Determining critical exponent in spontaneous magnetization of 2D Ising model I have been solving a problem related to Onsager's relation in Ising model. The initial relation is:
$<s>^8=1-sinh(\frac{2J}{k_BT})^{-4}$
I got that critical temperature is $\frac{2J}{k_BT}=sinh^{-1}(1)\approx0.88$. Then
$<s>^8=1-sinh(\frac{0.88}{1+t})^{-4},$ where $t=\frac{\delta T}{T_C}$ and $T=T_C+\delta T$
From that
$sinh(\frac{0.88}{1+t})^{-4} \approx 1-4.995t-9.01t^2-...$ at t=0
Then follows
$ <s> \sim |\frac{T-T_C}{T_C}|^{1/8}$ $\beta=1/8$
However, somehow this answer should get to
$<s>=B(-t)^{1/8}[1+b(-t)+..]$
Meanwhile I get
$<s>=(-4.995t-9.01t^2-...)^{1/8}=(-4.995t)^{1/8}[1+9/5t+...]^{1/8}$
However, I am not sure where the power of 1/8 dissapears in the series and I would be grateful if someone knows, how to get to that or if I have made a mistake somehwere above. And how is it possible to determine a temperature range valid for this expansion? I get that it should be near to $t=0$, but how to do that exactly?
Edit: the answer states that $B=(8\sqrt2K_C)^{1/8}$ and $b=\frac{1-\frac{9K_C}{\sqrt{2}}}{8}$ where $K_C=\frac{J}{kT_C}$
 A: Let me guess: Yeoman's textbook, right? I had the same problem too, and after long and tedious effort I finally got the answer to this problem!
There are several readings I can give (in total I have 7) but I'll give you the essential ones: The spontaneous magnetization of a 2D Ising Model by CN Yang (1951) and Statistical Mechanics, 3rd Edition by Paul D. Beale and Pathria R.K. (2011).
But basically, to solve this problem, you need:
Step 1

*

*Define $t=(T-Tc)/Tc$

*Define $Kc=J/(k Tc)$

*Express $\langle s\rangle$ in terms of $t$
Now you have $$\langle s\rangle=\left(1-\sinh\left(\frac{2Kc}{1+t}\right)^{-4}\right)^{1/8}$$
Investigate this near $t=0$
Step 2:
Expand $\langle s\rangle$ by Maclaurin series in terms of powers of $t$ (which makes sense if you think about it - "near $t=0$" means "small $t$", hence justifying this expansion!)
Step 3:
From the 7 readings (textbooks and papers) I have gathered, by experiments we know $\sinh(2Kc)=1$ (this is an important result). Plug this result into the above Maclaurin expansion.
Step 4:
Manipulate your expansion into the form of:
$$\langle s\rangle = B(−t)^{1/8}(1+b(−t)+...)$$

Note: The Maclaurin series must be expanded to second order in $t$ (i.e. you need to expand up to $t^2$). This is tedious - you have to be careful because you need to take the derivative of a messy thing. First derivative of $\sinh^{-4}$ was okay, but the second derivative is where things might get messy!
