f^-1(x)

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.)
 
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?

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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

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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

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testing the copy and paste method - -

f⁻¹(x)

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

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Reference: compart.com
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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.
 
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

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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

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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.

 
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