DifferentialRoot

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

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:

Evaluate numerically to any precision:

Compare the result with the built-in Sin function:

Solve a differential equation:

Numerical values:

Scope (23)

Numerical Evaluation (7)

Evaluate at machine precision:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

DifferentialRoot takes complex number parameters and arguments:

DifferentialRoot takes inexact input parameters:

Evaluate DifferentialRoot efficiently at high precision:

DifferentialRoot threads elementwise over lists and matrices:

Function Properties (5)

DifferentialRoot objects have all the standard features of a mathematical function:

Integrate the function:

Differentiate it:

Find its series expansion:

Plot it on the reals:

Plot it in the complex plane:

Simple exact values are generated automatically:

Use FunctionExpand to attempt to convert a DifferentialRoot object to a built-in mathematical function:

DifferentialRoot works on equations with rational coefficients:

Inhomogeneous holonomic equations are automatically transformed to higher-order homogeneous ones:

Differentiation (4)

The derivative of DifferentialRoot is a DifferentialRoot function:

Differentiate a DifferentialRoot object with respect to a parameter:

Compute higher-order derivatives of a DifferentialRoot object:

Differentiate a DifferentialRoot object:

Specific values of :

Plot of :

Integration (4)

The integral of a DifferentialRoot object is a DifferentialRoot object:

Compute higher-order integrals of a DifferentialRoot object:

Compute the definite integral of a DifferentialRoot object:

Integrate a DifferentialRoot object:

Specific values of :

Plot of :

Series Expansions (3)

Calculate the series expansion of a DifferentialRoot object:

Find the coefficient of the Taylor expansion of a DifferentialRoot object:

Calculate the first 9 coefficients:

Compare with the Sin function expansion coefficients:

Calculate the series expansion of a DifferentialRoot object with a parameter:

Generalizations & Extensions (1)

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

Applications (4)

Generate a DifferentialRoot object from a special function:

Integrate it:

DifferentialRoot objects have all the standard features of a mathematical function:

Find the coefficients of the series expansion of a DifferentialRoot object:

Calculate the first 5 coefficients of the expansion explicitly:

Compute arbitrary-order derivatives of a DifferentialRoot object:

Integrate the DifferentialRoot object:

Extract the differential equation and initial conditions of the function that is the integral of f:

Plot the function f, its integral and derivative functions:

Use DifferentialRoot to homogenize a differential equation:

Extract the homogenized equation:

Generate a DifferentialRoot object that is a combination of two mathematical functions:

Extract the differential equation and initial conditions that this function obeys:

Properties & Relations (5)

DifferentialRootReduce generates DifferentialRoot objects:

DSolve generates a DifferentialRoot object if the solution is not available in known functions:

GeneratingFunction may generate a DifferentialRoot object:

Integrate returns a DifferentialRoot object for general holonomic functions:

D returns a DifferentialRoot object for general holonomic functions:

Possible Issues (3)

DifferentialRoot takes only linear differential equations with polynomial coefficients:

DifferentialRoot will not evaluate if the initial values are given at a singular point:

The branch cut structure of a built-in function may differ from the automatically computed branch cut structure:

For some regions of the complex plane, the value of f differs from corresponding built-in function value:

For other regions, DifferentialRoot will give the same result:

Neat Examples (1)

Solve a differential equation that is unsolved in known mathematical functions:

Calculate the numerical values of this solution:

Plot this solution:

Differentiate this solution:

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DifferentialRoot

Details

Examples

Basic Examples (2)