({{, (RefLink[MathematicalFunctionData, paclet:ref/MathematicalFunctionData])[entity,property]
gives data corresponding to property for the mathematical function specified by entity.}, {, (RefLink[MathematicalFunctionData, paclet:ref/MathematicalFunctionData])[entprop,annotation]
gives data corresponding to the given entity or property in the format specified by annotation.}, {, (RefLink[MathematicalFunctionData, paclet:ref/MathematicalFunctionData])[entity,property,annotation] 
gives data for the given entity-property pair in the format specified by annotation.	}, {, (RefLink[MathematicalFunctionData, paclet:ref/MathematicalFunctionData])[entity,property,{(Subscript[qual, 1])->(Subscript[val, 1]),(Subscript[qual, 2])->(Subscript[val, 2]),…}]
gives data for the given entity-property pair with property qualifiers (Subscript[qual, 1]), (Subscript[qual, 2]), … set to the given values.}, {, (RefLink[MathematicalFunctionData, paclet:ref/MathematicalFunctionData])[entity,property,annotation,{(Subscript[qual, 1])->(Subscript[val, 1]),(Subscript[qual, 2])->(Subscript[val, 2]),…}]
gives data corresponding to the given entity, property, and annotation format, with property qualifiers (Subscript[qual, 1]), (Subscript[qual, 2]), … set to the given values.}})

OpenersNotesSection(Spacer1)Details

(RefLink[MathematicalFunctionData, paclet:ref/MathematicalFunctionData]) can be used to access identities involving mathematical functions.

(RefLink[MathematicalFunctionData, paclet:ref/MathematicalFunctionData])[] or (RefLink[MathematicalFunctionData, paclet:ref/MathematicalFunctionData])["Entities"] gives a list of available mathematical function entities.

(RefLink[MathematicalFunctionData, paclet:ref/MathematicalFunctionData])["Properties"] gives a list of available properties.

The specified entity in (RefLink[MathematicalFunctionData, paclet:ref/MathematicalFunctionData]) can be an (RefLink[Entity, paclet:ref/Entity]), (RefLink[EntityClass, paclet:ref/EntityClass]), entity canonical name or list of entities.

The specified property can be an (RefLink[EntityProperty, paclet:ref/EntityProperty]), (RefLink[EntityPropertyClass, paclet:ref/EntityPropertyClass]), property canonical name or list of properties.

(RefLink[MathematicalFunctionData, paclet:ref/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 (RefLink[Missing, paclet:ref/Missing])[…].

Properties include:

Some data is available for (RefLink[MathematicalFunctionData, paclet:ref/MathematicalFunctionData]) as a whole and can be given using the form (RefLink[MathematicalFunctionData, paclet:ref/MathematicalFunctionData])[property]. Such domain-level properties include:

The following annotation strings can be used in the second or third argument of (RefLink[MathematicalFunctionData, paclet:ref/MathematicalFunctionData]), where applicable, to specify in which format the data should be returned:

(RefLink[MathematicalFunctionData, paclet:ref/MathematicalFunctionData])[(RefLink[EntityProperty, paclet:ref/EntityProperty])[…],subproperty] can be used to lookup up property metadata. Available metadata strings that can be used in the second argument are:

(RefLink[MathematicalFunctionData, paclet:ref/MathematicalFunctionData]) relies on the internet to retrieve data from the Wolfram servers.

OpenersPrimaryExamplesSection(Spacer1)Examples  (23)

OpenersExampleSection0(Spacer1)Basic Examples  (8)

Display known addition formulas for (RefLink[Sin, paclet:ref/Sin]):

MathematicalFunctionData["Sin","AdditionFormulas"]

Return integral representations for (RefLink[Sin, paclet:ref/Sin]):

MathematicalFunctionData["Sin","IntegralRepresentations"]

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

MathematicalFunctionData[(Entity[(RowDefault["incomplete elliptic integral", " ", (Row["E", "(", "z", " \[VerticalSeparator] ", "m", ")"])]), Entity["MathematicalFunction","EllipticE"], "Entity[\"MathematicalFunction\", \"EllipticE\"]", "mathematical function"]),"ArgumentSimplifications"]

Give the residues of the binomial coefficient for symbolic arguments:

MathematicalFunctionData[(Entity[(RowDefault["binomial coefficient", " ", (Row[(, ({{"n"}, {"k"}}), )])]), Entity["MathematicalFunction","Binomial"], "Entity[\"MathematicalFunction\", \"Binomial\"]", "mathematical function"]),(EntityProperty["residues", EntityProperty["MathematicalFunction","Residues"], "EntityProperty[\"MathematicalFunction\", \"Residues\"]"])]

Return a list of sample function entities:

MathematicalFunctionData["SampleEntities"]

Return an entity association over an entity class:

MathematicalFunctionData["FresnelIntegrals","ArgumentSimplifications","EntityAssociation"]

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

MathematicalFunctionData[(Entity["Digital Library of Mathematical Functions", Entity["Source","DigitalLibraryOfMathematicalFunctions"], "Entity[\"Source\", \"DigitalLibraryOfMathematicalFunctions\"]", "Source"])->"27.5.5"]

Use (Key1[ctrl, ==]) to discover properties of a function:

LinguisticAssistant

OpenersExampleSection1(Spacer1)Scope  (7)

OpenersExampleSubsection0(Spacer1)Properties  (2)

Most (RefLink[MathematicalFunctionData, paclet:ref/MathematicalFunctionData]) properties return a list of identities, identity rules, etc.:

MathematicalFunctionData["Exp","NamedIdentities"]//Take[#,-3]&

Grid[List@@@%,Dividers->All,Alignment->Left]

A few properties return a single value:

MathematicalFunctionData["WhittakerW","Name"]

MathematicalFunctionData["WhittakerW","ArgumentPattern"]

MathematicalFunctionData["WhittakerW","WolframFunctionsSiteLink"]

Not all properties have known values:

MathematicalFunctionData["ProductLog","NamedIdentities"]

OpenersExampleSubsection1(Spacer1)Property Qualifiers  (5)

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

MathematicalFunctionData["InverseJacobiSC","SampleIndefiniteIntegrals"]

Length[%]

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

MathematicalFunctionData["InverseJacobiSC","SampleIndefiniteIntegrals","ValidGenerically"->True]

Length[%]

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

MathematicalFunctionData["LucasL","SeriesRepresentations","IncludedSubexpressions"->{HoldPattern[Binomial[_,_]]}]

Length[%]

With no qualifier, all available identities are returned:

MathematicalFunctionData["LucasL","SeriesRepresentations"]//Length

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

MathematicalFunctionData[(Entity[(RowDefault["hyperbolic cosecant function", " ", (Row["csch", "(", "z", ")"])]), Entity["MathematicalFunction","Csch"], "Entity[\"MathematicalFunction\", \"Csch\"]", "mathematical function"]),(EntityProperty["product\[Hyphen]of\[Hyphen]functions formulas", EntityProperty["MathematicalFunction","ProductOfFunctionsFormulas"], "EntityProperty[\"MathematicalFunction\", \"ProductOfFunctionsFormulas\"]"]),"ExcludedSubexpressions"->{Cosh[_]}]

Length[%]

With no qualifier, all available identities are returned:

MathematicalFunctionData[(Entity[(RowDefault["hyperbolic cosecant function", " ", (Row["csch", "(", "z", ")"])]), Entity["MathematicalFunction","Csch"], "Entity[\"MathematicalFunction\", \"Csch\"]", "mathematical function"]),(EntityProperty["product\[Hyphen]of\[Hyphen]functions formulas", EntityProperty["MathematicalFunction","ProductOfFunctionsFormulas"], "EntityProperty[\"MathematicalFunction\", \"ProductOfFunctionsFormulas\"]"])]//Length

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

MathematicalFunctionData["Sin","AdditionFormulas","TraditionalFormPresentation"->True]

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

MapThread[Rule,{
MathematicalFunctionData["MangoldtLambda","AlternativeRepresentations"],
MathematicalFunctionData["MangoldtLambda","AlternativeRepresentations","CrossReferences"->All]
}][[2]]

OpenersExampleSection2(Spacer1)Generalizations & Extensions  (1)

Where possible, mathematical expressions appearing in the first argument of (RefLink[MathematicalFunctionData, paclet:ref/MathematicalFunctionData]) are automatically mapped to their corresponding entities:

MathematicalFunctionData[Sin[z],"Zeros"]

OpenersExampleSection3(Spacer1)Applications  (2)

Verify that an identity yields (RefLink[True, paclet:ref/True]) by substituting variables into the pure function, applying (RefLink[Activate, paclet:ref/Activate]), and simplifying if necessary:

myIdent=MathematicalFunctionData[(Entity[(RowDefault["Chebyshev polynomial", " ", (Row[(Subscript["T", "n"]), "(", "z", ")"])]), Entity["MathematicalFunction","ChebyshevT:Polynomial"], "Entity[\"MathematicalFunction\", \"ChebyshevT:Polynomial\"]", "mathematical function"]),"FunctionalEquations"]//First

myIdent[n,z]

myIdent[n,z]//Activate

%//FullSimplify

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

myIdent=Select[MathematicalFunctionData["Sin","AdditionFormulas"],MemberQ[#,_Sum,∞]&]//First

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

myIdent[#^2&,3]//Activate

%//Simplify

OpenersExampleSection4(Spacer1)Properties & Relations  (3)

Use (RefLink[ToEntity, paclet:ref/ToEntity]) for entity discovery:

ToEntity[Sin[x],"MathematicalFunction"]

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

ToEntity[HermiteH,"MathematicalFunction"]

ToEntity[HermiteH[2,x],"MathematicalFunction"]

ToEntity[HermiteH[2+3I,x],"MathematicalFunction"]

Use (RefLink[FromEntity, paclet:ref/FromEntity]) to discover how to express a given mathematical function entity in the Wolfram Language:

FromEntity[(Entity[(RowDefault["Jacobi theta function", " ", (Row[(Subscript["ϑ", "1"]), "(", "z", ", ", "q", ")"])]), Entity["MathematicalFunction","EllipticTheta1"], "Entity[\"MathematicalFunction\", \"EllipticTheta1\"]", "mathematical function"])]

FromEntity[(Entity[(RowDefault["hypergeometric function", " ", (Row[(Subscript["", "3"]), (Subscript["F", "2"]), "", "(", (Subscript["a", "1"]), ", ", (Subscript["a", "2"]), ", ", (Subscript["a", "3"]), "; ", (Subscript["b", "1"]), ", ", (Subscript["b", "2"]), "; ", "z", ")"])]), Entity["MathematicalFunction","HypergeometricPFQ:3F2"], "Entity[\"MathematicalFunction\", \"HypergeometricPFQ:3F2\"]", "mathematical function"])]

OpenersExampleSection5(Spacer1)Possible Issues  (2)

An identity may yield (RefLink[Undefined, paclet:ref/Undefined]) if it contains a (RefLink[ConditionalExpression, paclet:ref/ConditionalExpression]) whose condition is not satisfied by the substituted values:

myIdent=MathematicalFunctionData[(Entity[(RowDefault["Chebyshev polynomial", " ", (Row[(Subscript["T", "n"]), "(", "z", ")"])]), Entity["MathematicalFunction","ChebyshevT:Polynomial"], "Entity[\"MathematicalFunction\", \"ChebyshevT:Polynomial\"]", "mathematical function"]),"SymbolicDerivatives"]//First

myIdent[n,z,m]//Activate

%/.{m->5/2}

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:

MathematicalFunctionData[(Entity[(RowDefault["floor function", " ", (Row["⌊", "z", "⌋"])]), Entity["MathematicalFunction","Floor"], "Entity[\"MathematicalFunction\", \"Floor\"]", "mathematical function"]),"SeriesRepresentations"]

Through[%[x]]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[EntityValue, paclet:ref/EntityValue])  ▪  (RefLink[FormulaData, paclet:ref/FormulaData])  ▪  (RefLink[FormulaLookup, paclet:ref/FormulaLookup])  ▪  (RefLink[FindFormula, paclet:ref/FindFormula])  ▪  (RefLink[ToEntity, paclet:ref/ToEntity])  ▪  (RefLink[FromEntity, paclet:ref/FromEntity])  ▪  (RefLink[WolframLanguageData, paclet:ref/WolframLanguageData])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Mathematical Functions, paclet:guide/MathematicalFunctions])}})}})

({{(Spacer1), ({{(Spacer1)Related Links}, {(WebLink[DocumentationFooterIconsRelatedLinks-Functions.png(Spacer1)http://functions.wolfram.com, http://functions.wolfram.com/])}, {(WebLink[DocumentationFooterIconsRelatedLinks-MathWorld.png(Spacer1)http://mathworld.wolfram.com, http://mathworld.wolfram.com/])}})}})