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:

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 ^(th) 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:

Wolfram Research (2008), DifferentialRoot, Wolfram Language function, https://reference.wolfram.com/language/ref/DifferentialRoot.html (updated 2020).

Text

Wolfram Research (2008), DifferentialRoot, Wolfram Language function, https://reference.wolfram.com/language/ref/DifferentialRoot.html (updated 2020).

CMS

Wolfram Language. 2008. "DifferentialRoot." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2020. https://reference.wolfram.com/language/ref/DifferentialRoot.html.

APA

Wolfram Language. (2008). DifferentialRoot. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DifferentialRoot.html

BibTeX

@misc{reference.wolfram_2024_differentialroot, author="Wolfram Research", title="{DifferentialRoot}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/DifferentialRoot.html}", note=[Accessed: 22-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_differentialroot, organization={Wolfram Research}, title={DifferentialRoot}, year={2020}, url={https://reference.wolfram.com/language/ref/DifferentialRoot.html}, note=[Accessed: 22-December-2024 ]}