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...") |
|||
Line 6: | Line 6: | ||
== ProductLog(z) == | == ProductLog(z) == | ||
− | Returns the value x that solves < | + | Returns the value ''x'' that solves |
+ | :<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. | ||
Line 12: | Line 15: | ||
=== Real-valued z === | === Real-valued z === | ||
− | + | :[[image: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 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. | + | 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 === | === 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. | + | 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 == | == See Also == | ||
− | * [[Ln]] | + | * [[Ln]] |
+ | * [[LogTen]] | ||
+ | * [[Product]] | ||
+ | * [[Exp]] | ||
+ | * [[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
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.
Enable comment auto-refresher