 'Z-transform (discrete)'

class transforms.transform_engine.DisplayMode(*values)

Bases: str, Enum

How to display the transform result.

CURVE = 'Curve (f vs x)'

COEFFICIENTS = 'Coefficients (aᵢ vs i)'

transforms.transform_engine.compute_function_samples(func, x_min, x_max, n_points=1024)

Sample a function over a range.

Parameters:

• func (Callable[[ndarray], ndarray]) – Vectorized callable f(x) -> y.

• x_min (float) – Lower bound.

• x_max (float) – Upper bound.

• n_points (int) – Number of sample points.

Return type:

tuple[ndarray, ndarray]

Returns:

Tuple of (x, y) arrays.

transforms.transform_engine.apply_transform(func, kind, x_min, x_max, n_points=1024, *, taylor_order=5, taylor_center=None, laplace_s_min=0.1, laplace_s_max=10.0, laplace_n_points=200, fourier_amp_threshold=0.01, z_transform_amp_threshold=0.01, laplace_amp_threshold=0.01)

Apply a transformation to a function and return (x, y) for plotting.

Parameters:

• func (Callable[[ndarray], ndarray]) – Vectorized callable f(x) -> y.

• kind (TransformKind) – Type of transformation.

• x_min (float) – Lower bound of the original domain.

• x_max (float) – Upper bound of the original domain.

• n_points (int) – Number of sample points for Original/Fourier/Taylor.

• taylor_order (int) – Order of Taylor expansion (for Taylor kind).

• taylor_center (float | None) – Center point for Taylor (default: midpoint).

• laplace_s_min (float) – Minimum s for Laplace (real axis).

• laplace_s_max (float) – Maximum s for Laplace.

• laplace_n_points (int) – Number of s values for Laplace.

• fourier_amp_threshold (float) – Relative amplitude threshold (fraction of max) to trim Fourier spectrum at low and high frequencies. Frequencies with |F(ω)| below this fraction of the maximum are excluded from the displayed range. Default 0.01.

• z_transform_amp_threshold (float) – Same as fourier_amp_threshold but for Z-transform magnitude spectrum. Default 0.01.

• laplace_amp_threshold (float) – Same for Laplace: trim s range where |L(s)| is below this fraction of max. Default 0.01.

Return type:

tuple[ndarray, ndarray, str, str]

Returns:

Tuple of (x_axis, y_values, x_label, y_label).

Raises:

ValueError – If kind is not a known transform type.

transforms.transform_engine.get_transform_coefficients(func, kind, x_min, x_max, n_points=512, *, taylor_order=5, taylor_center=None, laplace_s_min=0.1, laplace_s_max=10.0, laplace_n_points=50, fourier_amp_threshold=0.01, z_transform_amp_threshold=0.01, laplace_amp_threshold=0.01, hilbert_amp_threshold=0.01)

Return coefficient representation with physical axis and metadata.

Uses frequency ω (or s for Laplace) as x-axis when meaningful for interpretation.

Parameters:

• func (Callable[[ndarray], ndarray]) – Vectorized callable f(x) -> y.

• kind (TransformKind) – Transform kind.

• x_min (float) – Lower bound.

• x_max (float) – Upper bound.

• n_points (int) – Sample count.

• taylor_order (int) – Taylor order.

• taylor_center (float | None) – Taylor center.

• laplace_n_points (int) – Number of s values for Laplace.

• fourier_amp_threshold (float) – Amplitude threshold for Fourier coefficient trimming.

• z_transform_amp_threshold (float) – Amplitude threshold for Z-transform coefficient trimming.

• laplace_amp_threshold (float) – Amplitude threshold for Laplace coefficient trimming.

• hilbert_amp_threshold (float) – Amplitude threshold for Hilbert coefficient trimming.

• laplace_s_min (float)

• laplace_s_max (float)

Return type:

tuple[ndarray, ndarray, str, str, dict[str, object]]

Returns:

Tuple of (x_axis, coefficients, x_label, y_label, metadata). metadata: dict with domain, n_points, and transform-specific params.

Raises:

ValueError – If kind is not a known transform type.