The Novel Family of Transcendental Leal-Functions

[KJW]

While attempting to solve the differential equation:

[math]\dfrac{dr'}{dr} = \dfrac{1}{\sqrt{1 - \dfrac{2GM}{c^2 r}}}[/math]

expressing [math]r[/math] in terms of [math]r'[/math], I encountered a novel family of transcendental functions called "Leal-functions". These functions are similar to the Lambert W function (the function [math]W(x)[/math] that solves [math]W(x)e^{W(x)} = x[/math]), but (apparently) can't be derived from it.

The link to the full article about these functions:

https://www.sciencedirect.com/science/article/pii/S2405844020322611

The link to the section that defines these functions:

https://www.sciencedirect.com/science/article/pii/S2405844020322611#se0040

Below is a list of Leal functions and their definitions:

[math]y(x) = \textrm{Lsinh}(x)[/math]     [math]\iff[/math]     [math]y(x) \sinh(y(x)) = x[/math]

[math]y(x) = \textrm{Lcosh}(x)[/math]     [math]\iff[/math]     [math]y(x) \cosh(y(x)) = x[/math]

[math]y(x) = \textrm{Ltanh}(x)[/math]     [math]\iff[/math]     [math]y(x) \tanh(y(x)) = x[/math]

[math]y(x) = \textrm{Lcsch}(x)[/math]     [math]\iff[/math]     [math]y(x) \textrm{ csch}(y(x)) = x[/math]

[math]y(x) = \textrm{Lsech}(x)[/math]     [math]\iff[/math]     [math]y(x) \textrm{ sech}(y(x)) = x[/math]

[math]y(x) = \textrm{Lcoth}(x)[/math]     [math]\iff[/math]     [math]y(x) \coth(y(x)) = x[/math]

[math]y(x) = \textrm{Lln}(x)[/math]     [math]\iff[/math]     [math]y(x) \ln(y(x) + 1) = x[/math]

[math]y(x) = \textrm{Ltan}(x)[/math]     [math]\iff[/math]     [math]y(x) \tan(y(x)) = x[/math]

[math]y(x) = \textrm{Lsinh}_2(x)[/math]     [math]\iff[/math]     [math]y(x) + \sinh(y(x)) = x[/math]

[math]y(x) = \textrm{Lcosh}_2(x)[/math]     [math]\iff[/math]     [math]y(x) + \cosh(y(x)) = x[/math]

The authors say that the Leal family of functions can be extended to solve other transcendental equations, and provide examples of other similar functions. They even say that users can propose their own functions, applying the methodology used in the article.

It turns out that the solution to the above differential equation for the coordinate transformation of the [math]g_{rr}[/math] component of the Schwarzschild metric to [math]g_{r'r'} = -1[/math] involves the [math]\textrm{Lsinh}_2(x)[/math] Leal-function defined above.

 

 

Edited May 11 by KJW

[KJW]

Consider the differential equation:

[math]\dfrac{dy}{dx} = \dfrac{1}{\sqrt{1 - 2/x}} = \dfrac{\sqrt{x}}{\sqrt{x - 2}}[/math]

Let:  [math]x = u + 1[/math]     ;     [math]dx = du[/math]

[math]\dfrac{dy}{du} = \dfrac{\sqrt{u + 1}}{\sqrt{u - 1}} =  \dfrac{\sqrt{u + 1}}{\sqrt{u - 1}}  \dfrac{\sqrt{u + 1}}{\sqrt{u + 1}}[/math]

[math]= \dfrac{u + 1}{\sqrt{u^2 - 1}}[/math]

[math]y - C = \sqrt{u^2 - 1} + \textrm{arccosh}(u)[/math]

Let:  [math]u = \cosh(v)[/math]

[math]y - C = \sinh(v) + v[/math]

[math]v = \textrm{Lsinh}_2(y - C)[/math]

[math]u = \cosh(\textrm{Lsinh}_2(y - C))[/math]

Therefore:

[math]x = \cosh(\textrm{Lsinh}_2(y - C)) + 1[/math]

 

 

Edited May 11 by KJW

[Mordred]

Hrrm I can see these functions could have a broad range of applications. Thanks for sharing this, gives me something new to study myself. I always keep an eye out for useful mathematical methods that I could employ. I will have to look more into the Leal functions.

You might this listing handy

https://personal.math.ubc.ca/~cbm/aands/abramowitz_and_stegun.pdf

There is a section listing  transcendental functions. (The article simply has a good listing of various functions for the purpose of quick reference . It doesn't go into any particular details on any of them.

I found it handy in the past you might as well.

Edit doesn't seem to be a whole lot of information on those functions beyond the links you already posted.

Edited May 28 by Mordred

[KJW]

On 5/28/2024 at 10:51 AM, Mordred said:

I can see these functions could have a broad range of applications.

Functions of this type do seem to crop up every now and then in various places. They seem to highlight the notion of how few the functions are that can be expressed in terms of elementary functions.

 

 

On 5/28/2024 at 10:51 AM, Mordred said:

You might this listing handy

https://personal.math.ubc.ca/~cbm/aands/abramowitz_and_stegun.pdf

Thanks.

 

 

On 5/28/2024 at 10:51 AM, Mordred said:

Edit doesn't seem to be a whole lot of information on those functions beyond the links you already posted.

I think the authors invented the Leal-functions. I imagine being a contemporary of someone like Euler wondering if this newfangled notation will just be a flash in the pan or whether it will become a permanent part of mathematics.
 

 

[StringJunky]

@KJW Is this the paper you looked at?

The novel family of transcendental Leal-functions with applications to
science and engineering - Leal et al

https://www.sciencedirect.com/science/article/pii/S2405844020322611

Full pdf link is in there to the paper.

 

Edited 12 hours ago by StringJunky

[KJW]

3 minutes ago, StringJunky said:

@KJW Is this the paper you looked at?

The novel family of transcendental Leal-functions with applications to
science and engineering - Leal et al

https://www.sciencedirect.com/science/article/pii/S2405844020322611

Full pdf link is in there to the paper.

Checking the link... yes. The link opens to a webpage that asks if you are a robot. Verifying you are human takes you the webpage of the paper. From the webpage of the paper, you can view the PDF file, from which it can be saved.

 

[Mordred]

1 hour ago, KJW said:

 

I think the authors invented the Leal-functions. I imagine being a contemporary of someone like Euler wondering if this newfangled notation will just be a flash in the pan or whether it will become a permanent part of mathematics.
 

 

That's the impression I got as well.