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MathematicalFunctionData
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MathematicalFunctionData

For general access to the Wolfram Knowledgebase, use entities of type MathematicalFunction instead of MathematicalFunctionData.

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MathematicalFunctionData[entity,property]

gives data corresponding to property for the mathematical function specified by entity.

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MathematicalFunctionData[entprop,annotation]

gives data corresponding to the given entity or property in the format specified by annotation.

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MathematicalFunctionData[entity,property,annotation]

gives data for the given entity-property pair in the format specified by annotation.    

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MathematicalFunctionData[entity,property,{qual1val1,qual2val2,…}]

gives data for the given entity-property pair with property qualifiers qual1, qual2, … set to the given values.

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MathematicalFunctionData[entity,property,annotation,{qual1val1,qual2val2,…}]

gives data corresponding to the given entity, property and annotation format, with property qualifiers qual1, qual2, … set to the given values.

Details

  • MathematicalFunctionData can be used to access identities involving mathematical functions.
  • MathematicalFunctionData[] or MathematicalFunctionData["Entities"] gives a list of available mathematical function entities.
  • MathematicalFunctionData["Properties"] gives a list of available properties.
  • The specified entity in MathematicalFunctionData can be an Entity, EntityClass, entity canonical name or list of entities.
  • The specified property can be an EntityProperty, EntityPropertyClass, property canonical name or list of properties.
  • MathematicalFunctionData entity-property values are generally lists of pure functions that can be applied to user-supplied expressions.
  • Properties that do not apply or are not known in a particular case are indicated by Missing[…].
  • Properties include:
  • "AdditionFormulas"addition formulas
    "AlternativeRepresentations"alternative representations
    "ArgumentPattern"argument pattern
    "ArgumentSimplifications"argument simplifications
    "AsymptoticExpansions"asymptotic expansions
    "Classes"classes
    "ComplexCharacteristics"complex characteristics
    "ContinuedFractionRepresentations"continued fraction representations
    "DifferenceEquations"difference equations
    "DifferentialEquations"differential equations
    "FourierTransforms"Fourier transforms
    "FractionalDerivatives"fractional derivatives
    "FunctionalEquations"functional equations
    "GeneratingFunctions"generating functions
    "HalfArgumentFormulas"half‐argument formulas
    "HankelTransforms"Hankel transforms
    "HypergeometricRepresentations"hypergeometric representations
    "IntegralRepresentations"integral representations
    "InverseFourierTransforms"inverse Fourier transforms
    "InverseFunctionRelations"inverse function relations
    "LaplaceTransforms"Laplace transforms
    "LimitRepresentations"limit representations
    "LowOrderDerivatives"low‐order derivatives
    "MeijerGRepresentations"Meijer G representations
    "MellinTransforms"Mellin transforms
    "MultipliedArgumentFormulas"multiplied‐argument formulas
    "Name"function name
    "NamedIdentities"named identities
    "ParticularValues"particular values
    "ProductOfFunctionsFormulas"product‐of‐functions formulas
    "ProductRepresentations"product representations
    "ReflectionSymmetries"reflection symmetries
    "RelatedFunctionRepresentations"related function representations
    "RelatedFunctions"related functions
    "RelatedIdentities"related identities
    "RelatedInequalities"related inequalities
    "ResidueRepresentations"residue representations
    "Residues"residues
    "SampleDefiniteIntegrals"sample definite integrals
    "SampleFiniteProducts"sample finite products
    "SampleFiniteSums"sample finite sums
    "SampleIndefiniteIntegrals"sample indefinite integrals
    "SampleInfiniteProducts"sample infinite products
    "SampleInfiniteSums"sample infinite sums
    "SampleIntegrals"sample integrals
    "SeriesRepresentations"series representations
    "SummedTaylorSeriesLimits"summed Taylor series limits
    "SumOfFunctionsFormulas"sum‐of‐functions formulas
    "SymbolicDerivatives"symbolic derivatives
    "TraditionalFormBoxes"traditional form boxes
    "WolframFunctionsSiteLink"Wolfram Functions Site link
    "Wronskians"Wronskians
    "Zeros"zeros
  • Some data is available for MathematicalFunctionData as a whole and can be given using the form MathematicalFunctionData[property]. Such domain‐level properties include:
  • "Entities"all available entities
    "EntityCount"total number of available entities
    "EntityCanonicalNames"list of all entity canonical names
    "SampleEntities"list of sample entities
    "EntityClasses"all available entity classes
    "EntityClassCount"total number of available entity classes
    "EntityClassCanonicalNames"list of all entity class canonical names
    "SampleEntityClasses"list of sample entity classes
    "Properties"all available properties
    "PropertyCount"total number of available properties
    "PropertyCanonicalNames"list of all property canonical names
    "PropertyClasses"all available property classes
    "PropertyClassCount"total number of available property classes
    "PropertyClassCanonicalNames"list of all property class canonical names
    "RandomEntity"pseudorandom sample entity
    "RandomEntities"list of 10 pseudorandom sample entities
    {"RandomEntities",n}n pseudorandom entities
    "RandomEntityClass"pseudorandom sample entity class
    "RandomEntityClasses"pseudorandom sample entity classes
    {"RandomEntityClasses",n}n pseudorandom entity classes
  • The following annotation strings can be used in the second or third argument of MathematicalFunctionData, where applicable, to specify in which format the data should be returned:
  • "EntityAssociation"an association of entities and entity-property values
    "PropertyAssociation"an association of properties and entity-property values
    "EntityPropertyAssociation"an association in which the specified entities are keys, and values are a nested association of properties and entity-property values
    "PropertyEntityAssociation"an association in which the specified properties are keys, and values are a nested association of entities and entity-property values
    "Dataset"a dataset in which the specified entities are keys, and values are an association of property names and entity-property values
    "NonMissingEntities"a list of entities for which the given property does not return Missing[…]
    "NonMissingProperties"a list of properties for which the given entity does not return Missing[…]
    "NonMissingEntityAssociation"an association of entities and entity-property values with entities returning Missing[…] eliminated
    "NonMissingPropertyAssociation"an association of entities and entity-property values with properties returning Missing[…] eliminated
  • MathematicalFunctionData[EntityProperty[…],subproperty] can be used to look up property metadata. Available metadata strings that can be used in the second argument are:
  • "Description"a textual definition of the property
    "Label"the label of the property
    "Qualifiers"the list of possible qualifiers for the property
  • MathematicalFunctionData relies on the internet to retrieve data from the Wolfram servers.

Examples

open allclose all

Basic Examples  (8)Summary of the most common use cases

Display known addition formulas for Sin:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-4bb804
Out[1]=1

Return integral representations for Sin:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-chac03
Out[1]=1

Show argument simplifications for the incomplete elliptic integral of the second kind:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-gnbmw1
Out[1]=1

Give the residues of the binomial coefficient for symbolic arguments:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-t9qys
Out[1]=1

Return a list of sample function entities:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-fvv9w9
Out[1]=1

Return an entity association over an entity class:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-msdr7e
Out[1]=1

Return identities corresponding to an external cross-reference, if available:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-dduhig
Out[1]=1

Use to discover properties of a function:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-45cwj
Out[1]=1

Scope  (7)Survey of the scope of standard use cases

Properties  (2)

Most MathematicalFunctionData properties return a list of identities, identity rules, etc.:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-7bhzaw
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bnhpwteg9em6-st6ruo
Out[2]=2

A few properties return a single value:

In[2]:=2
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https://wolfram.com/xid/0bnhpwteg9em6-k7532m
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0bnhpwteg9em6-bigee7
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0bnhpwteg9em6-b9eb59
Out[4]=4

Not all properties have known values:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-0rhrsx
Out[1]=1

Property Qualifiers  (5)

By default, all known identities are returned, including some that may hold only over a subset of variable and parameter values:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-b9naqc
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bnhpwteg9em6-24n2ve
Out[2]=2

The "ValidGenerically" qualifier returns only identities that hold generically (i.e. for all complex values, possibly excepting sets of measure zero):

In[3]:=3
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https://wolfram.com/xid/0bnhpwteg9em6-gmrhvm
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0bnhpwteg9em6-2k4b3c
Out[4]=4

The "IncludedSubexpressions" qualifier returns only those identities that contain one or more user-supplied subexpressions:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-saehsc
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bnhpwteg9em6-7vecvf
Out[2]=2

With no qualifier, all available identities are returned:

In[3]:=3
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https://wolfram.com/xid/0bnhpwteg9em6-ifxkec
Out[3]=3

The "ExcludedSubexpressions" qualifier returns only those identities that do not contain one or more user-supplied excluded subexpressions:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-d3hhj3
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bnhpwteg9em6-l89tam
Out[2]=2

With no qualifier, all available identities are returned:

In[3]:=3
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https://wolfram.com/xid/0bnhpwteg9em6-epif6
Out[3]=3

The "TraditionalFormPresentation" qualifier returns results using traditional mathematical typesetting:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-ctft58

The "CrossReferences" qualifier returns known cross-references from a given source or sources corresponding to the specified identities:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-buqkse
Out[1]=1

Generalizations & Extensions  (1)Generalized and extended use cases

Where possible, mathematical expressions appearing in the first argument of MathematicalFunctionData are automatically mapped to their corresponding entities:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-ouoif3
Out[1]=1

Applications  (2)Sample problems that can be solved with this function

Verify that an identity yields True by substituting variables into the pure function, applying Activate, and simplifying if necessary:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-dbyt04
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bnhpwteg9em6-c52jsk
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0bnhpwteg9em6-2j8ot
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0bnhpwteg9em6-bapvw5
Out[4]=4

Some arguments to the pure functions that are returned as entity-property values may be supplied with values that are themselves pure functions:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-i4m3zt
Out[1]=1

Since a takes a single argument (a[k]) in the body of this result, substitute in a pure function for its value:

In[2]:=2
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https://wolfram.com/xid/0bnhpwteg9em6-nq5w1x
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0bnhpwteg9em6-bmo38w
Out[3]=3

Properties & Relations  (3)Properties of the function, and connections to other functions

Use ToEntity for entity discovery:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-vdsp1y
Out[1]=1

The same Wolfram Language symbol may correspond to different function entities:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-pmmodg
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bnhpwteg9em6-dompg5
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0bnhpwteg9em6-lcpulf
Out[3]=3

Use FromEntity to discover how to express a given mathematical function entity in the Wolfram Language:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-uevkp
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bnhpwteg9em6-bq6a3w
Out[2]=2

Possible Issues  (2)Common pitfalls and unexpected behavior

An identity may yield Undefined if it contains a ConditionalExpression whose condition is not satisfied by the substituted values:

In[1]:=1
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https://wolfram.com/xid/0bnhpwteg9em6-bhifl9
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bnhpwteg9em6-cj61je
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0bnhpwteg9em6-idytc4
Out[3]=3

The number of arguments taken by the pure functions within a entity-property list can vary, so care must be taken to ensure that they are applied to the correct number of arguments:

In[1]:=1
✖
https://wolfram.com/xid/0bnhpwteg9em6-brgtpv
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bnhpwteg9em6-cuad9f
Out[2]=2
Wolfram Research (2015), MathematicalFunctionData, Wolfram Language function, https://reference.wolfram.com/language/ref/MathematicalFunctionData.html (updated 2019).
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Wolfram Research (2015), MathematicalFunctionData, Wolfram Language function, https://reference.wolfram.com/language/ref/MathematicalFunctionData.html (updated 2019).

Text

Wolfram Research (2015), MathematicalFunctionData, Wolfram Language function, https://reference.wolfram.com/language/ref/MathematicalFunctionData.html (updated 2019).

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Wolfram Research (2015), MathematicalFunctionData, Wolfram Language function, https://reference.wolfram.com/language/ref/MathematicalFunctionData.html (updated 2019).

CMS

Wolfram Language. 2015. "MathematicalFunctionData." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2019. https://reference.wolfram.com/language/ref/MathematicalFunctionData.html.

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Wolfram Language. 2015. "MathematicalFunctionData." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2019. https://reference.wolfram.com/language/ref/MathematicalFunctionData.html.

APA

Wolfram Language. (2015). MathematicalFunctionData. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MathematicalFunctionData.html

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Wolfram Language. (2015). MathematicalFunctionData. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MathematicalFunctionData.html

BibTeX

@misc{reference.wolfram_2025_mathematicalfunctiondata, author="Wolfram Research", title="{MathematicalFunctionData}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/MathematicalFunctionData.html}", note=[Accessed: 19-April-2025 ]}

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@misc{reference.wolfram_2025_mathematicalfunctiondata, author="Wolfram Research", title="{MathematicalFunctionData}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/MathematicalFunctionData.html}", note=[Accessed: 19-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_mathematicalfunctiondata, organization={Wolfram Research}, title={MathematicalFunctionData}, year={2019}, url={https://reference.wolfram.com/language/ref/MathematicalFunctionData.html}, note=[Accessed: 19-April-2025 ]}

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@online{reference.wolfram_2025_mathematicalfunctiondata, organization={Wolfram Research}, title={MathematicalFunctionData}, year={2019}, url={https://reference.wolfram.com/language/ref/MathematicalFunctionData.html}, note=[Accessed: 19-April-2025 ]}