DifferentialRoot
✖
DifferentialRoot
gives the holonomic function , specified by the linear differential equation lde[h,x].
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
open allclose allBasic Examples (2)Summary of the most common use cases
Define f to be the sin function:

https://wolfram.com/xid/05fp67k07m-eytp8e


https://wolfram.com/xid/05fp67k07m-bl4eia

Evaluate numerically to any precision:

https://wolfram.com/xid/05fp67k07m-f3pz8c

Compare the result with the built-in Sin function:

https://wolfram.com/xid/05fp67k07m-7sdasy

Solve a differential equation:

https://wolfram.com/xid/05fp67k07m-er5nt0


https://wolfram.com/xid/05fp67k07m-v6wp87

Scope (23)Survey of the scope of standard use cases
Numerical Evaluation (7)
Evaluate at machine precision:

https://wolfram.com/xid/05fp67k07m-jua23i


https://wolfram.com/xid/05fp67k07m-uqurky

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

https://wolfram.com/xid/05fp67k07m-lw9h0n

DifferentialRoot takes complex number parameters and arguments:

https://wolfram.com/xid/05fp67k07m-64g5bd


https://wolfram.com/xid/05fp67k07m-diu7a2

DifferentialRoot takes inexact input parameters:

https://wolfram.com/xid/05fp67k07m-wvphoo


https://wolfram.com/xid/05fp67k07m-iwnnn6

Evaluate DifferentialRoot efficiently at high precision:

https://wolfram.com/xid/05fp67k07m-2c7v5i

DifferentialRoot threads elementwise over lists and matrices:

https://wolfram.com/xid/05fp67k07m-g2zcvw


https://wolfram.com/xid/05fp67k07m-lw7hpu

Function Properties (5)
DifferentialRoot objects have all the standard features of a mathematical function:

https://wolfram.com/xid/05fp67k07m-sjlbcm


https://wolfram.com/xid/05fp67k07m-7sdg2t


https://wolfram.com/xid/05fp67k07m-16vzpe


https://wolfram.com/xid/05fp67k07m-lbjm64


https://wolfram.com/xid/05fp67k07m-if27yd


https://wolfram.com/xid/05fp67k07m-zxwxeo

Simple exact values are generated automatically:

https://wolfram.com/xid/05fp67k07m-4gb6nx

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

https://wolfram.com/xid/05fp67k07m-fyvp49


https://wolfram.com/xid/05fp67k07m-26w3cp

DifferentialRoot works on equations with rational coefficients:

https://wolfram.com/xid/05fp67k07m-uryei4

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

https://wolfram.com/xid/05fp67k07m-79urvk

Differentiation (4)
The derivative of DifferentialRoot is a DifferentialRoot function:

https://wolfram.com/xid/05fp67k07m-41n6dj


https://wolfram.com/xid/05fp67k07m-eqbaum

Differentiate a DifferentialRoot object with respect to a parameter:

https://wolfram.com/xid/05fp67k07m-s7jlhu


https://wolfram.com/xid/05fp67k07m-i14vak

Compute higher-order derivatives of a DifferentialRoot object:

https://wolfram.com/xid/05fp67k07m-yloxzq


https://wolfram.com/xid/05fp67k07m-k9vdyl


https://wolfram.com/xid/05fp67k07m-4rowba


https://wolfram.com/xid/05fp67k07m-ugy1ya

Differentiate a DifferentialRoot object:

https://wolfram.com/xid/05fp67k07m-c46jps


https://wolfram.com/xid/05fp67k07m-8jykqv


https://wolfram.com/xid/05fp67k07m-llhv1n

Integration (4)
The integral of a DifferentialRoot object is a DifferentialRoot object:

https://wolfram.com/xid/05fp67k07m-dvob6a


https://wolfram.com/xid/05fp67k07m-hwytkk

Compute higher-order integrals of a DifferentialRoot object:

https://wolfram.com/xid/05fp67k07m-7wzkzx


https://wolfram.com/xid/05fp67k07m-u7r3br

Compute the definite integral of a DifferentialRoot object:

https://wolfram.com/xid/05fp67k07m-0a608r


https://wolfram.com/xid/05fp67k07m-ri4baw

Integrate a DifferentialRoot object:

https://wolfram.com/xid/05fp67k07m-ivwiks


https://wolfram.com/xid/05fp67k07m-tq2c8y


https://wolfram.com/xid/05fp67k07m-5vh16u

Series Expansions (3)
Calculate the series expansion of a DifferentialRoot object:

https://wolfram.com/xid/05fp67k07m-j4uy95

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

https://wolfram.com/xid/05fp67k07m-xckda7

Calculate the first 9 coefficients:

https://wolfram.com/xid/05fp67k07m-5knd2g

Compare with the Sin function expansion coefficients:

https://wolfram.com/xid/05fp67k07m-7k9qop

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

https://wolfram.com/xid/05fp67k07m-ez4xmx

Generalizations & Extensions (1)Generalized and extended use cases
Applications (4)Sample problems that can be solved with this function
Generate a DifferentialRoot object from a special function:

https://wolfram.com/xid/05fp67k07m-o9dteh


https://wolfram.com/xid/05fp67k07m-yks48n

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

https://wolfram.com/xid/05fp67k07m-ty1dx8

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

https://wolfram.com/xid/05fp67k07m-4m361q

Calculate the first 5 coefficients of the expansion explicitly:

https://wolfram.com/xid/05fp67k07m-v21u0

Compute arbitrary-order derivatives of a DifferentialRoot object:

https://wolfram.com/xid/05fp67k07m-j8auuo


https://wolfram.com/xid/05fp67k07m-0l5r6h

Integrate the DifferentialRoot object:

https://wolfram.com/xid/05fp67k07m-3xmnfj

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

https://wolfram.com/xid/05fp67k07m-7e8jbu

Plot the function f, its integral and derivative functions:

https://wolfram.com/xid/05fp67k07m-cyk1kn

Use DifferentialRoot to homogenize a differential equation:

https://wolfram.com/xid/05fp67k07m-b0cvci

Extract the homogenized equation:

https://wolfram.com/xid/05fp67k07m-8a2ftx

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

https://wolfram.com/xid/05fp67k07m-nzbigm

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

https://wolfram.com/xid/05fp67k07m-b8fo8f

Properties & Relations (5)Properties of the function, and connections to other functions
DifferentialRootReduce generates DifferentialRoot objects:

https://wolfram.com/xid/05fp67k07m-7y5qji

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

https://wolfram.com/xid/05fp67k07m-753j8v

GeneratingFunction may generate a DifferentialRoot object:

https://wolfram.com/xid/05fp67k07m-rzaqfr

Integrate returns a DifferentialRoot object for general holonomic functions:

https://wolfram.com/xid/05fp67k07m-f99vbv

D returns a DifferentialRoot object for general holonomic functions:

https://wolfram.com/xid/05fp67k07m-c4kj81

Possible Issues (3)Common pitfalls and unexpected behavior
DifferentialRoot takes only linear differential equations with polynomial coefficients:

https://wolfram.com/xid/05fp67k07m-7uy6i0


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

https://wolfram.com/xid/05fp67k07m-jijszb


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

https://wolfram.com/xid/05fp67k07m-mb0pmv

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

https://wolfram.com/xid/05fp67k07m-fpoh9e


https://wolfram.com/xid/05fp67k07m-l8fxmo

For other regions, DifferentialRoot will give the same result:

https://wolfram.com/xid/05fp67k07m-d9tbcv


https://wolfram.com/xid/05fp67k07m-s9zxhc

Neat Examples (1)Surprising or curious use cases
Solve a differential equation that is unsolved in known mathematical functions:

https://wolfram.com/xid/05fp67k07m-o6yx6

Calculate the numerical values of this solution:

https://wolfram.com/xid/05fp67k07m-kqiziv


https://wolfram.com/xid/05fp67k07m-fe277f


https://wolfram.com/xid/05fp67k07m-vo8cq0

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).
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.
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
Wolfram Language. (2008). DifferentialRoot. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DifferentialRoot.html
BibTeX
@misc{reference.wolfram_2025_differentialroot, author="Wolfram Research", title="{DifferentialRoot}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/DifferentialRoot.html}", note=[Accessed: 09-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_differentialroot, organization={Wolfram Research}, title={DifferentialRoot}, year={2020}, url={https://reference.wolfram.com/language/ref/DifferentialRoot.html}, note=[Accessed: 09-July-2025
]}