Mixing
Some infinite series involving hyperbolic functions
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I have been working on a problem in Quantum Field Theory that involves some sums that I haven't been able to get a closed form. They all involve hyperbolic functions:
begin{gather}
sum_{m=1}^{infty}frac{1}{m^3} coth left(frac{mpi}{2x}right),,nonumber\
sum_{m=1}^{infty}frac{1}{m^2} operatorname{csch} ^{2}left(frac{mpi}{2x}right),,\
sum_{m=1}^{infty}frac{1}{m} coth left(frac{mpi}{2x}right) operatorname{csch}^{2}left(frac{mpi}{2x}right)nonumber,.
end{gather}
I have wondered if anyone knows how to evaluate them and could help me out.
Any references that may address the computation of them are very welcome, too.
sequences-and-series hyperbolic-functions
edited Jan 19 at 23:52
AEngineer
1,5441317
asked Jan 19 at 23:41
HeitorGalacianHeitorGalacian
113
$endgroup$
add a comment |
$begingroup$
I have been working on a problem in Quantum Field Theory that involves some sums that I haven't been able to get a closed form. They all involve hyperbolic functions:
begin{gather}
sum_{m=1}^{infty}frac{1}{m^3} coth left(frac{mpi}{2x}right),,nonumber\
sum_{m=1}^{infty}frac{1}{m^2} operatorname{csch} ^{2}left(frac{mpi}{2x}right),,\
sum_{m=1}^{infty}frac{1}{m} coth left(frac{mpi}{2x}right) operatorname{csch}^{2}left(frac{mpi}{2x}right)nonumber,.
end{gather}
I have wondered if anyone knows how to evaluate them and could help me out.
Any references that may address the computation of them are very welcome, too.
sequences-and-series hyperbolic-functions
edited Jan 19 at 23:52
AEngineer
1,5441317
asked Jan 19 at 23:41
HeitorGalacianHeitorGalacian
113
$endgroup$
$begingroup$
Maybe you could find some luck with replacing the hyperbolic functions with their logarithmic definitions. Best possible case -- you get nice telescoping series. Worst case -- it won't help you, but it's at least better to work with. It is very probable that if the closed form exists, it is in terms of gamma, zeta, and other non-elementary functikns after further analysis
$endgroup$
– KKZiomek
Jan 20 at 0:33
1
$begingroup$
For the first sum, note that the limit as $x$ approaches $0$ is $zeta(3)$. That kind of puts the squash on any hope for an elementary form for that sum.
$endgroup$
– Oscar Lanzi
Jan 20 at 1:35
add a comment |
$begingroup$
I have been working on a problem in Quantum Field Theory that involves some sums that I haven't been able to get a closed form. They all involve hyperbolic functions:
begin{gather}
sum_{m=1}^{infty}frac{1}{m^3} coth left(frac{mpi}{2x}right),,nonumber\
sum_{m=1}^{infty}frac{1}{m^2} operatorname{csch} ^{2}left(frac{mpi}{2x}right),,\
sum_{m=1}^{infty}frac{1}{m} coth left(frac{mpi}{2x}right) operatorname{csch}^{2}left(frac{mpi}{2x}right)nonumber,.
end{gather}
I have wondered if anyone knows how to evaluate them and could help me out.
Any references that may address the computation of them are very welcome, too.
sequences-and-series hyperbolic-functions
edited Jan 19 at 23:52
AEngineer
1,5441317
asked Jan 19 at 23:41
HeitorGalacianHeitorGalacian
113
$endgroup$
I have been working on a problem in Quantum Field Theory that involves some sums that I haven't been able to get a closed form. They all involve hyperbolic functions:
begin{gather}
sum_{m=1}^{infty}frac{1}{m^3} coth left(frac{mpi}{2x}right),,nonumber\
sum_{m=1}^{infty}frac{1}{m^2} operatorname{csch} ^{2}left(frac{mpi}{2x}right),,\
sum_{m=1}^{infty}frac{1}{m} coth left(frac{mpi}{2x}right) operatorname{csch}^{2}left(frac{mpi}{2x}right)nonumber,.
end{gather}
I have wondered if anyone knows how to evaluate them and could help me out.
Any references that may address the computation of them are very welcome, too.
sequences-and-series hyperbolic-functions
sequences-and-series hyperbolic-functions
edited Jan 19 at 23:52
AEngineer
1,5441317
asked Jan 19 at 23:41
HeitorGalacianHeitorGalacian
113
edited Jan 19 at 23:52
AEngineer
1,5441317
asked Jan 19 at 23:41
HeitorGalacianHeitorGalacian
113
edited Jan 19 at 23:52
AEngineer
1,5441317
edited Jan 19 at 23:52
AEngineer
1,5441317
edited Jan 19 at 23:52
AEngineer
1,5441317
1,5441317
asked Jan 19 at 23:41
HeitorGalacianHeitorGalacian
113
asked Jan 19 at 23:41
HeitorGalacianHeitorGalacian
113
asked Jan 19 at 23:41
HeitorGalacianHeitorGalacian
113
113
$begingroup$
Maybe you could find some luck with replacing the hyperbolic functions with their logarithmic definitions. Best possible case -- you get nice telescoping series. Worst case -- it won't help you, but it's at least better to work with. It is very probable that if the closed form exists, it is in terms of gamma, zeta, and other non-elementary functikns after further analysis
$endgroup$
– KKZiomek
Jan 20 at 0:33
1
$begingroup$
For the first sum, note that the limit as $x$ approaches $0$ is $zeta(3)$. That kind of puts the squash on any hope for an elementary form for that sum.
$endgroup$
– Oscar Lanzi
Jan 20 at 1:35
add a comment |
$begingroup$
Maybe you could find some luck with replacing the hyperbolic functions with their logarithmic definitions. Best possible case -- you get nice telescoping series. Worst case -- it won't help you, but it's at least better to work with. It is very probable that if the closed form exists, it is in terms of gamma, zeta, and other non-elementary functikns after further analysis
$endgroup$
– KKZiomek
Jan 20 at 0:33
1
$begingroup$
For the first sum, note that the limit as $x$ approaches $0$ is $zeta(3)$. That kind of puts the squash on any hope for an elementary form for that sum.
$endgroup$
– Oscar Lanzi
Jan 20 at 1:35
$begingroup$
Maybe you could find some luck with replacing the hyperbolic functions with their logarithmic definitions. Best possible case -- you get nice telescoping series. Worst case -- it won't help you, but it's at least better to work with. It is very probable that if the closed form exists, it is in terms of gamma, zeta, and other non-elementary functikns after further analysis
$endgroup$
– KKZiomek
Jan 20 at 0:33
$begingroup$
Maybe you could find some luck with replacing the hyperbolic functions with their logarithmic definitions. Best possible case -- you get nice telescoping series. Worst case -- it won't help you, but it's at least better to work with. It is very probable that if the closed form exists, it is in terms of gamma, zeta, and other non-elementary functikns after further analysis
$endgroup$
– KKZiomek
Jan 20 at 0:33
1
1
$begingroup$
For the first sum, note that the limit as $x$ approaches $0$ is $zeta(3)$. That kind of puts the squash on any hope for an elementary form for that sum.
$endgroup$
– Oscar Lanzi
Jan 20 at 1:35
$begingroup$
For the first sum, note that the limit as $x$ approaches $0$ is $zeta(3)$. That kind of puts the squash on any hope for an elementary form for that sum.
$endgroup$
– Oscar Lanzi
Jan 20 at 1:35
add a comment |
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$begingroup$
Maybe you could find some luck with replacing the hyperbolic functions with their logarithmic definitions. Best possible case -- you get nice telescoping series. Worst case -- it won't help you, but it's at least better to work with. It is very probable that if the closed form exists, it is in terms of gamma, zeta, and other non-elementary functikns after further analysis
$endgroup$
– KKZiomek
Jan 20 at 0:33
1
$begingroup$
For the first sum, note that the limit as $x$ approaches $0$ is $zeta(3)$. That kind of puts the squash on any hope for an elementary form for that sum.
$endgroup$
– Oscar Lanzi
Jan 20 at 1:35