What is the infinite product series for $exp(sin(x))-1$?
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$e^{sin(x)}-1$ has the same roots as $sin(x)$. What is the difference between infinite product series expansions of $sin(x)$ and $e^{sin(x)}-1$ if they both have same infinite roots ?
calculus
edited Nov 17 at 15:09
mrtaurho
2,5541827
asked Nov 17 at 15:07
Omid Motahed
364
|
show 1 more comment
up vote
7
down vote
favorite
5
$e^{sin(x)}-1$ has the same roots as $sin(x)$. What is the difference between infinite product series expansions of $sin(x)$ and $e^{sin(x)}-1$ if they both have same infinite roots ?
calculus
edited Nov 17 at 15:09
mrtaurho
2,5541827
asked Nov 17 at 15:07
Omid Motahed
364
1
They may have the same roots, but effectively, sin(x) differs everywhere else in many ways from your function. Simple derivative analysis can show you that fact. They have different critical points, points of inflection, etc.. So, the series will be much different
– KKZiomek
Nov 17 at 15:48
1
I think that this Wikipedia article can give some insight about this...
– rafa11111
Nov 17 at 15:51
If you mean the real variable $x$ then they differ by a function without zeros $frac{e^{sin x}-1}{sin x}$ (compare with $p_1(x)=x-1$ and $p_2(x)=2x-2$). For complex $x$ the first function has more zeros, e.g. a solution to $sin x=2pi i$ is a zero of the first function that is not zero of the second one.
– A.Γ.
Nov 17 at 15:54
Is there a factorization formula for it like math.stackexchange.com/questions/157372/… ?
– Omid Motahed
Nov 17 at 16:30
If $f(z)/g(z)$ is entire and has no zeros then $f(z) = g(z) e^{h(z)}$ with $h(z)$ entire. This is also the way to express any meromorphic function as a series over its poles and any entire function as a product over its zeros
– reuns
Nov 18 at 2:50
|
show 1 more comment
up vote
7
down vote
favorite
5
up vote
7
down vote
favorite
5
5
$e^{sin(x)}-1$ has the same roots as $sin(x)$. What is the difference between infinite product series expansions of $sin(x)$ and $e^{sin(x)}-1$ if they both have same infinite roots ?
calculus
edited Nov 17 at 15:09
mrtaurho
2,5541827
asked Nov 17 at 15:07
Omid Motahed
364
$e^{sin(x)}-1$ has the same roots as $sin(x)$. What is the difference between infinite product series expansions of $sin(x)$ and $e^{sin(x)}-1$ if they both have same infinite roots ?
calculus
calculus
edited Nov 17 at 15:09
mrtaurho
2,5541827
asked Nov 17 at 15:07
Omid Motahed
364
edited Nov 17 at 15:09
mrtaurho
2,5541827
asked Nov 17 at 15:07
Omid Motahed
364
edited Nov 17 at 15:09
mrtaurho
2,5541827
edited Nov 17 at 15:09
mrtaurho
2,5541827
edited Nov 17 at 15:09
mrtaurho
2,5541827
2,5541827
asked Nov 17 at 15:07
Omid Motahed
364
asked Nov 17 at 15:07
Omid Motahed
364
asked Nov 17 at 15:07
Omid Motahed
364
364
1
They may have the same roots, but effectively, sin(x) differs everywhere else in many ways from your function. Simple derivative analysis can show you that fact. They have different critical points, points of inflection, etc.. So, the series will be much different
– KKZiomek
Nov 17 at 15:48
1
I think that this Wikipedia article can give some insight about this...
– rafa11111
Nov 17 at 15:51
If you mean the real variable $x$ then they differ by a function without zeros $frac{e^{sin x}-1}{sin x}$ (compare with $p_1(x)=x-1$ and $p_2(x)=2x-2$). For complex $x$ the first function has more zeros, e.g. a solution to $sin x=2pi i$ is a zero of the first function that is not zero of the second one.
– A.Γ.
Nov 17 at 15:54
Is there a factorization formula for it like math.stackexchange.com/questions/157372/… ?
– Omid Motahed
Nov 17 at 16:30
If $f(z)/g(z)$ is entire and has no zeros then $f(z) = g(z) e^{h(z)}$ with $h(z)$ entire. This is also the way to express any meromorphic function as a series over its poles and any entire function as a product over its zeros
– reuns
Nov 18 at 2:50
|
show 1 more comment
1
They may have the same roots, but effectively, sin(x) differs everywhere else in many ways from your function. Simple derivative analysis can show you that fact. They have different critical points, points of inflection, etc.. So, the series will be much different
– KKZiomek
Nov 17 at 15:48
1
I think that this Wikipedia article can give some insight about this...
– rafa11111
Nov 17 at 15:51
If you mean the real variable $x$ then they differ by a function without zeros $frac{e^{sin x}-1}{sin x}$ (compare with $p_1(x)=x-1$ and $p_2(x)=2x-2$). For complex $x$ the first function has more zeros, e.g. a solution to $sin x=2pi i$ is a zero of the first function that is not zero of the second one.
– A.Γ.
Nov 17 at 15:54
Is there a factorization formula for it like math.stackexchange.com/questions/157372/… ?
– Omid Motahed
Nov 17 at 16:30
If $f(z)/g(z)$ is entire and has no zeros then $f(z) = g(z) e^{h(z)}$ with $h(z)$ entire. This is also the way to express any meromorphic function as a series over its poles and any entire function as a product over its zeros
– reuns
Nov 18 at 2:50
1
1
They may have the same roots, but effectively, sin(x) differs everywhere else in many ways from your function. Simple derivative analysis can show you that fact. They have different critical points, points of inflection, etc.. So, the series will be much different
– KKZiomek
Nov 17 at 15:48
They may have the same roots, but effectively, sin(x) differs everywhere else in many ways from your function. Simple derivative analysis can show you that fact. They have different critical points, points of inflection, etc.. So, the series will be much different
– KKZiomek
Nov 17 at 15:48
1
1
I think that this Wikipedia article can give some insight about this...
– rafa11111
Nov 17 at 15:51
I think that this Wikipedia article can give some insight about this...
– rafa11111
Nov 17 at 15:51
If you mean the real variable $x$ then they differ by a function without zeros $frac{e^{sin x}-1}{sin x}$ (compare with $p_1(x)=x-1$ and $p_2(x)=2x-2$). For complex $x$ the first function has more zeros, e.g. a solution to $sin x=2pi i$ is a zero of the first function that is not zero of the second one.
– A.Γ.
Nov 17 at 15:54
If you mean the real variable $x$ then they differ by a function without zeros $frac{e^{sin x}-1}{sin x}$ (compare with $p_1(x)=x-1$ and $p_2(x)=2x-2$). For complex $x$ the first function has more zeros, e.g. a solution to $sin x=2pi i$ is a zero of the first function that is not zero of the second one.
– A.Γ.
Nov 17 at 15:54
Is there a factorization formula for it like math.stackexchange.com/questions/157372/… ?
– Omid Motahed
Nov 17 at 16:30
Is there a factorization formula for it like math.stackexchange.com/questions/157372/… ?
– Omid Motahed
Nov 17 at 16:30
If $f(z)/g(z)$ is entire and has no zeros then $f(z) = g(z) e^{h(z)}$ with $h(z)$ entire. This is also the way to express any meromorphic function as a series over its poles and any entire function as a product over its zeros
– reuns
Nov 18 at 2:50
If $f(z)/g(z)$ is entire and has no zeros then $f(z) = g(z) e^{h(z)}$ with $h(z)$ entire. This is also the way to express any meromorphic function as a series over its poles and any entire function as a product over its zeros
– reuns
Nov 18 at 2:50
|
show 1 more comment
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1
They may have the same roots, but effectively, sin(x) differs everywhere else in many ways from your function. Simple derivative analysis can show you that fact. They have different critical points, points of inflection, etc.. So, the series will be much different
– KKZiomek
Nov 17 at 15:48
1
I think that this Wikipedia article can give some insight about this...
– rafa11111
Nov 17 at 15:51
If you mean the real variable $x$ then they differ by a function without zeros $frac{e^{sin x}-1}{sin x}$ (compare with $p_1(x)=x-1$ and $p_2(x)=2x-2$). For complex $x$ the first function has more zeros, e.g. a solution to $sin x=2pi i$ is a zero of the first function that is not zero of the second one.
– A.Γ.
Nov 17 at 15:54
Is there a factorization formula for it like math.stackexchange.com/questions/157372/… ?
– Omid Motahed
Nov 17 at 16:30
If $f(z)/g(z)$ is entire and has no zeros then $f(z) = g(z) e^{h(z)}$ with $h(z)$ entire. This is also the way to express any meromorphic function as a series over its poles and any entire function as a product over its zeros
– reuns
Nov 18 at 2:50