Superconductor specific heat capacity
$begingroup$
I would like to obtain expression for heat capacity jump of superconductor. During calculation, I can not deal with the followiwng integral:
$$int_{0}^{infty}frac{dx}{x^2}left(frac{1}{cosh^2 x}-frac{tanh{x}}{x}right),$$
which comes from the expression for gap near $T_c$. My question is how to evaluate this integral. I tried to calculated residues of
$$f(z)=frac{z-sinh zcosh z}{z^3cosh^2z}$$
and use Cauchy theorem. But it was unsuccesful. The desired answer is $7zeta(3)/(8pi^2)$
statistical-mechanics
asked Dec 5 '18 at 8:37
Artem AlexandrovArtem Alexandrov
263
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migrated from physics.stackexchange.com Dec 6 '18 at 19:57
This question came from our site for active researchers, academics and students of physics.
|
show 1 more comment
$begingroup$
I would like to obtain expression for heat capacity jump of superconductor. During calculation, I can not deal with the followiwng integral:
$$int_{0}^{infty}frac{dx}{x^2}left(frac{1}{cosh^2 x}-frac{tanh{x}}{x}right),$$
which comes from the expression for gap near $T_c$. My question is how to evaluate this integral. I tried to calculated residues of
$$f(z)=frac{z-sinh zcosh z}{z^3cosh^2z}$$
and use Cauchy theorem. But it was unsuccesful. The desired answer is $7zeta(3)/(8pi^2)$
statistical-mechanics
asked Dec 5 '18 at 8:37
Artem AlexandrovArtem Alexandrov
263
$endgroup$
migrated from physics.stackexchange.com Dec 6 '18 at 19:57
This question came from our site for active researchers, academics and students of physics.
1
$begingroup$
Are you sure that this integral is correct? It is divergent at x = 0.
$endgroup$
– Gec
Dec 5 '18 at 8:44
$begingroup$
@Gec hm, I have checked it with Wolfram and there is the finite answer wolfram
$endgroup$
– Artem Alexandrov
Dec 5 '18 at 8:51
$begingroup$
The integral in this topic is different from that given to Wolfram.
$endgroup$
– Gec
Dec 5 '18 at 8:59
$begingroup$
Extending Gec's comment, this is what Wolfram says about the posted integral. Either way, might Mathematics be better suited for the math question?
$endgroup$
– Kyle Kanos
Dec 5 '18 at 11:15
$begingroup$
@Gec sorry! I made mistake in question :(
$endgroup$
– Artem Alexandrov
Dec 6 '18 at 8:27
|
show 1 more comment
$begingroup$
I would like to obtain expression for heat capacity jump of superconductor. During calculation, I can not deal with the followiwng integral:
$$int_{0}^{infty}frac{dx}{x^2}left(frac{1}{cosh^2 x}-frac{tanh{x}}{x}right),$$
which comes from the expression for gap near $T_c$. My question is how to evaluate this integral. I tried to calculated residues of
$$f(z)=frac{z-sinh zcosh z}{z^3cosh^2z}$$
and use Cauchy theorem. But it was unsuccesful. The desired answer is $7zeta(3)/(8pi^2)$
statistical-mechanics
asked Dec 5 '18 at 8:37
Artem AlexandrovArtem Alexandrov
263
$endgroup$
I would like to obtain expression for heat capacity jump of superconductor. During calculation, I can not deal with the followiwng integral:
$$int_{0}^{infty}frac{dx}{x^2}left(frac{1}{cosh^2 x}-frac{tanh{x}}{x}right),$$
which comes from the expression for gap near $T_c$. My question is how to evaluate this integral. I tried to calculated residues of
$$f(z)=frac{z-sinh zcosh z}{z^3cosh^2z}$$
and use Cauchy theorem. But it was unsuccesful. The desired answer is $7zeta(3)/(8pi^2)$
statistical-mechanics
statistical-mechanics
asked Dec 5 '18 at 8:37
Artem AlexandrovArtem Alexandrov
263
asked Dec 5 '18 at 8:37
Artem AlexandrovArtem Alexandrov
263
asked Dec 5 '18 at 8:37
Artem AlexandrovArtem Alexandrov
263
asked Dec 5 '18 at 8:37
Artem AlexandrovArtem Alexandrov
263
asked Dec 5 '18 at 8:37
Artem AlexandrovArtem Alexandrov
263
263
migrated from physics.stackexchange.com Dec 6 '18 at 19:57
This question came from our site for active researchers, academics and students of physics.
migrated from physics.stackexchange.com Dec 6 '18 at 19:57
This question came from our site for active researchers, academics and students of physics.
1
$begingroup$
Are you sure that this integral is correct? It is divergent at x = 0.
$endgroup$
– Gec
Dec 5 '18 at 8:44
$begingroup$
@Gec hm, I have checked it with Wolfram and there is the finite answer wolfram
$endgroup$
– Artem Alexandrov
Dec 5 '18 at 8:51
$begingroup$
The integral in this topic is different from that given to Wolfram.
$endgroup$
– Gec
Dec 5 '18 at 8:59
$begingroup$
Extending Gec's comment, this is what Wolfram says about the posted integral. Either way, might Mathematics be better suited for the math question?
$endgroup$
– Kyle Kanos
Dec 5 '18 at 11:15
$begingroup$
@Gec sorry! I made mistake in question :(
$endgroup$
– Artem Alexandrov
Dec 6 '18 at 8:27
|
show 1 more comment
1
$begingroup$
Are you sure that this integral is correct? It is divergent at x = 0.
$endgroup$
– Gec
Dec 5 '18 at 8:44
$begingroup$
@Gec hm, I have checked it with Wolfram and there is the finite answer wolfram
$endgroup$
– Artem Alexandrov
Dec 5 '18 at 8:51
$begingroup$
The integral in this topic is different from that given to Wolfram.
$endgroup$
– Gec
Dec 5 '18 at 8:59
$begingroup$
Extending Gec's comment, this is what Wolfram says about the posted integral. Either way, might Mathematics be better suited for the math question?
$endgroup$
– Kyle Kanos
Dec 5 '18 at 11:15
$begingroup$
@Gec sorry! I made mistake in question :(
$endgroup$
– Artem Alexandrov
Dec 6 '18 at 8:27
1
1
$begingroup$
Are you sure that this integral is correct? It is divergent at x = 0.
$endgroup$
– Gec
Dec 5 '18 at 8:44
$begingroup$
Are you sure that this integral is correct? It is divergent at x = 0.
$endgroup$
– Gec
Dec 5 '18 at 8:44
$begingroup$
@Gec hm, I have checked it with Wolfram and there is the finite answer wolfram
$endgroup$
– Artem Alexandrov
Dec 5 '18 at 8:51
$begingroup$
@Gec hm, I have checked it with Wolfram and there is the finite answer wolfram
$endgroup$
– Artem Alexandrov
Dec 5 '18 at 8:51
$begingroup$
The integral in this topic is different from that given to Wolfram.
$endgroup$
– Gec
Dec 5 '18 at 8:59
$begingroup$
The integral in this topic is different from that given to Wolfram.
$endgroup$
– Gec
Dec 5 '18 at 8:59
$begingroup$
Extending Gec's comment, this is what Wolfram says about the posted integral. Either way, might Mathematics be better suited for the math question?
$endgroup$
– Kyle Kanos
Dec 5 '18 at 11:15
$begingroup$
Extending Gec's comment, this is what Wolfram says about the posted integral. Either way, might Mathematics be better suited for the math question?
$endgroup$
– Kyle Kanos
Dec 5 '18 at 11:15
$begingroup$
@Gec sorry! I made mistake in question :(
$endgroup$
– Artem Alexandrov
Dec 6 '18 at 8:27
$begingroup$
@Gec sorry! I made mistake in question :(
$endgroup$
– Artem Alexandrov
Dec 6 '18 at 8:27
|
show 1 more comment
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1
$begingroup$
Are you sure that this integral is correct? It is divergent at x = 0.
$endgroup$
– Gec
Dec 5 '18 at 8:44
$begingroup$
@Gec hm, I have checked it with Wolfram and there is the finite answer wolfram
$endgroup$
– Artem Alexandrov
Dec 5 '18 at 8:51
$begingroup$
The integral in this topic is different from that given to Wolfram.
$endgroup$
– Gec
Dec 5 '18 at 8:59
$begingroup$
Extending Gec's comment, this is what Wolfram says about the posted integral. Either way, might Mathematics be better suited for the math question?
$endgroup$
– Kyle Kanos
Dec 5 '18 at 11:15
$begingroup$
@Gec sorry! I made mistake in question :(
$endgroup$
– Artem Alexandrov
Dec 6 '18 at 8:27