({{, (RefLink[ComplexInfinity, paclet:ref/ComplexInfinity])
represents a quantity with infinite magnitude, but undetermined complex phase. }})

OpenersNotesSection(Spacer1)Details

(RefLink[ComplexInfinity, paclet:ref/ComplexInfinity]) is converted to (RefLink[DirectedInfinity, paclet:ref/DirectedInfinity])[].

In (RefLink[OutputForm, paclet:ref/OutputForm]), (RefLink[DirectedInfinity, paclet:ref/DirectedInfinity])[] is printed as (RefLink[ComplexInfinity, paclet:ref/ComplexInfinity]).

OpenersPrimaryExamplesSection(Spacer1)Examples  (19)

OpenersExampleSection0(Spacer1)Basic Examples  (1)

Division by 0:

1/0

1/%

OpenersExampleSection1(Spacer1)Scope  (4)

Use (RefLink[ComplexInfinity, paclet:ref/ComplexInfinity]) in numerical functions:

Sin[ComplexInfinity]

ArcSec[ComplexInfinity]

Sin[Abs[ComplexInfinity]]

(RefLink[ComplexInfinity, paclet:ref/ComplexInfinity]) absorbs finite real, complex, and symbolic quantities:

ComplexInfinity+2

3 ComplexInfinity

ComplexInfinity + 3 I

ComplexInfinity  x

ComplexInfinity+ x

Do arithmetic with (RefLink[ComplexInfinity, paclet:ref/ComplexInfinity]):

ComplexInfinity/ComplexInfinity

ComplexInfinity ComplexInfinity

Use (RefLink[ComplexInfinity, paclet:ref/ComplexInfinity]) as an expansion point for series:

Series[1/(x^2+1),{x,ComplexInfinity,2}]

Series[ArcSin[x],{x,ComplexInfinity,1}]

OpenersExampleSection2(Spacer1)Applications  (2)

Set up a seemingly "analytic" function that is infinite in the whole left half-plane:

w[z_]:=1/(z-Sqrt[z^2]);

{w[2], w[Pi], w[Pi+2I]}//Quiet

Plotting shows details of the numerical calculation:

Plot3D[Re[w[x+I y]],{x,-2,2},{y,-2,2}]//Quiet

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

Series[LogGamma[a],{a,ComplexInfinity,2}]

OpenersExampleSection3(Spacer1)Properties & Relations  (6)

Use (RefLink[Quiet, paclet:ref/Quiet]) to suppress messages:

1/0//Quiet

1/0

(RefLink[ComplexInfinity, paclet:ref/ComplexInfinity]) can be generated by (RefLink[Simplify, paclet:ref/Simplify]) and (RefLink[FullSimplify, paclet:ref/FullSimplify]):

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

FullSimplify[1/(Sin[Pi/16]-Sqrt[2-Sqrt[2+Sqrt[2]]]/2)]

(RefLink[ComplexInfinity, paclet:ref/ComplexInfinity]) has indeterminate real and imaginary parts:

{Re[ComplexInfinity], Im[ComplexInfinity]}

{Abs[ComplexInfinity], Arg[ComplexInfinity]}

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

NumberQ[ComplexInfinity]

Obtain (RefLink[ComplexInfinity, paclet:ref/ComplexInfinity]) from limits:

Limit[Exp[-(-2+2I)/x],x->0]

(RefLink[ComplexInfinity, paclet:ref/ComplexInfinity]) behaves like a constant in differentiation:

D[ComplexInfinity,{z, 2}]

OpenersExampleSection4(Spacer1)Possible Issues  (4)

(RefLink[ComplexInfinity, paclet:ref/ComplexInfinity]) is not a numeric quantity:

NumericQ[ComplexInfinity]

(RefLink[ComplexInfinity, paclet:ref/ComplexInfinity]) is a symbol with infinite precision:

Precision[ComplexInfinity]

(RefLink[ComplexInfinity, paclet:ref/ComplexInfinity]) evaluates to (RefLink[DirectedInfinity, paclet:ref/DirectedInfinity]):

FullForm[ComplexInfinity]

Use (RefLink[ComplexInfinity, paclet:ref/ComplexInfinity]) with care in boundary conditions of differential equations:

DSolve[{z'[x]==-2x z[x]^2,z[ComplexInfinity]==0},z[x],x]

OpenersExampleSection5(Spacer1)Neat Examples  (2)

Infinite arguments of undetermined phase in all elementary functions:

HoldForm[#1[ComplexInfinity]]==#1[ComplexInfinity]&/@{Exp, Log, Sinc, Sin, Cos, Tan, Cot, Sec, Csc,
         ArcSin, ArcCos, ArcTan, ArcCot,ArcSec, ArcCsc,
         Sinh, Cosh, Tanh, Coth, Sech, Csch,
         ArcSinh, ArcCosh, ArcTanh, ArcCoth,ArcSech, ArcCsch}//Column

Behavior of the exponential function at (RefLink[ComplexInfinity, paclet:ref/ComplexInfinity]) shown on the Riemann sphere:

ParametricPlot3D[{0,0,1/2}+{Cos[ϕ] Sin[θ],Sin[ϕ] Sin[θ],Cos[θ]}*(1/2+0.5 ArcTan[Re[Exp[Cot[θ/2]( Cos[ϕ] I Sin[ϕ])/0.1]]]/Pi),{ϕ,0,2Pi},{θ,0,Pi},MaxRecursion->3]//Quiet

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

({{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/ComplexInfinity.html])}, {DocumentationFooterIconsRelatedLinks-Functions.png(Spacer1)The Wolfram Functions Site}})}})