Smooth Elementary Function that Outgrows All Tower Functions?
This started off with me goofing off on Twitter and quickly led to this question to which I didn't know the answer.
Let $T_2(x) = x^x$, $T_3(x) = x^{x{^x}}$, $T_4(x) = x^{x^{x^x}}$, and so forth.
Is there a $C^{infty}$ elementary function $f(x)$ (it can be piecewise-defined) with $displaystyle limlimits_{x to infty} frac{f(x)}{T_k(x)} = infty$ for all $k$? If so, what is a (possibly piecewise-defined) formula for such an $f$?
elementary-functions power-towers hyperoperation
asked Jan 4 at 8:58
Jeffrey RollandJeffrey Rolland
20917
add a comment |
This started off with me goofing off on Twitter and quickly led to this question to which I didn't know the answer.
Let $T_2(x) = x^x$, $T_3(x) = x^{x{^x}}$, $T_4(x) = x^{x^{x^x}}$, and so forth.
Is there a $C^{infty}$ elementary function $f(x)$ (it can be piecewise-defined) with $displaystyle limlimits_{x to infty} frac{f(x)}{T_k(x)} = infty$ for all $k$? If so, what is a (possibly piecewise-defined) formula for such an $f$?
elementary-functions power-towers hyperoperation
asked Jan 4 at 8:58
Jeffrey RollandJeffrey Rolland
20917
(I tried $f(x) = T_k(x)$ on [k+(1/4), k+(3/4)] and linearly interpolating between endpoints, but I couldn't find a way to make that both smooth (convoluting with Gaussians would do that, but it wouldn't be elementary then) and elementary (it's already elementary but not smooth.)
– Jeffrey Rolland
Jan 4 at 9:03
2
Possibly related: math.stackexchange.com/questions/1892870/…
– Martin R
Jan 4 at 9:03
(The Smooth Approximation Theorem from differential topology also does not guarantee an elementary solution.)
– Jeffrey Rolland
Jan 4 at 9:04
@Martin R. Mmm, the second answer in that question in close, but doesn't allow piecewise-defined functions; I already found an elementary piecewise-defined function that contradicts the second answer.
– Jeffrey Rolland
Jan 4 at 9:09
1
I am no expert on this topic at all. My guess would be that a similar approach as in this comment might work: Define $f$ on the positive integers as $f(k) = T_k(k)$, and smoothly interpolate between the integers.
– Martin R
Jan 4 at 9:22
add a comment |
This started off with me goofing off on Twitter and quickly led to this question to which I didn't know the answer.
Let $T_2(x) = x^x$, $T_3(x) = x^{x{^x}}$, $T_4(x) = x^{x^{x^x}}$, and so forth.
Is there a $C^{infty}$ elementary function $f(x)$ (it can be piecewise-defined) with $displaystyle limlimits_{x to infty} frac{f(x)}{T_k(x)} = infty$ for all $k$? If so, what is a (possibly piecewise-defined) formula for such an $f$?
elementary-functions power-towers hyperoperation
asked Jan 4 at 8:58
Jeffrey RollandJeffrey Rolland
20917
This started off with me goofing off on Twitter and quickly led to this question to which I didn't know the answer.
Let $T_2(x) = x^x$, $T_3(x) = x^{x{^x}}$, $T_4(x) = x^{x^{x^x}}$, and so forth.
Is there a $C^{infty}$ elementary function $f(x)$ (it can be piecewise-defined) with $displaystyle limlimits_{x to infty} frac{f(x)}{T_k(x)} = infty$ for all $k$? If so, what is a (possibly piecewise-defined) formula for such an $f$?
elementary-functions power-towers hyperoperation
elementary-functions power-towers hyperoperation
asked Jan 4 at 8:58
Jeffrey RollandJeffrey Rolland
20917
asked Jan 4 at 8:58
Jeffrey RollandJeffrey Rolland
20917
asked Jan 4 at 8:58
Jeffrey RollandJeffrey Rolland
20917
asked Jan 4 at 8:58
Jeffrey RollandJeffrey Rolland
20917
asked Jan 4 at 8:58
Jeffrey RollandJeffrey Rolland
20917
20917
(I tried $f(x) = T_k(x)$ on [k+(1/4), k+(3/4)] and linearly interpolating between endpoints, but I couldn't find a way to make that both smooth (convoluting with Gaussians would do that, but it wouldn't be elementary then) and elementary (it's already elementary but not smooth.)
– Jeffrey Rolland
Jan 4 at 9:03
2
Possibly related: math.stackexchange.com/questions/1892870/…
– Martin R
Jan 4 at 9:03
(The Smooth Approximation Theorem from differential topology also does not guarantee an elementary solution.)
– Jeffrey Rolland
Jan 4 at 9:04
@Martin R. Mmm, the second answer in that question in close, but doesn't allow piecewise-defined functions; I already found an elementary piecewise-defined function that contradicts the second answer.
– Jeffrey Rolland
Jan 4 at 9:09
1
I am no expert on this topic at all. My guess would be that a similar approach as in this comment might work: Define $f$ on the positive integers as $f(k) = T_k(k)$, and smoothly interpolate between the integers.
– Martin R
Jan 4 at 9:22
add a comment |
(I tried $f(x) = T_k(x)$ on [k+(1/4), k+(3/4)] and linearly interpolating between endpoints, but I couldn't find a way to make that both smooth (convoluting with Gaussians would do that, but it wouldn't be elementary then) and elementary (it's already elementary but not smooth.)
– Jeffrey Rolland
Jan 4 at 9:03
2
Possibly related: math.stackexchange.com/questions/1892870/…
– Martin R
Jan 4 at 9:03
(The Smooth Approximation Theorem from differential topology also does not guarantee an elementary solution.)
– Jeffrey Rolland
Jan 4 at 9:04
@Martin R. Mmm, the second answer in that question in close, but doesn't allow piecewise-defined functions; I already found an elementary piecewise-defined function that contradicts the second answer.
– Jeffrey Rolland
Jan 4 at 9:09
1
I am no expert on this topic at all. My guess would be that a similar approach as in this comment might work: Define $f$ on the positive integers as $f(k) = T_k(k)$, and smoothly interpolate between the integers.
– Martin R
Jan 4 at 9:22
(I tried $f(x) = T_k(x)$ on [k+(1/4), k+(3/4)] and linearly interpolating between endpoints, but I couldn't find a way to make that both smooth (convoluting with Gaussians would do that, but it wouldn't be elementary then) and elementary (it's already elementary but not smooth.)
– Jeffrey Rolland
Jan 4 at 9:03
(I tried $f(x) = T_k(x)$ on [k+(1/4), k+(3/4)] and linearly interpolating between endpoints, but I couldn't find a way to make that both smooth (convoluting with Gaussians would do that, but it wouldn't be elementary then) and elementary (it's already elementary but not smooth.)
– Jeffrey Rolland
Jan 4 at 9:03
2
2
Possibly related: math.stackexchange.com/questions/1892870/…
– Martin R
Jan 4 at 9:03
Possibly related: math.stackexchange.com/questions/1892870/…
– Martin R
Jan 4 at 9:03
(The Smooth Approximation Theorem from differential topology also does not guarantee an elementary solution.)
– Jeffrey Rolland
Jan 4 at 9:04
(The Smooth Approximation Theorem from differential topology also does not guarantee an elementary solution.)
– Jeffrey Rolland
Jan 4 at 9:04
@Martin R. Mmm, the second answer in that question in close, but doesn't allow piecewise-defined functions; I already found an elementary piecewise-defined function that contradicts the second answer.
– Jeffrey Rolland
Jan 4 at 9:09
@Martin R. Mmm, the second answer in that question in close, but doesn't allow piecewise-defined functions; I already found an elementary piecewise-defined function that contradicts the second answer.
– Jeffrey Rolland
Jan 4 at 9:09
1
1
I am no expert on this topic at all. My guess would be that a similar approach as in this comment might work: Define $f$ on the positive integers as $f(k) = T_k(k)$, and smoothly interpolate between the integers.
– Martin R
Jan 4 at 9:22
I am no expert on this topic at all. My guess would be that a similar approach as in this comment might work: Define $f$ on the positive integers as $f(k) = T_k(k)$, and smoothly interpolate between the integers.
– Martin R
Jan 4 at 9:22
add a comment |
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(I tried $f(x) = T_k(x)$ on [k+(1/4), k+(3/4)] and linearly interpolating between endpoints, but I couldn't find a way to make that both smooth (convoluting with Gaussians would do that, but it wouldn't be elementary then) and elementary (it's already elementary but not smooth.)
– Jeffrey Rolland
Jan 4 at 9:03
2
Possibly related: math.stackexchange.com/questions/1892870/…
– Martin R
Jan 4 at 9:03
(The Smooth Approximation Theorem from differential topology also does not guarantee an elementary solution.)
– Jeffrey Rolland
Jan 4 at 9:04
@Martin R. Mmm, the second answer in that question in close, but doesn't allow piecewise-defined functions; I already found an elementary piecewise-defined function that contradicts the second answer.
– Jeffrey Rolland
Jan 4 at 9:09
1
I am no expert on this topic at all. My guess would be that a similar approach as in this comment might work: Define $f$ on the positive integers as $f(k) = T_k(k)$, and smoothly interpolate between the integers.
– Martin R
Jan 4 at 9:22