# Wolfram Language & System 10.4 (2016)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# DifferentialRoot

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

DifferentialRoot[lde]
represents a pure holonomic function .

## DetailsDetails

• 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 , where can contain equations and inequalities.

## ExamplesExamplesopen allclose all

### Basic Examples  (2)Basic Examples  (2)

Define f to be the sin function:

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Plot its result:

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Solve a differential equation:

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Numerical values:

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