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Definition of $sin$ function on vector space
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Can we define $sin$ function on any vector space?
I used the logic for $sin$ define on complex numbers
$sin(x+iy)$ but how to generalize it?
linear-algebra
edited Aug 1 at 7:00
Robert Frost
3,885936
asked Aug 1 at 1:20
S.Chauhan
62
add a comment |Â
up vote
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down vote
favorite
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Can we define $sin$ function on any vector space?
I used the logic for $sin$ define on complex numbers
$sin(x+iy)$ but how to generalize it?
linear-algebra
edited Aug 1 at 7:00
Robert Frost
3,885936
asked Aug 1 at 1:20
S.Chauhan
62
You probably cannot define a sine function on any ol' vector space, but it is likely that you can extend the definition to a reasonable class of fields (or algebras) that possess a complete metric structure. The basic idea would be to work with power series. For example, in $p$-adic land, we can define an exponential function via the formal power series $$ exp_p(x) = sum_j=0^infty fracx^jj!, $$ where $j!$ is the usual factorial of a positive integer, which can then be embedded into $mathbbQ_p$ or $mathbbC_p$ in the "obvious" manner.
– Xander Henderson
Aug 1 at 1:28
Even so, the actual problem is quite a lot more delicate. For example, the above power series does not converge on the entire space. Indeed, it only converges on a disk near zero. Wikipedia gives a reasonable discussion of the problem. I would expect that similar problems of convergence pop up in other settings.
– Xander Henderson
Aug 1 at 1:29
Also, just for your information, I have voted to close this question, as I think that it lacks context. You talk about generalizing the sine function to the complex numbers as $sin(x+iy)$, but you don't explain how you have defined $sin(x+iy)$. Via a power series? Via Euler's formula? Something else? What structures, exactly, are allowing this generalization. I think that there is a really interesting question buried in here somewhere, but I am not sure that the question I'm thinking about is the question that you are trying to ask.
– Xander Henderson
Aug 1 at 1:42
I imagine the answer to this depends on what triangles looks like in the space in question. Where the space isn't Euclidean then I'm pretty sure there would need to exist some map to a Euclidean space because Sine is inherently linked to the concepts of a straight line and a right angle.
– Robert Frost
Aug 1 at 6:59
add a comment |Â
up vote
1
down vote
favorite
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up vote
1
down vote
favorite
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1
Can we define $sin$ function on any vector space?
I used the logic for $sin$ define on complex numbers
$sin(x+iy)$ but how to generalize it?
linear-algebra
edited Aug 1 at 7:00
Robert Frost
3,885936
asked Aug 1 at 1:20
S.Chauhan
62
Can we define $sin$ function on any vector space?
I used the logic for $sin$ define on complex numbers
$sin(x+iy)$ but how to generalize it?
linear-algebra
edited Aug 1 at 7:00
Robert Frost
3,885936
asked Aug 1 at 1:20
S.Chauhan
62
edited Aug 1 at 7:00
Robert Frost
3,885936
edited Aug 1 at 7:00
Robert Frost
3,885936
edited Aug 1 at 7:00
Robert Frost
3,885936
3,885936
asked Aug 1 at 1:20
S.Chauhan
62
asked Aug 1 at 1:20
S.Chauhan
62
asked Aug 1 at 1:20
S.Chauhan
62
62
You probably cannot define a sine function on any ol' vector space, but it is likely that you can extend the definition to a reasonable class of fields (or algebras) that possess a complete metric structure. The basic idea would be to work with power series. For example, in $p$-adic land, we can define an exponential function via the formal power series $$ exp_p(x) = sum_j=0^infty fracx^jj!, $$ where $j!$ is the usual factorial of a positive integer, which can then be embedded into $mathbbQ_p$ or $mathbbC_p$ in the "obvious" manner.
– Xander Henderson
Aug 1 at 1:28
Even so, the actual problem is quite a lot more delicate. For example, the above power series does not converge on the entire space. Indeed, it only converges on a disk near zero. Wikipedia gives a reasonable discussion of the problem. I would expect that similar problems of convergence pop up in other settings.
– Xander Henderson
Aug 1 at 1:29
Also, just for your information, I have voted to close this question, as I think that it lacks context. You talk about generalizing the sine function to the complex numbers as $sin(x+iy)$, but you don't explain how you have defined $sin(x+iy)$. Via a power series? Via Euler's formula? Something else? What structures, exactly, are allowing this generalization. I think that there is a really interesting question buried in here somewhere, but I am not sure that the question I'm thinking about is the question that you are trying to ask.
– Xander Henderson
Aug 1 at 1:42
I imagine the answer to this depends on what triangles looks like in the space in question. Where the space isn't Euclidean then I'm pretty sure there would need to exist some map to a Euclidean space because Sine is inherently linked to the concepts of a straight line and a right angle.
– Robert Frost
Aug 1 at 6:59
add a comment |Â
You probably cannot define a sine function on any ol' vector space, but it is likely that you can extend the definition to a reasonable class of fields (or algebras) that possess a complete metric structure. The basic idea would be to work with power series. For example, in $p$-adic land, we can define an exponential function via the formal power series $$ exp_p(x) = sum_j=0^infty fracx^jj!, $$ where $j!$ is the usual factorial of a positive integer, which can then be embedded into $mathbbQ_p$ or $mathbbC_p$ in the "obvious" manner.
– Xander Henderson
Aug 1 at 1:28
Even so, the actual problem is quite a lot more delicate. For example, the above power series does not converge on the entire space. Indeed, it only converges on a disk near zero. Wikipedia gives a reasonable discussion of the problem. I would expect that similar problems of convergence pop up in other settings.
– Xander Henderson
Aug 1 at 1:29
Also, just for your information, I have voted to close this question, as I think that it lacks context. You talk about generalizing the sine function to the complex numbers as $sin(x+iy)$, but you don't explain how you have defined $sin(x+iy)$. Via a power series? Via Euler's formula? Something else? What structures, exactly, are allowing this generalization. I think that there is a really interesting question buried in here somewhere, but I am not sure that the question I'm thinking about is the question that you are trying to ask.
– Xander Henderson
Aug 1 at 1:42
I imagine the answer to this depends on what triangles looks like in the space in question. Where the space isn't Euclidean then I'm pretty sure there would need to exist some map to a Euclidean space because Sine is inherently linked to the concepts of a straight line and a right angle.
– Robert Frost
Aug 1 at 6:59
You probably cannot define a sine function on any ol' vector space, but it is likely that you can extend the definition to a reasonable class of fields (or algebras) that possess a complete metric structure. The basic idea would be to work with power series. For example, in $p$-adic land, we can define an exponential function via the formal power series $$ exp_p(x) = sum_j=0^infty fracx^jj!, $$ where $j!$ is the usual factorial of a positive integer, which can then be embedded into $mathbbQ_p$ or $mathbbC_p$ in the "obvious" manner.
– Xander Henderson
Aug 1 at 1:28
You probably cannot define a sine function on any ol' vector space, but it is likely that you can extend the definition to a reasonable class of fields (or algebras) that possess a complete metric structure. The basic idea would be to work with power series. For example, in $p$-adic land, we can define an exponential function via the formal power series $$ exp_p(x) = sum_j=0^infty fracx^jj!, $$ where $j!$ is the usual factorial of a positive integer, which can then be embedded into $mathbbQ_p$ or $mathbbC_p$ in the "obvious" manner.
– Xander Henderson
Aug 1 at 1:28
Even so, the actual problem is quite a lot more delicate. For example, the above power series does not converge on the entire space. Indeed, it only converges on a disk near zero. Wikipedia gives a reasonable discussion of the problem. I would expect that similar problems of convergence pop up in other settings.
– Xander Henderson
Aug 1 at 1:29
Even so, the actual problem is quite a lot more delicate. For example, the above power series does not converge on the entire space. Indeed, it only converges on a disk near zero. Wikipedia gives a reasonable discussion of the problem. I would expect that similar problems of convergence pop up in other settings.
– Xander Henderson
Aug 1 at 1:29
Also, just for your information, I have voted to close this question, as I think that it lacks context. You talk about generalizing the sine function to the complex numbers as $sin(x+iy)$, but you don't explain how you have defined $sin(x+iy)$. Via a power series? Via Euler's formula? Something else? What structures, exactly, are allowing this generalization. I think that there is a really interesting question buried in here somewhere, but I am not sure that the question I'm thinking about is the question that you are trying to ask.
– Xander Henderson
Aug 1 at 1:42
Also, just for your information, I have voted to close this question, as I think that it lacks context. You talk about generalizing the sine function to the complex numbers as $sin(x+iy)$, but you don't explain how you have defined $sin(x+iy)$. Via a power series? Via Euler's formula? Something else? What structures, exactly, are allowing this generalization. I think that there is a really interesting question buried in here somewhere, but I am not sure that the question I'm thinking about is the question that you are trying to ask.
– Xander Henderson
Aug 1 at 1:42
I imagine the answer to this depends on what triangles looks like in the space in question. Where the space isn't Euclidean then I'm pretty sure there would need to exist some map to a Euclidean space because Sine is inherently linked to the concepts of a straight line and a right angle.
– Robert Frost
Aug 1 at 6:59
I imagine the answer to this depends on what triangles looks like in the space in question. Where the space isn't Euclidean then I'm pretty sure there would need to exist some map to a Euclidean space because Sine is inherently linked to the concepts of a straight line and a right angle.
– Robert Frost
Aug 1 at 6:59
add a comment |Â
1 Answer
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Notice that in the complex numbers we can multiply elements of our vector space. That, together with a notion of convergence, allows you to use the standard Taylor series for $sin(z)$:
$$
sin(z) = sum_i=0^infty (-1)^i fracz^2i+1(2i+1)!.
$$
There are many other algebras (vector spaces with a natural multiplication) where we have a notion of convergence. These are the settings in which we expect to have a useful notion of $sin(v)$.
answered Aug 1 at 1:29
Jamie Radcliffe
33625
add a comment |Â
1 Answer
1
active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Notice that in the complex numbers we can multiply elements of our vector space. That, together with a notion of convergence, allows you to use the standard Taylor series for $sin(z)$:
$$
sin(z) = sum_i=0^infty (-1)^i fracz^2i+1(2i+1)!.
$$
There are many other algebras (vector spaces with a natural multiplication) where we have a notion of convergence. These are the settings in which we expect to have a useful notion of $sin(v)$.
answered Aug 1 at 1:29
Jamie Radcliffe
33625
add a comment |Â
up vote
1
down vote
Notice that in the complex numbers we can multiply elements of our vector space. That, together with a notion of convergence, allows you to use the standard Taylor series for $sin(z)$:
$$
sin(z) = sum_i=0^infty (-1)^i fracz^2i+1(2i+1)!.
$$
There are many other algebras (vector spaces with a natural multiplication) where we have a notion of convergence. These are the settings in which we expect to have a useful notion of $sin(v)$.
answered Aug 1 at 1:29
Jamie Radcliffe
33625
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Notice that in the complex numbers we can multiply elements of our vector space. That, together with a notion of convergence, allows you to use the standard Taylor series for $sin(z)$:
$$
sin(z) = sum_i=0^infty (-1)^i fracz^2i+1(2i+1)!.
$$
There are many other algebras (vector spaces with a natural multiplication) where we have a notion of convergence. These are the settings in which we expect to have a useful notion of $sin(v)$.
answered Aug 1 at 1:29
Jamie Radcliffe
33625
Notice that in the complex numbers we can multiply elements of our vector space. That, together with a notion of convergence, allows you to use the standard Taylor series for $sin(z)$:
$$
sin(z) = sum_i=0^infty (-1)^i fracz^2i+1(2i+1)!.
$$
There are many other algebras (vector spaces with a natural multiplication) where we have a notion of convergence. These are the settings in which we expect to have a useful notion of $sin(v)$.
answered Aug 1 at 1:29
Jamie Radcliffe
33625
answered Aug 1 at 1:29
Jamie Radcliffe
33625
answered Aug 1 at 1:29
Jamie Radcliffe
33625
answered Aug 1 at 1:29
Jamie Radcliffe
33625
33625
add a comment |Â
add a comment |Â
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You probably cannot define a sine function on any ol' vector space, but it is likely that you can extend the definition to a reasonable class of fields (or algebras) that possess a complete metric structure. The basic idea would be to work with power series. For example, in $p$-adic land, we can define an exponential function via the formal power series $$ exp_p(x) = sum_j=0^infty fracx^jj!, $$ where $j!$ is the usual factorial of a positive integer, which can then be embedded into $mathbbQ_p$ or $mathbbC_p$ in the "obvious" manner.
– Xander Henderson
Aug 1 at 1:28
Even so, the actual problem is quite a lot more delicate. For example, the above power series does not converge on the entire space. Indeed, it only converges on a disk near zero. Wikipedia gives a reasonable discussion of the problem. I would expect that similar problems of convergence pop up in other settings.
– Xander Henderson
Aug 1 at 1:29
Also, just for your information, I have voted to close this question, as I think that it lacks context. You talk about generalizing the sine function to the complex numbers as $sin(x+iy)$, but you don't explain how you have defined $sin(x+iy)$. Via a power series? Via Euler's formula? Something else? What structures, exactly, are allowing this generalization. I think that there is a really interesting question buried in here somewhere, but I am not sure that the question I'm thinking about is the question that you are trying to ask.
– Xander Henderson
Aug 1 at 1:42
I imagine the answer to this depends on what triangles looks like in the space in question. Where the space isn't Euclidean then I'm pretty sure there would need to exist some map to a Euclidean space because Sine is inherently linked to the concepts of a straight line and a right angle.
– Robert Frost
Aug 1 at 6:59