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



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Basic Examples  (1)

Division by 0:

Scope  (4)

Use ComplexInfinity in numerical functions:

ComplexInfinity absorbs finite real, complex, and symbolic quantities:

Do arithmetic with ComplexInfinity:

Use ComplexInfinity as an expansion point for series:

Applications  (2)

Set up a seemingly "analytic" function that is infinite in the whole left halfplane:

Plotting shows details of the numerical calculation:

Asymptotics of the LogGamma function at ComplexInfinity:

Properties & Relations  (6)

Use Quiet to suppress messages:

ComplexInfinity can be generated by Simplify and FullSimplify:

ComplexInfinity has indeterminate real and imaginary parts:

ComplexInfinity is not a number:

Obtain ComplexInfinity from limits:

ComplexInfinity behaves like a constant in differentiation:

Possible Issues  (4)

ComplexInfinity is not a numeric quantity:

ComplexInfinity is a symbol with infinite precision:

ComplexInfinity evaluates to DirectedInfinity:

Use ComplexInfinity with care in boundary conditions of differential equations:

Neat Examples  (2)

Infinite arguments of undetermined phase in all elementary functions:

Behavior of the exponential function at ComplexInfinity shown on the Riemann sphere:

Wolfram Research (1988), ComplexInfinity, Wolfram Language function,


Wolfram Research (1988), ComplexInfinity, Wolfram Language function,


Wolfram Language. 1988. "ComplexInfinity." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (1988). ComplexInfinity. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2023_complexinfinity, author="Wolfram Research", title="{ComplexInfinity}", year="1988", howpublished="\url{}", note=[Accessed: 19-April-2024 ]}


@online{reference.wolfram_2023_complexinfinity, organization={Wolfram Research}, title={ComplexInfinity}, year={1988}, url={}, note=[Accessed: 19-April-2024 ]}