Mixing
Proving a Problem has a Closed Form Solution
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I have been working on how deduce the radius of a circle based only on knowing the length of a chord within the circle and the area of the segment the chord creates. This restricts the radius to only one possibility but I can't seem to find a closed form solution for finding the radius using the given information. I am not interested in the answer to the question but I am interested in how one would go about proving whether or not this problem and others like it have a closed form solution. What field should I be looking in to or papers should I be reading in order to work on proving whether or not this problem and others like it have a closed form solution or not?
geometry
asked Apr 3 '13 at 2:47
Jesse SternJesse Stern
9117
$endgroup$
add a comment |
$begingroup$
I have been working on how deduce the radius of a circle based only on knowing the length of a chord within the circle and the area of the segment the chord creates. This restricts the radius to only one possibility but I can't seem to find a closed form solution for finding the radius using the given information. I am not interested in the answer to the question but I am interested in how one would go about proving whether or not this problem and others like it have a closed form solution. What field should I be looking in to or papers should I be reading in order to work on proving whether or not this problem and others like it have a closed form solution or not?
geometry
asked Apr 3 '13 at 2:47
Jesse SternJesse Stern
9117
$endgroup$
$begingroup$
@bryan Is calculus being used to prove it is solvable or not solvable?
$endgroup$
– Jesse Stern
Apr 3 '13 at 3:05
$begingroup$
I would consider that a closed form solution. I would be interested to see how you found the answer so quickly. Would you post the solution as a comment or as an answer?
$endgroup$
– Jesse Stern
Apr 3 '13 at 3:16
$begingroup$
Naturally I do not know the exact way you are going to approach it but I have tried both the calculus and trig approach and they have both yielded formulas that cannot be simplified. I think if you actually try it you may find this to be the case with your approach
$endgroup$
– Jesse Stern
Apr 3 '13 at 3:30
$begingroup$
I take that back: as you suggest, I didn’t look quite far enough ahead.
$endgroup$
– Brian M. Scott
Apr 3 '13 at 3:39
add a comment |
$begingroup$
I have been working on how deduce the radius of a circle based only on knowing the length of a chord within the circle and the area of the segment the chord creates. This restricts the radius to only one possibility but I can't seem to find a closed form solution for finding the radius using the given information. I am not interested in the answer to the question but I am interested in how one would go about proving whether or not this problem and others like it have a closed form solution. What field should I be looking in to or papers should I be reading in order to work on proving whether or not this problem and others like it have a closed form solution or not?
geometry
asked Apr 3 '13 at 2:47
Jesse SternJesse Stern
9117
$endgroup$
I have been working on how deduce the radius of a circle based only on knowing the length of a chord within the circle and the area of the segment the chord creates. This restricts the radius to only one possibility but I can't seem to find a closed form solution for finding the radius using the given information. I am not interested in the answer to the question but I am interested in how one would go about proving whether or not this problem and others like it have a closed form solution. What field should I be looking in to or papers should I be reading in order to work on proving whether or not this problem and others like it have a closed form solution or not?
geometry
geometry
asked Apr 3 '13 at 2:47
Jesse SternJesse Stern
9117
asked Apr 3 '13 at 2:47
Jesse SternJesse Stern
9117
asked Apr 3 '13 at 2:47
Jesse SternJesse Stern
9117
asked Apr 3 '13 at 2:47
Jesse SternJesse Stern
9117
asked Apr 3 '13 at 2:47
Jesse SternJesse Stern
9117
9117
$begingroup$
@bryan Is calculus being used to prove it is solvable or not solvable?
$endgroup$
– Jesse Stern
Apr 3 '13 at 3:05
$begingroup$
I would consider that a closed form solution. I would be interested to see how you found the answer so quickly. Would you post the solution as a comment or as an answer?
$endgroup$
– Jesse Stern
Apr 3 '13 at 3:16
$begingroup$
Naturally I do not know the exact way you are going to approach it but I have tried both the calculus and trig approach and they have both yielded formulas that cannot be simplified. I think if you actually try it you may find this to be the case with your approach
$endgroup$
– Jesse Stern
Apr 3 '13 at 3:30
$begingroup$
I take that back: as you suggest, I didn’t look quite far enough ahead.
$endgroup$
– Brian M. Scott
Apr 3 '13 at 3:39
add a comment |
$begingroup$
@bryan Is calculus being used to prove it is solvable or not solvable?
$endgroup$
– Jesse Stern
Apr 3 '13 at 3:05
$begingroup$
I would consider that a closed form solution. I would be interested to see how you found the answer so quickly. Would you post the solution as a comment or as an answer?
$endgroup$
– Jesse Stern
Apr 3 '13 at 3:16
$begingroup$
Naturally I do not know the exact way you are going to approach it but I have tried both the calculus and trig approach and they have both yielded formulas that cannot be simplified. I think if you actually try it you may find this to be the case with your approach
$endgroup$
– Jesse Stern
Apr 3 '13 at 3:30
$begingroup$
I take that back: as you suggest, I didn’t look quite far enough ahead.
$endgroup$
– Brian M. Scott
Apr 3 '13 at 3:39
$begingroup$
@bryan Is calculus being used to prove it is solvable or not solvable?
$endgroup$
– Jesse Stern
Apr 3 '13 at 3:05
$begingroup$
@bryan Is calculus being used to prove it is solvable or not solvable?
$endgroup$
– Jesse Stern
Apr 3 '13 at 3:05
$begingroup$
I would consider that a closed form solution. I would be interested to see how you found the answer so quickly. Would you post the solution as a comment or as an answer?
$endgroup$
– Jesse Stern
Apr 3 '13 at 3:16
$begingroup$
I would consider that a closed form solution. I would be interested to see how you found the answer so quickly. Would you post the solution as a comment or as an answer?
$endgroup$
– Jesse Stern
Apr 3 '13 at 3:16
$begingroup$
Naturally I do not know the exact way you are going to approach it but I have tried both the calculus and trig approach and they have both yielded formulas that cannot be simplified. I think if you actually try it you may find this to be the case with your approach
$endgroup$
– Jesse Stern
Apr 3 '13 at 3:30
$begingroup$
Naturally I do not know the exact way you are going to approach it but I have tried both the calculus and trig approach and they have both yielded formulas that cannot be simplified. I think if you actually try it you may find this to be the case with your approach
$endgroup$
– Jesse Stern
Apr 3 '13 at 3:30
$begingroup$
I take that back: as you suggest, I didn’t look quite far enough ahead.
$endgroup$
– Brian M. Scott
Apr 3 '13 at 3:39
$begingroup$
I take that back: as you suggest, I didn’t look quite far enough ahead.
$endgroup$
– Brian M. Scott
Apr 3 '13 at 3:39
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
You are asking for a general method for solving a given equation in closed form. There is no comprehensive method known. Here are some references for the Elementary functions.
The problem of existence of elementary inverses of elementary functions is solved in Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90 and in Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J.
Math. 101 (1979) (4) 743-759.
The question for closed-form solutions of equations that are values of elementary functions is asked in Chow, T. Y.: What is a Closed-Form Number? Amer. Math. Monthly 106 (1999) (5) 440-448.
The problem of solving a given ordinary equation of some kind in a differential field (e.g. in the Elementary functions or in the Liouvillian functions) is solved in Rosenlicht, M.: On the explicit solvability of certain transcendental equations. Publications mathématiques de l'IHÉS 36 (1969) 15-22.
answered Nov 16 '17 at 22:34
IV_IV_
1,138522
$endgroup$
add a comment |
$begingroup$
It is hard to make a general statement to this effect. In your case, without writing down equations, I can already tell that your equations have a combination of $sin{theta}$ and $theta$, which rarely produces closed form solutions. Then again, some equations can surprise: today, someone posted an awful equation to which some very bright person deduced a closed-form solution.
One result that may have some relevance is Liouville's Theorem which states what kind of functions have closed-form antiderivatives. Other than that, I know of no general statement on what may be solved with a closed form.
edited Apr 13 '17 at 12:21
Community♦
1
answered Apr 3 '13 at 2:57
Ron GordonRon Gordon
122k14154263
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add a comment |
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2 Answers
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2 Answers
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oldest
votes
$begingroup$
You are asking for a general method for solving a given equation in closed form. There is no comprehensive method known. Here are some references for the Elementary functions.
The problem of existence of elementary inverses of elementary functions is solved in Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90 and in Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J.
Math. 101 (1979) (4) 743-759.
The question for closed-form solutions of equations that are values of elementary functions is asked in Chow, T. Y.: What is a Closed-Form Number? Amer. Math. Monthly 106 (1999) (5) 440-448.
The problem of solving a given ordinary equation of some kind in a differential field (e.g. in the Elementary functions or in the Liouvillian functions) is solved in Rosenlicht, M.: On the explicit solvability of certain transcendental equations. Publications mathématiques de l'IHÉS 36 (1969) 15-22.
answered Nov 16 '17 at 22:34
IV_IV_
1,138522
$endgroup$
add a comment |
$begingroup$
You are asking for a general method for solving a given equation in closed form. There is no comprehensive method known. Here are some references for the Elementary functions.
The problem of existence of elementary inverses of elementary functions is solved in Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90 and in Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J.
Math. 101 (1979) (4) 743-759.
The question for closed-form solutions of equations that are values of elementary functions is asked in Chow, T. Y.: What is a Closed-Form Number? Amer. Math. Monthly 106 (1999) (5) 440-448.
The problem of solving a given ordinary equation of some kind in a differential field (e.g. in the Elementary functions or in the Liouvillian functions) is solved in Rosenlicht, M.: On the explicit solvability of certain transcendental equations. Publications mathématiques de l'IHÉS 36 (1969) 15-22.
answered Nov 16 '17 at 22:34
IV_IV_
1,138522
$endgroup$
add a comment |
$begingroup$
You are asking for a general method for solving a given equation in closed form. There is no comprehensive method known. Here are some references for the Elementary functions.
The problem of existence of elementary inverses of elementary functions is solved in Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90 and in Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J.
Math. 101 (1979) (4) 743-759.
The question for closed-form solutions of equations that are values of elementary functions is asked in Chow, T. Y.: What is a Closed-Form Number? Amer. Math. Monthly 106 (1999) (5) 440-448.
The problem of solving a given ordinary equation of some kind in a differential field (e.g. in the Elementary functions or in the Liouvillian functions) is solved in Rosenlicht, M.: On the explicit solvability of certain transcendental equations. Publications mathématiques de l'IHÉS 36 (1969) 15-22.
answered Nov 16 '17 at 22:34
IV_IV_
1,138522
$endgroup$
You are asking for a general method for solving a given equation in closed form. There is no comprehensive method known. Here are some references for the Elementary functions.
The problem of existence of elementary inverses of elementary functions is solved in Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90 and in Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J.
Math. 101 (1979) (4) 743-759.
The question for closed-form solutions of equations that are values of elementary functions is asked in Chow, T. Y.: What is a Closed-Form Number? Amer. Math. Monthly 106 (1999) (5) 440-448.
The problem of solving a given ordinary equation of some kind in a differential field (e.g. in the Elementary functions or in the Liouvillian functions) is solved in Rosenlicht, M.: On the explicit solvability of certain transcendental equations. Publications mathématiques de l'IHÉS 36 (1969) 15-22.
answered Nov 16 '17 at 22:34
IV_IV_
1,138522
edited Jan 1 at 15:54
answered Nov 16 '17 at 22:34
IV_IV_
1,138522
answered Nov 16 '17 at 22:34
IV_IV_
1,138522
answered Nov 16 '17 at 22:34
IV_IV_
1,138522
1,138522
add a comment |
add a comment |
$begingroup$
It is hard to make a general statement to this effect. In your case, without writing down equations, I can already tell that your equations have a combination of $sin{theta}$ and $theta$, which rarely produces closed form solutions. Then again, some equations can surprise: today, someone posted an awful equation to which some very bright person deduced a closed-form solution.
One result that may have some relevance is Liouville's Theorem which states what kind of functions have closed-form antiderivatives. Other than that, I know of no general statement on what may be solved with a closed form.
edited Apr 13 '17 at 12:21
Community♦
1
answered Apr 3 '13 at 2:57
Ron GordonRon Gordon
122k14154263
$endgroup$
add a comment |
$begingroup$
It is hard to make a general statement to this effect. In your case, without writing down equations, I can already tell that your equations have a combination of $sin{theta}$ and $theta$, which rarely produces closed form solutions. Then again, some equations can surprise: today, someone posted an awful equation to which some very bright person deduced a closed-form solution.
One result that may have some relevance is Liouville's Theorem which states what kind of functions have closed-form antiderivatives. Other than that, I know of no general statement on what may be solved with a closed form.
edited Apr 13 '17 at 12:21
Community♦
1
answered Apr 3 '13 at 2:57
Ron GordonRon Gordon
122k14154263
$endgroup$
add a comment |
$begingroup$
It is hard to make a general statement to this effect. In your case, without writing down equations, I can already tell that your equations have a combination of $sin{theta}$ and $theta$, which rarely produces closed form solutions. Then again, some equations can surprise: today, someone posted an awful equation to which some very bright person deduced a closed-form solution.
One result that may have some relevance is Liouville's Theorem which states what kind of functions have closed-form antiderivatives. Other than that, I know of no general statement on what may be solved with a closed form.
edited Apr 13 '17 at 12:21
Community♦
1
answered Apr 3 '13 at 2:57
Ron GordonRon Gordon
122k14154263
$endgroup$
It is hard to make a general statement to this effect. In your case, without writing down equations, I can already tell that your equations have a combination of $sin{theta}$ and $theta$, which rarely produces closed form solutions. Then again, some equations can surprise: today, someone posted an awful equation to which some very bright person deduced a closed-form solution.
One result that may have some relevance is Liouville's Theorem which states what kind of functions have closed-form antiderivatives. Other than that, I know of no general statement on what may be solved with a closed form.
edited Apr 13 '17 at 12:21
Community♦
1
answered Apr 3 '13 at 2:57
Ron GordonRon Gordon
122k14154263
edited Apr 13 '17 at 12:21
Community♦
1
edited Apr 13 '17 at 12:21
Community♦
1
edited Apr 13 '17 at 12:21
Community♦
1
1
answered Apr 3 '13 at 2:57
Ron GordonRon Gordon
122k14154263
answered Apr 3 '13 at 2:57
Ron GordonRon Gordon
122k14154263
answered Apr 3 '13 at 2:57
Ron GordonRon Gordon
122k14154263
122k14154263
add a comment |
add a comment |
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$begingroup$
@bryan Is calculus being used to prove it is solvable or not solvable?
$endgroup$
– Jesse Stern
Apr 3 '13 at 3:05
$begingroup$
I would consider that a closed form solution. I would be interested to see how you found the answer so quickly. Would you post the solution as a comment or as an answer?
$endgroup$
– Jesse Stern
Apr 3 '13 at 3:16
$begingroup$
Naturally I do not know the exact way you are going to approach it but I have tried both the calculus and trig approach and they have both yielded formulas that cannot be simplified. I think if you actually try it you may find this to be the case with your approach
$endgroup$
– Jesse Stern
Apr 3 '13 at 3:30
$begingroup$
I take that back: as you suggest, I didn’t look quite far enough ahead.
$endgroup$
– Brian M. Scott
Apr 3 '13 at 3:39