Can we evaluate noninteger hyperoparations?
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The hyperoperation $a[n]b$ is addition when $n=1$, multiplication when $n=2$, exponentiation when $n=3$, tetration when $n=4$, and so on.
What happens when $n$ is noninteger?
Can we evaluate, e.g. $a[2.5]b$, $a[pi]$b$, etc?
How about $frac{mathrm{d}}{mathrm{d}x}a[2.5]x$, $frac{mathrm{d}}{mathrm{d}x}x[2.5]b$, or $frac{mathrm{d}}{mathrm{d}x}a[x]b$? Integrals?
Is this a meaningful concept?
analytic-continuation hyperoperation
asked Nov 17 at 11:59
spraff
460213
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up vote
0
down vote
favorite
The hyperoperation $a[n]b$ is addition when $n=1$, multiplication when $n=2$, exponentiation when $n=3$, tetration when $n=4$, and so on.
What happens when $n$ is noninteger?
Can we evaluate, e.g. $a[2.5]b$, $a[pi]$b$, etc?
How about $frac{mathrm{d}}{mathrm{d}x}a[2.5]x$, $frac{mathrm{d}}{mathrm{d}x}x[2.5]b$, or $frac{mathrm{d}}{mathrm{d}x}a[x]b$? Integrals?
Is this a meaningful concept?
analytic-continuation hyperoperation
asked Nov 17 at 11:59
spraff
460213
The question is "can we assign some values to these operations which would work for any $a$ and $b$ in a consistent way and reduce to the usual operations for integer order". There may be several ways to do so (or no way, which is an interesting theorem to prove)
– Yuriy S
Nov 17 at 12:06
There is an answer of mine giving one example see math.stackexchange.com/a/1272791/1714 . Note, this is not "the only" way to do that but it was focused to the question in the sense of binary operations
– Gottfried Helms
Nov 18 at 23:22
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
The hyperoperation $a[n]b$ is addition when $n=1$, multiplication when $n=2$, exponentiation when $n=3$, tetration when $n=4$, and so on.
What happens when $n$ is noninteger?
Can we evaluate, e.g. $a[2.5]b$, $a[pi]$b$, etc?
How about $frac{mathrm{d}}{mathrm{d}x}a[2.5]x$, $frac{mathrm{d}}{mathrm{d}x}x[2.5]b$, or $frac{mathrm{d}}{mathrm{d}x}a[x]b$? Integrals?
Is this a meaningful concept?
analytic-continuation hyperoperation
asked Nov 17 at 11:59
spraff
460213
The hyperoperation $a[n]b$ is addition when $n=1$, multiplication when $n=2$, exponentiation when $n=3$, tetration when $n=4$, and so on.
What happens when $n$ is noninteger?
Can we evaluate, e.g. $a[2.5]b$, $a[pi]$b$, etc?
How about $frac{mathrm{d}}{mathrm{d}x}a[2.5]x$, $frac{mathrm{d}}{mathrm{d}x}x[2.5]b$, or $frac{mathrm{d}}{mathrm{d}x}a[x]b$? Integrals?
Is this a meaningful concept?
analytic-continuation hyperoperation
analytic-continuation hyperoperation
asked Nov 17 at 11:59
spraff
460213
asked Nov 17 at 11:59
spraff
460213
asked Nov 17 at 11:59
spraff
460213
asked Nov 17 at 11:59
spraff
460213
asked Nov 17 at 11:59
spraff
460213
460213
The question is "can we assign some values to these operations which would work for any $a$ and $b$ in a consistent way and reduce to the usual operations for integer order". There may be several ways to do so (or no way, which is an interesting theorem to prove)
– Yuriy S
Nov 17 at 12:06
There is an answer of mine giving one example see math.stackexchange.com/a/1272791/1714 . Note, this is not "the only" way to do that but it was focused to the question in the sense of binary operations
– Gottfried Helms
Nov 18 at 23:22
add a comment |
The question is "can we assign some values to these operations which would work for any $a$ and $b$ in a consistent way and reduce to the usual operations for integer order". There may be several ways to do so (or no way, which is an interesting theorem to prove)
– Yuriy S
Nov 17 at 12:06
There is an answer of mine giving one example see math.stackexchange.com/a/1272791/1714 . Note, this is not "the only" way to do that but it was focused to the question in the sense of binary operations
– Gottfried Helms
Nov 18 at 23:22
The question is "can we assign some values to these operations which would work for any $a$ and $b$ in a consistent way and reduce to the usual operations for integer order". There may be several ways to do so (or no way, which is an interesting theorem to prove)
– Yuriy S
Nov 17 at 12:06
The question is "can we assign some values to these operations which would work for any $a$ and $b$ in a consistent way and reduce to the usual operations for integer order". There may be several ways to do so (or no way, which is an interesting theorem to prove)
– Yuriy S
Nov 17 at 12:06
There is an answer of mine giving one example see math.stackexchange.com/a/1272791/1714 . Note, this is not "the only" way to do that but it was focused to the question in the sense of binary operations
– Gottfried Helms
Nov 18 at 23:22
There is an answer of mine giving one example see math.stackexchange.com/a/1272791/1714 . Note, this is not "the only" way to do that but it was focused to the question in the sense of binary operations
– Gottfried Helms
Nov 18 at 23:22
add a comment |
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The question is "can we assign some values to these operations which would work for any $a$ and $b$ in a consistent way and reduce to the usual operations for integer order". There may be several ways to do so (or no way, which is an interesting theorem to prove)
– Yuriy S
Nov 17 at 12:06
There is an answer of mine giving one example see math.stackexchange.com/a/1272791/1714 . Note, this is not "the only" way to do that but it was focused to the question in the sense of binary operations
– Gottfried Helms
Nov 18 at 23:22