GRALE: grale::PolynomialMassProfileLens Class Reference

GRALE

grale
PolynomialMassProfileLens

#include <polynomialmassprofilelens.h>

Inheritance diagram for grale::PolynomialMassProfileLens:

Inheritance graph

Collaboration diagram for grale::PolynomialMassProfileLens:

Collaboration graph

List of all members.

Additional Inherited Members
Protected Member Functions inherited from grale::SymmetricLens
	SymmetricLens (GravitationalLens::LensType t)
	Constructor meant to be used by subclasses.
Protected Member Functions inherited from grale::GravitationalLens
	GravitationalLens (LensType t)
	Meant to be used by a specific lens implementation.

Detailed Description

For the potential, we'll need to calculate something like:

$\int_{x_1}^{x_2}\frac{\sum_{k=0}^M a_k x^k}{x+b} dx$

To calculate this, the following relation can be helpful

$\sum_{k=0}^M a_k x^k = \sum_{k=0}^M f_k (x+b)^k$

in which

$f_k = \sum_{l=k}^M \left(\begin{array}{c} l \\ k \end{array}\right) a_l (-b)^{l-k}$

This can be proved by noting that

$x^N = [(x+b)-b]^N = \sum_{k=0}^N \left(\begin{array}{c} N \\ k \end{array}\right) (x+b)^k (-b)^{N-k}$

so that

$\sum_{l=0}^M a_l x^l = \sum_{l=0}^M \sum_{k=0}^l \left(\begin{array}{c} l \\ k \end{array}\right) a_l (x+b)^k (-b)^{l-k}$

The documentation for this class was generated from the following file:

src/lens/polynomialmassprofilelens.h