# DirectedInfinity

represents an infinite numerical quantity whose direction in the complex plane is unknown.

represents an infinite numerical quantity that is a positive real multiple of the complex number z.

# Details • You can think of 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:
•  Infinity -Infinity DirectedInfinity[-1] ComplexInfinity
• Certain arithmetic operations are performed on DirectedInfinity quantities.
• In OutputForm, is printed in terms of Infinity, and is printed as ComplexInfinity.

# Examples

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## Basic Examples(3)

Use as an expansion point and direction:

Use as an integration limit:

Use as a limiting point:

## Scope(6)

Some directions have a special StandardForm:

Use inf to enter :

Use Infinity as an alternative input form:

Multiplying by a number changes the direction:

Unspecified or Indeterminate direction represents ComplexInfinity:

Finite or symbolic quantities are absorbed:

Extended arithmetic with infinite quantities:

In this case the result depends on the directions x and y: Operations that cannot be unambiguously defined produce Indeterminate: In this case the result depends on the growth rates of the numerator and denominator:

Use in mathematical functions:

The value in different directions may vary:

## Applications(2)

Integrate along a line from the origin with direction :

Asymptotics of the LogGamma function at :

Plot asymptotic value compared to function value in different directions:

## Properties & Relations(3)

Simplify and FullSimplify can generate infinities:  A nested DirectedInfinity reduces to one DirectedInfinity:

is not a number:

## Possible Issues(3)

Symbolic quantities might get lost in operations:

The Accuracy and Precision for DirectedInfinity refer to the direction argument:

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