Inverse factorial function
I am wondering what is the inverse/opposite factorial function?
e.g inverse-factorial(6)=3
Furthermore, I am intrigued to know the answer to:
a!=π
find a
I would really appreciate if anyone could explain this to me as I have found nowhere online with a good explanation of inverse factorial functions.
Thanks
factorial inverse-function
asked Nov 14 '18 at 12:18
Cameron Gray
61
add a comment |
I am wondering what is the inverse/opposite factorial function?
e.g inverse-factorial(6)=3
Furthermore, I am intrigued to know the answer to:
a!=π
find a
I would really appreciate if anyone could explain this to me as I have found nowhere online with a good explanation of inverse factorial functions.
Thanks
factorial inverse-function
asked Nov 14 '18 at 12:18
Cameron Gray
61
1
See math.stackexchange.com/questions/931846/… and ams.org/journals/proc/2012-140-04/S0002-9939-2011-11023-2
– lhf
Nov 14 '18 at 12:21
Can you elaborate on what you want out of this? The most popular continuous version of factorial is the Gamma function, but I don't know if "ask a calculator to approximate the solution to $Gamma(x)=pi$ and add $1$ to the answer" is the sort of thing you're looking for. For detailed strategies for approximating answers, maybe see math.stackexchange.com/a/2739498/26369
– Mark S.
Nov 14 '18 at 12:29
No integer's factorial is $pi$ the only thing you have seen is $frac{1}{2}!=pi$ which is not true. See definition of Gamma function : en.wikipedia.org/wiki/Gamma_function and see what happened when we put $frac{1}{2}$
– Sujit Bhattacharyya
Nov 14 '18 at 12:33
also refer to math.stackexchange.com/questions/1624347
– G Cab
Nov 14 '18 at 12:53
add a comment |
I am wondering what is the inverse/opposite factorial function?
e.g inverse-factorial(6)=3
Furthermore, I am intrigued to know the answer to:
a!=π
find a
I would really appreciate if anyone could explain this to me as I have found nowhere online with a good explanation of inverse factorial functions.
Thanks
factorial inverse-function
asked Nov 14 '18 at 12:18
Cameron Gray
61
I am wondering what is the inverse/opposite factorial function?
e.g inverse-factorial(6)=3
Furthermore, I am intrigued to know the answer to:
a!=π
find a
I would really appreciate if anyone could explain this to me as I have found nowhere online with a good explanation of inverse factorial functions.
Thanks
factorial inverse-function
factorial inverse-function
asked Nov 14 '18 at 12:18
Cameron Gray
61
asked Nov 14 '18 at 12:18
Cameron Gray
61
asked Nov 14 '18 at 12:18
Cameron Gray
61
asked Nov 14 '18 at 12:18
Cameron Gray
61
asked Nov 14 '18 at 12:18
Cameron Gray
61
61
1
See math.stackexchange.com/questions/931846/… and ams.org/journals/proc/2012-140-04/S0002-9939-2011-11023-2
– lhf
Nov 14 '18 at 12:21
Can you elaborate on what you want out of this? The most popular continuous version of factorial is the Gamma function, but I don't know if "ask a calculator to approximate the solution to $Gamma(x)=pi$ and add $1$ to the answer" is the sort of thing you're looking for. For detailed strategies for approximating answers, maybe see math.stackexchange.com/a/2739498/26369
– Mark S.
Nov 14 '18 at 12:29
No integer's factorial is $pi$ the only thing you have seen is $frac{1}{2}!=pi$ which is not true. See definition of Gamma function : en.wikipedia.org/wiki/Gamma_function and see what happened when we put $frac{1}{2}$
– Sujit Bhattacharyya
Nov 14 '18 at 12:33
also refer to math.stackexchange.com/questions/1624347
– G Cab
Nov 14 '18 at 12:53
add a comment |
1
See math.stackexchange.com/questions/931846/… and ams.org/journals/proc/2012-140-04/S0002-9939-2011-11023-2
– lhf
Nov 14 '18 at 12:21
Can you elaborate on what you want out of this? The most popular continuous version of factorial is the Gamma function, but I don't know if "ask a calculator to approximate the solution to $Gamma(x)=pi$ and add $1$ to the answer" is the sort of thing you're looking for. For detailed strategies for approximating answers, maybe see math.stackexchange.com/a/2739498/26369
– Mark S.
Nov 14 '18 at 12:29
No integer's factorial is $pi$ the only thing you have seen is $frac{1}{2}!=pi$ which is not true. See definition of Gamma function : en.wikipedia.org/wiki/Gamma_function and see what happened when we put $frac{1}{2}$
– Sujit Bhattacharyya
Nov 14 '18 at 12:33
also refer to math.stackexchange.com/questions/1624347
– G Cab
Nov 14 '18 at 12:53
1
1
See math.stackexchange.com/questions/931846/… and ams.org/journals/proc/2012-140-04/S0002-9939-2011-11023-2
– lhf
Nov 14 '18 at 12:21
See math.stackexchange.com/questions/931846/… and ams.org/journals/proc/2012-140-04/S0002-9939-2011-11023-2
– lhf
Nov 14 '18 at 12:21
Can you elaborate on what you want out of this? The most popular continuous version of factorial is the Gamma function, but I don't know if "ask a calculator to approximate the solution to $Gamma(x)=pi$ and add $1$ to the answer" is the sort of thing you're looking for. For detailed strategies for approximating answers, maybe see math.stackexchange.com/a/2739498/26369
– Mark S.
Nov 14 '18 at 12:29
Can you elaborate on what you want out of this? The most popular continuous version of factorial is the Gamma function, but I don't know if "ask a calculator to approximate the solution to $Gamma(x)=pi$ and add $1$ to the answer" is the sort of thing you're looking for. For detailed strategies for approximating answers, maybe see math.stackexchange.com/a/2739498/26369
– Mark S.
Nov 14 '18 at 12:29
No integer's factorial is $pi$ the only thing you have seen is $frac{1}{2}!=pi$ which is not true. See definition of Gamma function : en.wikipedia.org/wiki/Gamma_function and see what happened when we put $frac{1}{2}$
– Sujit Bhattacharyya
Nov 14 '18 at 12:33
No integer's factorial is $pi$ the only thing you have seen is $frac{1}{2}!=pi$ which is not true. See definition of Gamma function : en.wikipedia.org/wiki/Gamma_function and see what happened when we put $frac{1}{2}$
– Sujit Bhattacharyya
Nov 14 '18 at 12:33
also refer to math.stackexchange.com/questions/1624347
– G Cab
Nov 14 '18 at 12:53
also refer to math.stackexchange.com/questions/1624347
– G Cab
Nov 14 '18 at 12:53
add a comment |
2 Answers
2
active
oldest
votes
Unfortunately there is not a closed form or nice series for the inverse of the factorial (or Gamma function).
First obstacle is that the factorial has a local minimum at $x:;psi(x)=0; to ; x=0.4616..$, so , considering only positive values of the argument, that gives you two values for the inverse.
For an analysis of the problem please refer to this and this papers.
A lighter look is given in this other paper.
Finally an interesting approximated function is given here.
answered Nov 14 '18 at 12:46
G Cab
17.9k31237
add a comment |
inverse functions are not well defined when it is not a $1:1$ function, and as there is a minimum where:
$$Gamma'(z+1)=0$$
that is:
$$int_0^inftypartial_nt^ze^{-t}dt=0$$
which can be numerically estimated.
answered Nov 14 '18 at 13:22
Henry Lee
1,739218
add a comment |
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2 Answers
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2 Answers
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Unfortunately there is not a closed form or nice series for the inverse of the factorial (or Gamma function).
First obstacle is that the factorial has a local minimum at $x:;psi(x)=0; to ; x=0.4616..$, so , considering only positive values of the argument, that gives you two values for the inverse.
For an analysis of the problem please refer to this and this papers.
A lighter look is given in this other paper.
Finally an interesting approximated function is given here.
answered Nov 14 '18 at 12:46
G Cab
17.9k31237
add a comment |
Unfortunately there is not a closed form or nice series for the inverse of the factorial (or Gamma function).
First obstacle is that the factorial has a local minimum at $x:;psi(x)=0; to ; x=0.4616..$, so , considering only positive values of the argument, that gives you two values for the inverse.
For an analysis of the problem please refer to this and this papers.
A lighter look is given in this other paper.
Finally an interesting approximated function is given here.
answered Nov 14 '18 at 12:46
G Cab
17.9k31237
add a comment |
Unfortunately there is not a closed form or nice series for the inverse of the factorial (or Gamma function).
First obstacle is that the factorial has a local minimum at $x:;psi(x)=0; to ; x=0.4616..$, so , considering only positive values of the argument, that gives you two values for the inverse.
For an analysis of the problem please refer to this and this papers.
A lighter look is given in this other paper.
Finally an interesting approximated function is given here.
answered Nov 14 '18 at 12:46
G Cab
17.9k31237
Unfortunately there is not a closed form or nice series for the inverse of the factorial (or Gamma function).
First obstacle is that the factorial has a local minimum at $x:;psi(x)=0; to ; x=0.4616..$, so , considering only positive values of the argument, that gives you two values for the inverse.
For an analysis of the problem please refer to this and this papers.
A lighter look is given in this other paper.
Finally an interesting approximated function is given here.
answered Nov 14 '18 at 12:46
G Cab
17.9k31237
answered Nov 14 '18 at 12:46
G Cab
17.9k31237
answered Nov 14 '18 at 12:46
G Cab
17.9k31237
answered Nov 14 '18 at 12:46
G Cab
17.9k31237
17.9k31237
add a comment |
add a comment |
inverse functions are not well defined when it is not a $1:1$ function, and as there is a minimum where:
$$Gamma'(z+1)=0$$
that is:
$$int_0^inftypartial_nt^ze^{-t}dt=0$$
which can be numerically estimated.
answered Nov 14 '18 at 13:22
Henry Lee
1,739218
add a comment |
inverse functions are not well defined when it is not a $1:1$ function, and as there is a minimum where:
$$Gamma'(z+1)=0$$
that is:
$$int_0^inftypartial_nt^ze^{-t}dt=0$$
which can be numerically estimated.
answered Nov 14 '18 at 13:22
Henry Lee
1,739218
add a comment |
inverse functions are not well defined when it is not a $1:1$ function, and as there is a minimum where:
$$Gamma'(z+1)=0$$
that is:
$$int_0^inftypartial_nt^ze^{-t}dt=0$$
which can be numerically estimated.
answered Nov 14 '18 at 13:22
Henry Lee
1,739218
inverse functions are not well defined when it is not a $1:1$ function, and as there is a minimum where:
$$Gamma'(z+1)=0$$
that is:
$$int_0^inftypartial_nt^ze^{-t}dt=0$$
which can be numerically estimated.
answered Nov 14 '18 at 13:22
Henry Lee
1,739218
answered Nov 14 '18 at 13:22
Henry Lee
1,739218
answered Nov 14 '18 at 13:22
Henry Lee
1,739218
answered Nov 14 '18 at 13:22
Henry Lee
1,739218
1,739218
add a comment |
add a comment |
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1
See math.stackexchange.com/questions/931846/… and ams.org/journals/proc/2012-140-04/S0002-9939-2011-11023-2
– lhf
Nov 14 '18 at 12:21
Can you elaborate on what you want out of this? The most popular continuous version of factorial is the Gamma function, but I don't know if "ask a calculator to approximate the solution to $Gamma(x)=pi$ and add $1$ to the answer" is the sort of thing you're looking for. For detailed strategies for approximating answers, maybe see math.stackexchange.com/a/2739498/26369
– Mark S.
Nov 14 '18 at 12:29
No integer's factorial is $pi$ the only thing you have seen is $frac{1}{2}!=pi$ which is not true. See definition of Gamma function : en.wikipedia.org/wiki/Gamma_function and see what happened when we put $frac{1}{2}$
– Sujit Bhattacharyya
Nov 14 '18 at 12:33
also refer to math.stackexchange.com/questions/1624347
– G Cab
Nov 14 '18 at 12:53