Generalized W Lambert function

In summary, the equation x²[A+B.exp(x)]=1 cannot be solved analytically using elementary functions. While there may be a specific function that could express the roots, it is not commonly used or implemented in mathematical software. However, numerical methods can be used to find solutions. One method is to first find a solution with B=0 and then use a small perturbation to find solutions for larger values of B. Another method is to assume B>>A, but this may not always be a valid assumption.
  • #1
MartiniBird
2
0
Hi everyone,
I'm currently trying to solve this equation : x²[A+B.exp(x)]=1 for A and B real numbers, and x a complex (this comes from physics, so in my case, Re(x)>0)

I know that x.exp(x)=a has a solution using Lambert function : x=W(a)
I know that x².exp(x)=a may be recast to use the Lambert function, the solution being something like x=2W(sqrt(a)/2)

But what about my equation ? I tried a lot of things to recast the equation and use the Lambert function, but nothing, so I'm asking to you guys ...

Thanks
 
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  • #2
MartiniBird said:
Hi everyone,
I'm currently trying to solve this equation : x²[A+B.exp(x)]=1 for A and B real numbers, and x a complex (this comes from physics, so in my case, Re(x)>0)

I know that x.exp(x)=a has a solution using Lambert function : x=W(a)
I know that x².exp(x)=a may be recast to use the Lambert function, the solution being something like x=2W(sqrt(a)/2)

But what about my equation ? I tried a lot of things to recast the equation and use the Lambert function, but nothing, so I'm asking to you guys ...

Thanks

The roots of the equation x²[A+B.exp(x)]=1 cannot be expressed as a combination of a finite number of elementary functions.
As far as I know, up to now, there is no standard special function which could help to analitically solve it.
Of course, it is possible that someone already defined a particular function in order to formally express the roots. Even if such a function was defined, it cannot be of general use, since the function is not implemented in mathematical softwares, nor common in math background.
Nevertheless, the equation can be solved, thanks to numerical methods. I think that it is presently the usual way to solve these kind of problems when they are encountered in Physics.
 
  • #3
Ok thanks. I know there are other methods to solve that. I have started to find x0 solution of my equation with B=0 (this case has a particular physical signification. For example, x has to be real, not complex in this case). And then, to solve my equation with B≠0, I assume that x=x0+dx and obtain solution with dx complex. But this is just for small variations ! I don't know I can do that for stronger perturbation dx.

I guess I cannot do something else... first I thought that I could do the assumption B>>A to remove the A in my equation because we know solutions for x².B.exp(x)=1 ... however it seems not correct because the condition to do that is in fact A<<B.cos(Re(x)) that may be not true for some values of x...

But thanks anyway
 

1. What is the Generalized W Lambert function?

The Generalized W Lambert function, also known as the Omega function, is a mathematical function that is the inverse of the function xex. It is used to solve equations of the form xex = y, where x is a complex number and y is a real number. It is denoted by Wn, where n is a positive integer.

2. What is the difference between the Generalized W Lambert function and the regular W Lambert function?

The Generalized W Lambert function is an extension of the regular W Lambert function, which is only defined for real numbers. The regular W Lambert function, denoted by W, is a special case of the Generalized W Lambert function when n = 1. The Generalized W Lambert function allows for the solution of equations with complex numbers, while the regular W Lambert function does not.

3. What are the applications of the Generalized W Lambert function?

The Generalized W Lambert function has various applications in mathematics, physics, and engineering. It is used to solve equations involving complex numbers, such as in quantum mechanics and electrical engineering. It is also used in mathematical modeling and optimization problems.

4. How is the Generalized W Lambert function calculated?

The Generalized W Lambert function is typically calculated using numerical methods, such as the Newton-Raphson method, due to the complexity of the equation. It can also be approximated using series expansions or by using specialized software programs.

5. Are there any special properties of the Generalized W Lambert function?

Yes, there are several special properties of the Generalized W Lambert function. One of the most notable is the fact that it is a multivalued function, meaning it can have multiple solutions for a given input. It also has a branch point at x = -1/e, where it is not defined. Additionally, it has connections to other mathematical functions, such as the trigonometric and hyperbolic functions.

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