({{, (RefLink[DirectedInfinity, paclet:ref/DirectedInfinity])[]
represents an infinite numerical quantity whose direction in the complex plane is unknown. }, {, (RefLink[DirectedInfinity, paclet:ref/DirectedInfinity])[z]
represents an infinite numerical quantity that is a positive real multiple of the complex number z. }})

OpenersNotesSection(Spacer1)Details

You can think of (RefLink[DirectedInfinity, paclet:ref/DirectedInfinity])[z] as representing a point in the complex plane reached by starting at the origin and going an infinite distance in the direction of the point z.

The following conversions are made:

Certain arithmetic operations are performed on (RefLink[DirectedInfinity, paclet:ref/DirectedInfinity]) quantities.

In (RefLink[OutputForm, paclet:ref/OutputForm]), (RefLink[DirectedInfinity, paclet:ref/DirectedInfinity])[z] is printed in terms of (RefLink[Infinity, paclet:ref/Infinity]), and (RefLink[DirectedInfinity, paclet:ref/DirectedInfinity])[] is printed as (RefLink[ComplexInfinity, paclet:ref/ComplexInfinity]).

OpenersPrimaryExamplesSection(Spacer1)Examples  (17)

OpenersExampleSection0(Spacer1)Basic Examples  (3)

Use as an expansion point and direction:

Series[ArcSin[z],{z,DirectedInfinity[I],1}]

Use as an integration limit:

Integrate[Exp[-x^2],{x,0,DirectedInfinity[1+I]}]

Use as a limiting point:

Limit[(x-1)/(2x+3),x->DirectedInfinity[]]

OpenersExampleSection1(Spacer1)Scope  (6)

Some directions have a special (RefLink[StandardForm, paclet:ref/StandardForm]):

Table[DirectedInfinity[i],{i,{1,-1,I,-I}}]

Use (Key0[esc]) inf (Key0[esc]) to enter ∞:

Limit[1/x,x->∞]

Use (RefLink[Infinity, paclet:ref/Infinity]) as an alternative input form:

Infinity//FullForm

Multiplying by a number changes the direction:

(1+I) Infinity

Unspecified or (RefLink[Indeterminate, paclet:ref/Indeterminate]) direction represents (RefLink[ComplexInfinity, paclet:ref/ComplexInfinity]):

{DirectedInfinity[],DirectedInfinity[Indeterminate]}

Finite or symbolic quantities are absorbed:

3 DirectedInfinity[z]

I DirectedInfinity[z]

DirectedInfinity[z]+x

Extended arithmetic with infinite quantities:

DirectedInfinity[x] DirectedInfinity[y]

DirectedInfinity[x]+DirectedInfinity[y]

In this case the result depends on the directions x and y:

DirectedInfinity[x]+DirectedInfinity[y]/.{x->1,y->2}

DirectedInfinity[x]+DirectedInfinity[y]/.{x->1,y->-1}

Operations that cannot be unambiguously defined produce (RefLink[Indeterminate, paclet:ref/Indeterminate]):

DirectedInfinity[x]/ DirectedInfinity[y]

In this case the result depends on the growth rates of the numerator and denominator:

Limit[(x t)/(y t^2),t->Infinity]

Limit[(x t)/(y (t+1)),t->Infinity]

Use in mathematical functions:

Sin[DirectedInfinity[]]

BesselJ[0,-Infinity]

The value in different directions may vary:

{Exp[Infinity],Exp[-Infinity]}

OpenersExampleSection2(Spacer1)Applications  (2)

Integrate along a line from the origin with direction π/4:

Integrate[Exp[-x^2]/(x^2-1),{x,0,DirectedInfinity[Exp[I Pi/4]]}]

Asymptotics of the (RefLink[LogGamma, paclet:ref/LogGamma]) function at (RefLink[DirectedInfinity, paclet:ref/DirectedInfinity])[z]:

Series[LogGamma[n],{n,DirectedInfinity[z],3}]

Plot asymptotic value compared to function value in different directions:

Plot[Evaluate[Abs[Normal[%]-LogGamma[n]]/.{n->121/3z}/.z->Exp[I ϕ]],{ϕ,-3,3}]

OpenersExampleSection3(Spacer1)Properties & Relations  (3)

(RefLink[Simplify, paclet:ref/Simplify]) and (RefLink[FullSimplify, paclet:ref/FullSimplify]) can generate infinities:

Simplify[1/((E+1)^2-(E^2+2E+1))]

FullSimplify[Log[Sin[Pi/16]-Sqrt[2-Sqrt[2+Sqrt[2]]]/2]]

A nested (RefLink[DirectedInfinity, paclet:ref/DirectedInfinity]) reduces to one (RefLink[DirectedInfinity, paclet:ref/DirectedInfinity]):

DirectedInfinity[DirectedInfinity[DirectedInfinity[I]]]

(RefLink[DirectedInfinity, paclet:ref/DirectedInfinity])[] is not a number:

NumberQ[DirectedInfinity[]]

OpenersExampleSection4(Spacer1)Possible Issues  (3)

Symbolic quantities might get lost in operations:

DirectedInfinity[z]+x

The (RefLink[Accuracy, paclet:ref/Accuracy]) and (RefLink[Precision, paclet:ref/Precision]) for (RefLink[DirectedInfinity, paclet:ref/DirectedInfinity]) refer to the direction argument:

Precision[DirectedInfinity[Exp[I Pi/4]]]

Precision[DirectedInfinity[N[Exp[I Pi/4],20]]]

Simplifications performed by the Wolfram Language assume symbols to represent numbers:

x/x

∞/∞

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[Indeterminate, paclet:ref/Indeterminate])  ▪  (RefLink[ComplexInfinity, paclet:ref/ComplexInfinity])}})}})

({{DocumentationFooterIconsRelatedTutorial.png, ({{Tutorials}, {▪(RefLinkPlain[Indeterminate and Infinite Results, paclet:tutorial/IndeterminateAndInfiniteResults])}})}})

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

({{(Spacer1), ({{(Spacer1)Related Links}, {(WebLink[DocumentationFooterIconsRelatedLinks-MathWorld.png(Spacer1)MathWorld, http://mathworld.wolfram.com/DirectedInfinity.html])}, {DocumentationFooterIconsRelatedLinks-Functions.png(Spacer1)The Wolfram Functions Site}})}})