Overview

Symbolica documentation for getting started, symbolic expressions, numerical evaluation, pattern matching, and APIs in Python and Rust.

`Exact integration`

Exact definite integration using hyperlogarithms.

integrate uses hyperlogarithms over [0, +infinity), in variable order; integrate_over accepts explicit directed intervals. Symbolica's Expression.integrate(x) continues to compute an antiderivative.

Browser execution is serial, including when parallel=True. HEPkit's existing native IBP tools are available separately through hepkit.ibp.

Classes


`AlgebraError`	Failure of an exact algebra operation.
`AlgebraicLetter`	Immutable metadata for one formal quadratic-root pair allocated by an integration.
`ContextError`
`DivergentIntegralError`
`DuplicateVariableError`
`InputError`	Invalid expression, integration-variable schedule, or options.
`IntegrationError`	Base class for errors raised by Hyperbolica.
`IntegrationOptions`	Options controlling exact hyperlogarithm integration
`IntegrationResult`	An inspectable exact integration result
`PreparedIntegral`	A lowered Atom input that can be integrated repeatedly without repeating Symbolica-to-ring and Hlog conversion
`UnsupportedFeatureError`	The requested integration feature is unavailable for this input.

Functions


`integrate`	Integrate a Symbolica expression over `[0, infinity)` in variable order.
`integrate_detailed`	Integrate and return an inspectable exact result.
`integrate_detailed_over`	Integrate over arbitrary directed intervals and retain exact metadata
`integrate_over`	Integrate over arbitrary directed intervals in variable order.
`mzv_symbol`	Return the registered multiple-zeta-value function symbol
`prepare`	Lower a Symbolica expression once for repeated integrations or option sweeps

`integrate`

integrate(
    expression: Expression,
    variables: typing.Sequence[Expression],
    options: typing.Optional[IntegrationOptions] = None,
) -> Expression

Integrate a Symbolica expression over [0, infinity) in variable order.

Parameters

expression A native symbolica.Expression from the shipped Symbolica kernel. It is never converted to text.
variables Plain Symbolica symbols in integration order.
options Optional IntegrationOptions.

Returns

symbolica.Expression The normalized exact result.

`integrate_detailed`

integrate_detailed(
    expression: Expression,
    variables: typing.Sequence[Expression],
    options: typing.Optional[IntegrationOptions] = None,
) -> IntegrationResult

Integrate and return an inspectable exact result.

Parameters

expression A native symbolica.Expression from the shipped Symbolica kernel.
variables Plain shipped Symbolica symbols in integration order.
options Optional IntegrationOptions.

Returns

IntegrationResult The returned IntegrationResult retains variables, indeterminates, algebraic-letter metadata, and the collected term count in addition to its expression.

`integrate_detailed_over`

integrate_detailed_over(
    expression: Expression,
    variables: typing.Sequence[Expression],
    intervals: typing.Sequence[tuple[Expression, Expression]],
    options: typing.Optional[IntegrationOptions] = None,
) -> IntegrationResult

Integrate over arbitrary directed intervals and retain exact metadata.

intervals contains one (from, to) pair per integration variable and accepts native real directed infinities through Symbol.INFINITY.

`integrate_over`

integrate_over(
    expression: Expression,
    variables: typing.Sequence[Expression],
    intervals: typing.Sequence[tuple[Expression, Expression]],
    options: typing.Optional[IntegrationOptions] = None,
) -> Expression

Integrate over arbitrary directed intervals in variable order.

Parameters

expression A native symbolica.Expression from the shipped Symbolica kernel.
variables Plain Symbolica symbols in integration order.
intervals One (from, to) pair of native expressions per variable. Use symbolica.Symbol.INFINITY (or its negation) for directed infinity.
options Optional IntegrationOptions.

Returns

symbolica.Expression The normalized exact result.

`mzv_symbol`

mzv_symbol() -> Expression

Return the registered multiple-zeta-value function symbol.

Call the returned Symbolica expression with indices, e.g. mzv_symbol()(3). This is the same head used in exact integration results. It is a formal symbol; for a depth-one numerical reference use Symbolica's built-in E("3").zeta().evaluate({}, decimal_digit_precision=40).

`prepare`

prepare(
    expression: Expression,
    variables: typing.Sequence[Expression],
    options: typing.Optional[IntegrationOptions] = None,
) -> PreparedIntegral

Lower a Symbolica expression once for repeated integrations or option sweeps.

The returned immutable PreparedIntegral owns the lowered input and an independent copy of options. Mathematical input remains a native Symbolica expression throughout preparation.

Parameters

expression A native symbolica.Expression from the shipped Symbolica kernel.
variables Plain shipped Symbolica symbols in integration order.
options Options captured as the prepared object's default integration options.

Returns

PreparedIntegral Reusable immutable lowered input.