BUILT-IN MATHEMATICA SYMBOL

DirectedInfinity

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

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

MORE INFORMATION

You can think of 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:

	Infinity	DirectedInfinity
	-Infinity	DirectedInfinity
	ComplexInfinity	DirectedInfinity

Certain arithmetic operations are performed on DirectedInfinity quantities.

In OutputForm, DirectedInfinity[z] is printed in terms of Infinity, and DirectedInfinity 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:

Use as an expansion point and direction:

In[1]:=

Out[1]=

Use as an integration limit:

In[1]:=

Out[1]=

Use as a limiting point:

In[1]:=

Out[1]=

Scope (6)

Some directions have a special StandardForm:

Use Esc inf Esc 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 DirectedInfinity[z]:

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:

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 Mathematica assume symbols to represent numbers: