You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter inertia
- Start date

One problem with this from a strictly mathematical perspective is that it fails the condition that

f⁻¹( f(x) ) = x or

f( f⁻¹(x) ) = x

(Let me know if some browsers do not render the Unicode superscript "-" and "1" in the above.)

Okay. I give up. How did you produce a superscript using the CARM provided word processor?

___

.

It is a superscript character, rather than a number that is transformed to be smaller and higher.Okay. I give up. How did you produce a superscript using the CARM provided word processor?

en.wikipedia.org

That's right. I copied and pasted from a Wikipedia page about Unicode.It is a superscript character, rather than a number that is transformed to be smaller and higher.

## List of Unicode characters - Wikipedia

en.wikipedia.org

That's right. I copied and pasted from a Wikipedia page about Unicode.

Using the Microsoft Word processor where superscripts are readily available it is easy. Here at CARM with the help of Word I can write:

Euler's identity e^(iπ) + 1 = 0 for example.

Now about that i*pi superscript --> stackoverflow

___

.

Last edited:

Using the Microsoft Word processor where superscripts are readily available it is easy. Here at CARM with the help of Word I can write:

Euler's identity e^(iπ) + 1 = 0 for example.

Now about that i*pi superscript --> stackoverflow

___

.

testing the copy and paste method - -

f⁻¹(x)

Okay, this worked. [ Noting that Wikipedia did not exhibit the number 1 in its table. ]

____

Reference: compart.com

.

Last edited:

Talking about inverses, I recently discovered the Lambert W function which is the inverse of f(x) = x*e^x. Or f⁻¹(x) = W(x). This function has no expression in the form of traditional finite functions but is very useful. It is just that in my years of math studies, I had never come across this function with its wide variety of applications.testing the copy and paste method - -

f⁻¹(x)

Okay, this worked. [ Noting that Wikipedia did not exhibit the number 1 in its table. ]

____

Reference: compart.com

.

Talking about inverses, I recently discovered the Lambert W function which is the inverse of f(x) = x*e^x. Or f⁻¹(x) = W(x). This function has no expression in the form of traditional finite functions but is very useful. It is just that in my years of math studies, I had never come across this function with its wide variety of applications.

I've never used it. So, I opened up my Handbook of Mathematical Functions by Abramowitz and Stegun. To my surprise, it wasn't listed anywhere. It wasn't listed in my Numerical Recipes book, my Foundations of Applied Mathematics book, or my book on Vectors and Tensors in Engineering and Physics.

All of that and then - I "fired up" my Matlab.

Code:

```
syms x W
solve(x == W*exp(W), W)
ans =
lambertw(0, x)
```

Another example:

Code:

```
A = [0 -1/exp(1); pi i];
lambertw(A)
lambertw(-1, A)
ans =
0.0000 + 0.0000i -1.0000 + 0.0000i
1.0737 + 0.0000i 0.3747 + 0.5764i
ans =
-Inf + 0.0000i -1.0000 + 0.0000i
-0.3910 - 4.6281i -1.0896 - 2.7664i
```

___

.

Other names for Lambert W are, product log, and omega function. In mathematica, instead of lambertw, it is productlog. Here is a short 10 minute tutorial.I've never used it. So, I opened up my Handbook of Mathematical Functions by Abramowitz and Stegun. To my surprise, it wasn't listed anywhere. It wasn't listed in my Numerical Recipes book, my Foundations of Applied Mathematics book, or my book on Vectors and Tensors in Engineering and Physics.

All of that and then - I "fired up" my Matlab.

Code:`syms x W solve(x == W*exp(W), W) ans = lambertw(0, x)`

Another example:

Code:`A = [0 -1/exp(1); pi i]; lambertw(A) lambertw(-1, A) ans = 0.0000 + 0.0000i -1.0000 + 0.0000i 1.0737 + 0.0000i 0.3747 + 0.5764i ans = -Inf + 0.0000i -1.0000 + 0.0000i -0.3910 - 4.6281i -1.0896 - 2.7664i`

___

.

Other names for Lambert W are, product log, and omega function. In mathematica, instead of lambertw, it is productlog. Here is a short 10 minute tutorial.

Thanks for the video. Now I have another tool in my toolbox.

____

.