What Is the Lambert W Function and How Does It Solve Transcendental Equations?

[pergradus]

Homework Statement

I came up against this equation doing some homework and couldn't figure out how to solve it. I need a numerical value for x.

Homework Equations

$$\frac{xe^x}{e^x -1} - 5 = 0$$Maybe I'm just forgetting some basic log rules or something, but how would you solve this?

 

[eumyang]

This doesn't look like it can be solved using (elementary) algebraic methods. Assuming the question was copied correctly, maybe solving this involves the Lambert function?

 

[Coto]

This is what is called a transcendental equation and there is no algebraic way to solve it. Instead you can consider the solution as the intersection of two graphs in this way:

Simplifying your equation, we obtain,

$$\frac{x}{5} = -e^{-x} + 1, x \neq 0,$$

one can then plot two graphs, one graph for the function on the LHS, and the other for the function on the RHS. Intersection points are your solutions.

 

[pergradus]

This doesn't look like it can be solved using (elementary) algebraic methods. Assuming the question was copied correctly, maybe solving this involves the Lambert function?

The solution is needed to derive Wein's Law by differentiating Plancks function with respect to lambda. So it's part of a larger problem. Not sure what the Lambert function is.

This is what is called a transcendental equation and there is no algebraic way to solve it. Instead you can consider the solution as the intersection of two graphs in this way:

Simplifying your equation, we obtain,

LaTeX Code: \\frac{x}{5} = -e^{-x} + 1, x \\neq 0

one can then plot two graphs, one graph for the function on the LHS, and the other for the function on the RHS. Intersection points are your solutions.

Hmm thanks for the tip. Is there no analytic way to solve it though? Also, how do you know it is transcendental by looking at it?

 

[Coto]

Hmm thanks for the tip. Is there no analytic way to solve it though? Also, how do you know it is transcendental by looking at it?

http://en.wikipedia.org/wiki/Transcendental_equation

 

[dextercioby]

The solution is needed to derive Wein's Law by differentiating Plancks function with respect to lambda. So it's part of a larger problem. Not sure what the Lambert function is.Hmm thanks for the tip. Is there no analytic way to solve it though? Also, how do you know it is transcendental by looking at it?

Please, also see here who the Lambert function is http://en.wikipedia.org/wiki/Lambert_function. The article is well written.

The graphic intersection solution is the one to use and using a calculator, you can get a 3,4 decimal approximation of the solution.

 

[HallsofIvy]

Homework Statement

I came up against this equation doing some homework and couldn't figure out how to solve it. I need a numerical value for x.

Homework Equations

$$\frac{xe^x}{e^x -1} - 5 = 0$$


Maybe I'm just forgetting some basic log rules or something, but how would you solve this?

Was this actually a homework problem? It seems a very peculiar problem for homework. Especially if you have not yet been introduced to "transcendental functions" or "transcendental equations". As dextercioby said, it can be solved using Lambert's W function which is defined as the inverse function to $xe^x$

That is, you can manipulate this equation to be $ye^y= constant$ and then say that $y= W(constant)$ where "W" is Lambert's W function.