Stumped on mathematical proof

AI Thread Summary
The discussion centers on a mathematical proof from Blundell and Blundell's textbook, specifically regarding the identity involving the ##\coth## function. The initial proof of $$1-e^{-\beta \omega}=2\sinh\left(\frac{\beta \omega}{2}\right)$$ is confirmed, but confusion arises over a subsequent equation, $$e^{\beta \omega} + 1 = 2\beta$$, which participants argue cannot hold for all values of ##\beta## and ##\omega##. It is suggested that there is a missing factor of ##\beta## in the textbook's expression, leading to further verification of the identities. Ultimately, the participants agree that the overall identity appears reasonable, despite the textbook's potential oversight. The conversation emphasizes the importance of critically evaluating textbook claims in mathematical proofs.
laser1
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From Blundell and Blundell Chapter 20 Problem 20.3.

I have proved that $$1-e^{-\beta \omega}=2\sinh\left(\frac{\beta \omega}{2}\right)$$ with no problem, but I am stuck on the ##\coth## term. I have tried to solve this but it gets messy and I'd rather not include them here. Thanks!
 
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If that ##\coth## identity holds, then:
$$e^{\beta \omega} + 1 = 2\beta$$
 
PeroK said:
If that ##\coth## identity holds, then:
$$e^{\beta \omega} + 1 = 2\beta$$
Okay, assuming that is true, surely that equation can't be true for all ##\beta## and ##\omega##?
 
laser1 said:
Okay, assuming that is true, surely that equation can't be true for all ##\beta## and ##\omega##?
That is clearly not an identity. Why do you think the textbook is infallible?
 
PeroK said:
That is clearly not an identity. Why do you think the textbook is infallible?
Alr fair enough. As a mere undergrad, I always assume I am wrong first rather than the textbook! Of course, if I can't reason with the textbook I will ask here/my lecturer.
 
laser1 said:
Alr fair enough. As a mere undergrad, I always assume I am wrong first rather than the textbook! Of course, if I can't reason with the textbook I will ask here/my lecturer.
It's not a question of assumptions. Are ##\beta## and ##\omega## related in that way?
 
PeroK said:
It's not a question of assumptions. Are ##\beta## and ##\omega## related in that way?
I don't think so.
 
laser1 said:
I don't think so.
Perhaps check that first identity!
 
laser1 said:
I have proved that $$1-e^{-\beta \omega}=2\sinh\left(\frac{\beta \omega}{2}\right)$$ with no problem,
That's the mistake!
 
  • #10
PeroK said:
That's the mistake!
WhatsApp Image 2024-10-26 at 10.28.54.jpeg



Edit: oh yeah I see the denominator now yikes
 
  • #12
PeroK said:
It's not a question of assumptions. Are ##\beta## and ##\omega## related in that way?
The expression just looked wrong to me. There is a factor of ##\beta## missing in (20.51). It should be:
$$\frac{\beta \omega}{2}\coth\big (\frac{\beta \omega}{2}\big )$$
 
  • #13
PeroK said:
The expression just looked wrong to me. There is a factor of ##\beta## missing in (20.51). It should be:
$$\frac{\beta \omega}{2}\coth\big (\frac{\beta \omega}{2}\big )$$
yeah I'm getting the same as you now. As in, I have the book answer, but the book is missing a factor of ##\beta## on the first term
 
  • #14
I have not done the explicit computation (just visualized how the terms would combine), but apart from the missing ##\beta## the overall identity looks quite reasonable to me.
 
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