What is the inverse function of f(x) = c/x^(1/n)?

[roupam]

Hi,

I have written a paper (attached)
I would be happy to get comments on it

Thanks
Roupam

PS.
(Since, there is no independent research in the math section, I have posted here)

 

[CRGreathouse]

On p. 2, in your equation for zeta, i is an unfortunate choice for an index variable since you're working with complex numbers. I suggest k.

On p. 3, "for some constant B and Li(x)" is unclear; it looks like you're making Li(x) constant. Reword if possible. Also "Cramérs conjecture" should be "Cramér's conjecture". (There are a number of minor grammar errors here and throughout, especially overuse of the comma; I'll omit those.)

On p. 6, your functions f and f^{-1} are not inverses. This may be fatal.

 

[roupam]

On p. 2, in your equation for zeta, i is an unfortunate choice for an index variable since you're working with complex numbers. I suggest k.

On p. 3, "for some constant B and Li(x)" is unclear; it looks like you're making Li(x) constant. Reword if possible. Also "Cramérs conjecture" should be "Cramér's conjecture". (There are a number of minor grammar errors here and throughout, especially overuse of the comma; I'll omit those.)

Being a Comp. Sc. Student, I have the habit of using "i"
Thanks, I will replace it with something like "a", or "r" etc.

On p. 6, your functions f and f^{-1} are not inverses. This may be fatal.

Can you please tell me why they are not inverses ?
Isnt the inverse formula correct?

 

[CRGreathouse]

Being a Comp. Sc. Student, I have the habit of using "i"
Thanks, I will replace it with something like "a", or "r" etc.

Fine habit, 90% of the time...

Can you please tell me why they are not inverses ?
Isnt the inverse formula correct?

You basically want
f(f^{-1}(x)) = x = f^{-1}(f(x))
for all x, and neither is the case.

 

[roupam]

You basically want
f(f^{-1}(x)) = x = f^{-1}(f(x))
for all x, and neither is the case.

Well, it seems right to me...

f^{-1}(x) = c^n/x^n
f(x) = c/x^(1/n)

which gives,
f(f^{-1}(x)) = x = f^{-1}(f(x))

And, I am not using the complex roots of x^(1/n), which I stated in the Preliminaries section of the paper that we are only working with positive reals...

Thanks
Roupam

 

[CRGreathouse]

Well, it seems right to me...

f^{-1}(x) = c^n/x^n
f(x) = c/x^(1/n)

which gives,
f(f^{-1}(x)) = x = f^{-1}(f(x))

No. If you show your calculations step-by-step we can show you where you're making a mistake.

 

[roupam]

No. If you show your calculations step-by-step we can show you where you're making a mistake.

Ok, here goes...

f^{-1}(x) = c^n/x^n
f(x) = c/x^(1/n)

which gives,
f(f^{-1}(x)) = c/(f^{-1}(x))^(1/n) = c/(c^n/x^n)^(1/n) = c/(c/x) = x
Again,
f^{-1}(f(x)) = f^{-1}(c/x^(1/n)) = c^n/(c/x^(1/n))^n = c^n/(c^n/x) = x