Transformation of expoential to hyperbolic

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Homework Help Overview

The discussion revolves around the transformation of an exponential function into a hyperbolic sine function within the context of statistical mechanics. The original poster presents a formula from a textbook involving parameters related to magnetization and temperature, specifically focusing on the expression exp(-MgbH/KT) and its equivalence to a hyperbolic sine ratio.

Discussion Character

  • Conceptual clarification, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants explore the derivation of the given formula, questioning the validity of the transformation and the assumptions behind the parameters involved. There are attempts to clarify the notation used in the expressions and the implications of specific values for the variables.

Discussion Status

The discussion is ongoing, with participants raising questions about potential typos in the original formula and the correctness of the transformations. Some participants express confusion regarding the implications of certain variable values and seek clarification on the notation used in the equations.

Contextual Notes

There are mentions of specific values for the variables, such as M and x, and the context of the problem involves assumptions about physical constants like the Boltzmann constant. The original poster acknowledges a mistake in the transcription of the formula, which may affect the interpretation of the problem.

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Homework Statement



The book has it exp(-MgbH/KT) =(sinh(2S+1)x/2)/(sinh(x/2)) for M=2S+1, and x = gbH/(kt).

Homework Equations


The Attempt at a Solution



I'd have it as cosh(Mx)-sinh(Mx). How did they get the above result? Help please. Thanks.
 
Last edited:
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If x=0 and y=1 then the book's formula gives exp(0)=0. 1=0 is nonsense. You don't need to ask where it comes from. Or is there a typo in there? Your formula is certainly correct.
 
Last edited:
Dick said:
If x=0 and y=1 then the book's formula gives exp(0)=0. 1=0 is nonsense. You don't need to ask where it comes from. Or is there a typo in there?

Sorry I have made a mistake in writing that down. The book actually have it as exp(-MgbH/KT). M = 2S+1. The book let gbH/KT be x. And then it went ahead to say exp(-MgbH/KT) = sinh[(2S+1)x/2]/sinh(x/2). Which is sinh[Mx/2]/sinh(x/2) . Sorry for the inconvenience.
 
Well for x=2 and M=1 that equation yields \exp(-2) = 1?

Here I'm assuming k=K.

BTW. Does (sinh(2S+1)x/2) mean (x/2) sinh(2S+1). Or do you mean sinh((2S+1)x/2) ?
 
uart said:
Well for x=2 and M=1 that equation yields \exp(-2) = 1?

Here I'm assuming k=K.

BTW. Does (sinh(2S+1)x/2) mean (x/2) sinh(2S+1). Or do you mean sinh((2S+1)x/2) ?

K is a Boltzmann constant. M is magnetisation. b = beta.
I meant the last one. The equation should read: exp(-MgbgH/KT) = sinh[(2S+1)x/2]/sinh(x/2) with M and x as defined above.
Cheers.
 
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