# DifferentialRoot

DifferentialRoot[lde][x]

gives the holonomic function , specified by the linear differential equation lde[h,x].

DifferentialRoot[lde]

represents a pure holonomic function .

# Details  • Mathematical function, suitable for both symbolic and numerical manipulation; also known as holonomic function and D-finite function.
• The holonomic function defined by a DifferentialRoot function satisfies a holonomic differential equation with polynomial coefficients and initial values .
• DifferentialRoot can be used like any other mathematical function.
• FunctionExpand will attempt to convert DifferentialRoot functions in terms of special functions.
• The functions representable by DifferentialRoot include a large number of special functions.
• DifferentialRootReduce can convert most special functions to DifferentialRoot functions.
• Holonomic functions are closed under many operations, including:
• constant multiple, integer power sums and products composition with polynomial, rational, and algebraic functions convolution derivatives and integrals
• DifferentialRoot is automatically generated by functions such as Integrate, DSolve, and GeneratingFunction.
• Functions such as Integrate, D, SeriesCoefficient, and DSolve work with DifferentialRoot inputs.
• DifferentialRoot can be evaluated to arbitrary numerical precision.
• DifferentialRoot automatically threads over lists.
• DifferentialRoot[lde,pred] represents a solution restricted to avoid cuts in the complex plane defined by pred[z], where pred[z] can contain equations and inequalities.

# Examples

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## Basic Examples(2)

Define f to be the sin function:

Plot its result:

Solve a differential equation:

Numerical values:

## Scope(5)

Simple exact values are generated automatically:

DifferentialRoot works on rational coefficients:

Inhomogeneous linear recurrences:

Solutions of a differential equation:

## Generalizations & Extensions(1)

Equations with holonomic constant terms are automatically lifted to polynomial coefficients:

## Applications(1)

Find the DifferentialRoot object of a special function:

Compute integrals:

## Properties & Relations(5)

Extract the differential equation from a DifferentialRoot object:

Extract branch cuts if any:

Use DifferentialRootReduce to generate DifferentialRoot objects:

Integrate a DifferentialRoot object:

Find coefficients of the expansion of a DifferentialRoot object:

Introduced in 2008
(7.0)