How is Dipole Anisotropy Expansion Derived in Weinberg's Cosmology?

[nicksauce]

In Weinberg's cosmology book, section 2.4 we have
$$T' = \frac{T}{\gamma(1+\beta\cos{\theta})}$$

He then claims, "Expanding in powers of beta, the temperature shift can be expressed as a sum of Legendre polynomials"
$$\Delta T = T' -T = T\left(-\frac{\beta^2}{6} - \beta P_1(\cos{\theta}) + \frac{2\beta^2}{3}P_2(\cos{\theta}) + ...\right)$$

Can someone help me fill in the algebra here? I really am having a hard time seeing where this is coming from.

 

[nicksauce]

Alright, so I am able to get it to work, if I expand the original function as a Taylor series, then go back and write it in terms of the Legendre Polynomials (up to second order). But I am hoping/wondering if there is a more elegant way to achieve the final result.

 

[Chalnoth]

Alright, so I am able to get it to work, if I expand the original function as a Taylor series, then go back and write it in terms of the Legendre Polynomials (up to second order). But I am hoping/wondering if there is a more elegant way to achieve the final result.

I'm not sure. I mean, the Taylor series expansion is exceedingly simple for this function, so it may be possible to make use of one of the recurrence relations of the Legendre polynomials to transform the Taylor expansion into an expansion in Legendre polynomials. But that would seem to be a fair bit of work.