Formal solution and Schwarzschild equation

The radiation transport equation (RTE) for time-independent plane-parallel atmosphere has well known form (e.g. Rutten, 2003):
\begin{equation}
\mu\,\frac{\mathrm{d}I_{\nu}}{\mathrm{d} \tau_{\nu}} = I_{\nu} - S_{\nu},
\end{equation}
 where $I_{\nu}$ is specific intensity, $S_{\nu}$ source function, $\tau_{\nu}$ is the radial optical depth and $\mu$ is the cosine of the angle between the line of sight and vertical axis. Mathematically, this is a linear first order differential equation with constant coefficients. Physically, it describes propagation of radiation through an interacting medium. Properties of the medium are hidden in the source function and the optical depth. However, to make the first step toward the solution of RTE, we will assume that the optical depth is independent variable and that the source function is a know function of it.

Gauss quadrature and Legendre polynomials

Numerical integration is among the most common tasks in astrophysics. Simple formulae, like trapezoidal, Newton-Cotes or Simpson, are often not enough accurate. Gauss quadrature provides a more accurate solution, but its implementation is a bit more difficult. Here I will demonstrate the implementation of that algorithm in IDL. The algorithm includes finding the coefficients of Legendre polynomials and their zeros. Both tasks are explicitly solved in the following as well.

But let's go step by step. Our task is to solve numerically the integral:
$$I = \int_a^b f(x)\, \mathrm{d} x.$$

Exponential integrals

The formal solution of the radiative transfer equation can be expressed using exponential integrals. These are mathematical functions defined as:
\begin{equation}
E_n(x) = \int_1^{\infty}\, \exp (-yx)\frac{\mathrm{d}x}{y^n}, \;\;\;\;n = 1, 2, ...
\end{equation}
There is several important properties of these functions: