({{, (RefLink[Assumptions, paclet:ref/Assumptions])
is an option for functions such as (RefLink[Simplify, paclet:ref/Simplify]), (RefLink[Refine, paclet:ref/Refine]), and (RefLink[Integrate, paclet:ref/Integrate]) that specifies default assumptions to be made about symbolic quantities. }})

OpenersNotesSection(Spacer1)Details

The typical default setting is Assumptions:>(RefLink[$Assumptions, paclet:ref/$Assumptions]).

The assumptions can be equations, inequalities, or domain specifications, or lists or logical combinations of these.

(RefLink[Assuming, paclet:ref/Assuming]) modifies (RefLink[$Assumptions, paclet:ref/$Assumptions]) and so modifies the value of default settings for (RefLink[Assumptions, paclet:ref/Assumptions]) options.

x∈(RefLink[Reals, paclet:ref/Reals]) can be used to specify that x should be treated as a real variable.

Quantities that appear algebraically in inequalities are always assumed to be real.

OpenersPrimaryExamplesSection(Spacer1)Examples  (4)

OpenersExampleSection0(Spacer1)Basic Examples  (1)

(RefLink[Refine, paclet:ref/Refine]):

Refine[Sqrt[x^2],Assumptions->x<0]

(RefLink[Simplify, paclet:ref/Simplify]):

Simplify[Cos[k Pi]^m,Assumptions->Element[k,Integers]&&Mod[m,2]==0]

(RefLink[FunctionExpand, paclet:ref/FunctionExpand]):

FunctionExpand[Factorial2[n],Assumptions->n∈Integers]

(RefLink[Integrate, paclet:ref/Integrate]):

Integrate[x^a,{x,0,1},Assumptions->a>0]

(RefLink[FourierTransform, paclet:ref/FourierTransform]):

FourierTransform[1/(x^2+a),x,w,Assumptions->a>0]

(RefLink[Limit, paclet:ref/Limit]):

Limit[x^a,x->Infinity,Assumptions->a<0]

(RefLink[Series, paclet:ref/Series]):

Series[Min[x^2,1+a]Cos[x]+Sin[x],{x,0,2},Assumptions->a>0]

OpenersExampleSection1(Spacer1)Properties & Relations  (3)

(RefLink[$Assumptions, paclet:ref/$Assumptions]) is the default value of (RefLink[Assumptions, paclet:ref/Assumptions]) for most functions that have the option:

Quiet[Select[ToExpression/@Names["System`*"],Head[#]===Symbol&&MemberQ[Options[#],Assumptions:>$Assumptions]&]]

(RefLink[Assuming, paclet:ref/Assuming]) locally sets the value of (RefLink[$Assumptions, paclet:ref/$Assumptions]):

Assuming[x>0,{$Assumptions,Refine[Sqrt[x^2]]}]

(RefLink[Block, paclet:ref/Block]) can also be used to locally set the value of (RefLink[$Assumptions, paclet:ref/$Assumptions]):

Block[{$Assumptions=x>0},{$Assumptions,Refine[Sqrt[x^2]]}]

Here the assumptions from (RefLink[Assuming, paclet:ref/Assuming]) are not used inside (RefLink[Block, paclet:ref/Block]):

Assuming[x≥0,Block[{$Assumptions=y<0},Refine[Sqrt[x^2y^2]]]]

Assumptions from nested (RefLink[Assuming, paclet:ref/Assuming]) are combined:

Assuming[x≥0,Assuming[y<0,Refine[Sqrt[x^2y^2]]]]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[Assuming, paclet:ref/Assuming])  ▪  (RefLink[$Assumptions, paclet:ref/$Assumptions])  ▪  (RefLink[GenerateConditions, paclet:ref/GenerateConditions])  ▪  (RefLink[VerifyConvergence, paclet:ref/VerifyConvergence])  ▪  (RefLink[Integrate, paclet:ref/Integrate])  ▪  (RefLink[DSolve, paclet:ref/DSolve])  ▪  (RefLink[Refine, paclet:ref/Refine])  ▪  (RefLink[Limit, paclet:ref/Limit])}})}})

({{DocumentationFooterIconsRelatedTutorial.png, ({{Tutorials}, {▪(RefLinkPlain[Definite Integrals, paclet:tutorial/DefiniteIntegrals])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Algebraic Transformations, paclet:guide/AlgebraicTransformations])}, {▪(RefLinkPlain[Series Expansions, paclet:guide/SeriesExpansions])}, {▪(RefLinkPlain[Assumptions and Domains, paclet:guide/AssumptionsAndDomains])}})}})