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ComplexInfinity

represents a quantity with infinite magnitude, but undetermined complex phase.

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ComplexInfinity

ComplexInfinity

represents a quantity with infinite magnitude, but undetermined complex phase.

Details

Examples

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Basic Examples  (1)

Division by 0:

Wolfram Language code: 1 / 0
Wolfram Language code: 1 / %

Scope  (4)

Use ComplexInfinity in numerical functions:

Wolfram Language code: Sin[ComplexInfinity]
Wolfram Language code: ArcSec[ComplexInfinity]
Wolfram Language code: Sin[Abs[ComplexInfinity]]

ComplexInfinity absorbs finite real, complex, and symbolic quantities:

Wolfram Language code: ComplexInfinity + 2
Wolfram Language code: 3 ComplexInfinity
Wolfram Language code: ComplexInfinity + 3 I
Wolfram Language code: ComplexInfinity x
Wolfram Language code: ComplexInfinity + x

Do arithmetic with ComplexInfinity:

Wolfram Language code: ComplexInfinity / ComplexInfinity
Wolfram Language code: ComplexInfinity ComplexInfinity

Use ComplexInfinity as an expansion point for series:

Wolfram Language code: Series[1 / (x ^ 2 + 1), {x, ComplexInfinity, 2}]
Wolfram Language code: Series[ArcSin[x], {x, ComplexInfinity, 1}]

Applications  (2)

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

Wolfram Language code: w[z_] := 1 / (z - Sqrt[z ^ 2]);
Wolfram Language code: {w[2], w[Pi], w[Pi + 2I]}//Quiet

Plotting shows details of the numerical calculation:

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

Asymptotics of the LogGamma function at ComplexInfinity:

Wolfram Language code: Series[LogGamma[a], {a, ComplexInfinity, 2}]

Properties & Relations  (6)

Use Quiet to suppress messages:

Wolfram Language code: 1 / 0//Quiet
Wolfram Language code: 1 / 0

ComplexInfinity can be generated by Simplify and FullSimplify:

Wolfram Language code: Simplify[1 / ((E + 1) ^ 2 - (E ^ 2 + 2E + 1))]
Wolfram Language code: FullSimplify[1 / (Sin[Pi / 16] - Sqrt[2 - Sqrt[2 + Sqrt[2]]] / 2)]

ComplexInfinity has indeterminate real and imaginary parts:

Wolfram Language code: {Re[ComplexInfinity], Im[ComplexInfinity]}
Wolfram Language code: {Abs[ComplexInfinity], Arg[ComplexInfinity]}

ComplexInfinity is not a number:

Wolfram Language code: NumberQ[ComplexInfinity]

Obtain ComplexInfinity from limits:

Wolfram Language code: Limit[Exp[-(-2 + 2I) / x], x -> 0]

ComplexInfinity behaves like a constant in differentiation:

Wolfram Language code: D[ComplexInfinity, {z, 2}]

Possible Issues  (4)

ComplexInfinity is not a numeric quantity:

Wolfram Language code: NumericQ[ComplexInfinity]

ComplexInfinity is a symbol with infinite precision:

Wolfram Language code: Precision[ComplexInfinity]

ComplexInfinity evaluates to DirectedInfinity:

Wolfram Language code: FullForm[ComplexInfinity]

Use ComplexInfinity with care in boundary conditions of differential equations:

Wolfram Language code: DSolve[{z'[x] == -2x z[x] ^ 2, z[ComplexInfinity] == 0}, z[x], x]

Neat Examples  (2)

Infinite arguments of undetermined phase in all elementary functions:

Wolfram Language code: 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 ComplexInfinity shown on the Riemann sphere:

Wolfram Language code: 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
Wolfram Research (1988), ComplexInfinity, Wolfram Language function, https://reference.wolfram.com/language/ref/ComplexInfinity.html.

Text

Wolfram Research (1988), ComplexInfinity, Wolfram Language function, https://reference.wolfram.com/language/ref/ComplexInfinity.html.

CMS

Wolfram Language. 1988. "ComplexInfinity." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ComplexInfinity.html.

APA

Wolfram Language. (1988). ComplexInfinity. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ComplexInfinity.html

BibTeX

@misc{reference.wolfram_2026_complexinfinity, author="Wolfram Research", title="{ComplexInfinity}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/ComplexInfinity.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_complexinfinity, organization={Wolfram Research}, title={ComplexInfinity}, year={1988}, url={https://reference.wolfram.com/language/ref/ComplexInfinity.html}, note=[Accessed: 12-June-2026]}