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DifferentialRoot
✖
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

open allclose all

Basic Examples  (2)Summary of the most common use cases

Define f to be the sin function:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-eytp8e
Out[1]=1

Plot its result:

In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-bl4eia
Out[2]=2

Evaluate numerically to any precision:

In[3]:=3
✖
https://wolfram.com/xid/05fp67k07m-f3pz8c
Out[3]=3

Compare the result with the built-in Sin function:

In[4]:=4
✖
https://wolfram.com/xid/05fp67k07m-7sdasy
Out[4]=4

Solve a differential equation:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-er5nt0
Out[1]=1

Numerical values:

In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-v6wp87
Out[2]=2

Scope  (23)Survey of the scope of standard use cases

Numerical Evaluation  (7)

Evaluate at machine precision:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-jua23i
Out[1]=1

Evaluate to high precision:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-uqurky
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-lw9h0n
Out[1]=1

DifferentialRoot takes complex number parameters and arguments:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-64g5bd
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-diu7a2
Out[2]=2

DifferentialRoot takes inexact input parameters:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-wvphoo
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-iwnnn6
Out[2]=2

Evaluate DifferentialRoot efficiently at high precision:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-2c7v5i
Out[1]=1

DifferentialRoot threads elementwise over lists and matrices:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-g2zcvw
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-lw7hpu
Out[2]=2

Function Properties  (5)

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

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-sjlbcm
Out[1]=1

Integrate the function:

In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-7sdg2t
Out[2]=2

Differentiate it:

In[3]:=3
✖
https://wolfram.com/xid/05fp67k07m-16vzpe
Out[3]=3

Find its series expansion:

In[4]:=4
✖
https://wolfram.com/xid/05fp67k07m-lbjm64
Out[4]=4

Plot it on the reals:

In[5]:=5
✖
https://wolfram.com/xid/05fp67k07m-if27yd
Out[5]=5

Plot it in the complex plane:

In[6]:=6
✖
https://wolfram.com/xid/05fp67k07m-zxwxeo
Out[6]=6

Simple exact values are generated automatically:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-4gb6nx
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-fyvp49
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-26w3cp
Out[2]=2

DifferentialRoot works on equations with rational coefficients:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-uryei4
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-79urvk
Out[1]=1

Differentiation  (4)

The derivative of DifferentialRoot is a DifferentialRoot function:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-41n6dj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-eqbaum
Out[2]=2

Differentiate a DifferentialRoot object with respect to a parameter:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-s7jlhu
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-i14vak
Out[2]=2

Compute higher-order derivatives of a DifferentialRoot object:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-yloxzq
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-k9vdyl
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/05fp67k07m-4rowba
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/05fp67k07m-ugy1ya
Out[4]=4

Differentiate a DifferentialRoot object:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-c46jps
Out[1]=1

Specific values of :

In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-8jykqv
Out[2]=2

Plot of :

In[3]:=3
✖
https://wolfram.com/xid/05fp67k07m-llhv1n
Out[3]=3

Integration  (4)

The integral of a DifferentialRoot object is a DifferentialRoot object:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-dvob6a
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-hwytkk
Out[2]=2

Compute higher-order integrals of a DifferentialRoot object:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-7wzkzx
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-u7r3br
Out[2]=2

Compute the definite integral of a DifferentialRoot object:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-0a608r
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-ri4baw
Out[2]=2

Integrate a DifferentialRoot object:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-ivwiks
Out[1]=1

Specific values of :

In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-tq2c8y
Out[2]=2

Plot of :

In[3]:=3
✖
https://wolfram.com/xid/05fp67k07m-5vh16u
Out[3]=3

Series Expansions  (3)

Calculate the series expansion of a DifferentialRoot object:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-j4uy95
Out[1]=1

Find the ^(th) coefficient of the Taylor expansion of a DifferentialRoot object:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-xckda7
Out[1]=1

Calculate the first 9 coefficients:

In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-5knd2g
Out[2]=2

Compare with the Sin function expansion coefficients:

In[3]:=3
✖
https://wolfram.com/xid/05fp67k07m-7k9qop
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-ez4xmx
Out[1]=1

Generalizations & Extensions  (1)Generalized and extended use cases

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

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-5cwiel
Out[1]=1

Applications  (4)Sample problems that can be solved with this function

Generate a DifferentialRoot object from a special function:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-o9dteh
Out[1]=1

Integrate it:

In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-yks48n
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-ty1dx8
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-4m361q
Out[2]=2

Calculate the first 5 coefficients of the expansion explicitly:

In[3]:=3
✖
https://wolfram.com/xid/05fp67k07m-v21u0
Out[3]=3

Compute arbitrary-order derivatives of a DifferentialRoot object:

In[4]:=4
✖
https://wolfram.com/xid/05fp67k07m-j8auuo
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/05fp67k07m-0l5r6h
Out[5]=5

Integrate the DifferentialRoot object:

In[6]:=6
✖
https://wolfram.com/xid/05fp67k07m-3xmnfj
Out[6]=6

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

In[7]:=7
✖
https://wolfram.com/xid/05fp67k07m-7e8jbu
Out[7]=7

Plot the function f, its integral and derivative functions:

In[8]:=8
✖
https://wolfram.com/xid/05fp67k07m-cyk1kn
Out[8]=8

Use DifferentialRoot to homogenize a differential equation:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-b0cvci
Out[1]=1

Extract the homogenized equation:

In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-8a2ftx
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-nzbigm
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-b8fo8f
Out[2]=2

Properties & Relations  (5)Properties of the function, and connections to other functions

DifferentialRootReduce generates DifferentialRoot objects:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-7y5qji
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-753j8v
Out[1]=1

GeneratingFunction may generate a DifferentialRoot object:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-rzaqfr
Out[1]=1

Integrate returns a DifferentialRoot object for general holonomic functions:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-f99vbv
Out[1]=1

D returns a DifferentialRoot object for general holonomic functions:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-c4kj81
Out[1]=1

Possible Issues  (3)Common pitfalls and unexpected behavior

DifferentialRoot takes only linear differential equations with polynomial coefficients:

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-7uy6i0
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-jijszb
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-mb0pmv
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-fpoh9e
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/05fp67k07m-l8fxmo
Out[3]=3

For other regions, DifferentialRoot will give the same result:

In[4]:=4
✖
https://wolfram.com/xid/05fp67k07m-d9tbcv
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/05fp67k07m-s9zxhc
Out[5]=5

Neat Examples  (1)Surprising or curious use cases

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

In[1]:=1
✖
https://wolfram.com/xid/05fp67k07m-o6yx6
Out[1]=1

Calculate the numerical values of this solution:

In[2]:=2
✖
https://wolfram.com/xid/05fp67k07m-kqiziv
Out[2]=2

Plot this solution:

In[3]:=3
✖
https://wolfram.com/xid/05fp67k07m-fe277f
Out[3]=3

Differentiate this solution:

In[4]:=4
✖
https://wolfram.com/xid/05fp67k07m-vo8cq0
Out[4]=4
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).

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 ]}

✖
@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 ]}

✖
@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 ]}