Difference between revisions of "ProductLog"

(Created page with "Category:Math Functions Category:Functions that operate on complex numbers ''new to Analytica 4.7'' == ProductLog(z) == Returns the value x that solves <code>z...")
 
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== ProductLog(z) ==
 
== ProductLog(z) ==
  
Returns the value x that solves <code>z = x * [[Exp]](x)</code>.
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Returns the value ''x'' that solves  
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:<math>z = x * Exp(x)</math>.
 +
 
 +
where [[Exp]] is  the exponential function.
  
 
This is also known as the [https://en.wikipedia.org/wiki/Lambert_W_function Lambert W function], often denoted as <math>W_0(z)</math> in mathematical publications. It appears in the analytic solution of many equations involving exponentials and non-exponentials in the same equation.
 
This is also known as the [https://en.wikipedia.org/wiki/Lambert_W_function Lambert W function], often denoted as <math>W_0(z)</math> in mathematical publications. It appears in the analytic solution of many equations involving exponentials and non-exponentials in the same equation.
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=== Real-valued z ===
 
=== Real-valued z ===
  
::[[image:ProductLog_graph.png]]
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:[[image:ProductLog_graph.png]]
  
 
The function is real-valued for a real-valued parameter z &ge; -exp(-1). When «z» is real but less than -exp(-1), it returns [[NaN]] when [[EnableComplexNumbers]] is 0.
 
The function is real-valued for a real-valued parameter z &ge; -exp(-1). When «z» is real but less than -exp(-1), it returns [[NaN]] when [[EnableComplexNumbers]] is 0.
  
The solution is unique when z&ge;0 or z=-exp(-1). There are two solutions when -exp(-1) < z < 0, in which case [[ProductLog]] returns the main branch, whose value is greater than -1. The secondary branch, whose values are less that -1 in the -exp(-1) < z < 0 interval, is not available from this function.
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The solution is unique when z &ge; 0 or z = -exp(-1). There are two solutions when -exp(-1) < z < 0, in which case [[ProductLog]] returns the main branch, whose value is greater than -1. The secondary branch, whose values are less that -1 in the -exp(-1) < z < 0 interval, is not available from this function.
  
 
=== Complex numbers ===
 
=== Complex numbers ===
  
When «z» is a complex number, or z<-exp(-1), the result is a complex number. To obtain the complex result when the parameter is real-valued, you need to set the [[EnableComplexNumbers]] system variable to 1.
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When «z» is a complex number, or z < -exp(-1), the result is a complex number. To obtain the complex result when the parameter is real-valued, you need to set the [[EnableComplexNumbers]] system variable to 1.
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 +
==History==
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Introduced in [[Analytica 4.7]].
  
 
== See Also ==
 
== See Also ==
* [[Ln]], [[LogTen]]
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* [[Ln]]
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* [[LogTen]]
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* [[Product]]
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* [[Exp]]
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* [[EnableComplexNumbers]]

Revision as of 21:43, 17 February 2016


new to Analytica 4.7

ProductLog(z)

Returns the value x that solves

[math]\displaystyle{ z = x * Exp(x) }[/math].

where Exp is the exponential function.

This is also known as the Lambert W function, often denoted as [math]\displaystyle{ W_0(z) }[/math] in mathematical publications. It appears in the analytic solution of many equations involving exponentials and non-exponentials in the same equation.

Real-valued z

ProductLog graph.png

The function is real-valued for a real-valued parameter z ≥ -exp(-1). When «z» is real but less than -exp(-1), it returns NaN when EnableComplexNumbers is 0.

The solution is unique when z ≥ 0 or z = -exp(-1). There are two solutions when -exp(-1) < z < 0, in which case ProductLog returns the main branch, whose value is greater than -1. The secondary branch, whose values are less that -1 in the -exp(-1) < z < 0 interval, is not available from this function.

Complex numbers

When «z» is a complex number, or z < -exp(-1), the result is a complex number. To obtain the complex result when the parameter is real-valued, you need to set the EnableComplexNumbers system variable to 1.

History

Introduced in Analytica 4.7.

See Also

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