I have this function that I am working with - Exner function
$$ \left(\frac{p}{p_0}\right)^\frac{R_d}{c_p} = \frac{T}{\theta} $$
and I am trying to solve for theta numerically
So
$$ \theta = T \,\cdot\, p_0^\frac{R_d}{C_p} \, \cdot\frac{1}{p^\frac{R_d}{C_p}}$$
In a numerical software that I am looking at they have approximated
$$ \frac{1}{p^\frac{R_d}{C_p}}$$
as $$ -\frac{R_d}{C_p} \cdot p$$
Since there is no documentation my first reaction was to assume that approximation was obtained used a Taylor series expansion (up to first order)
I am assuming they did they did a Taylor series expansion of the following expression
$$ \frac{1}{x^n} $$
It is still not exactly equal to that but it is close. Am I on the right track ?
As you wrote $$\left(\frac{p}{p_0}\right)^a=\frac{T}{\theta }\implies \theta =T \left(\frac{p}{p_0}\right)^{-a}$$
Now, use Taylor series around $p=p_0$ to get $$\theta=T-\frac{a T }{p}(p-p_0)+O\left((p-p_0)^2\right)=(1+a)T-\frac{a T}{p_0}p+O\left((p-p_0)^2\right)$$