Advanced Physics for Brachytherapy 2018

The method

Many orthogonal bases exist

In example:

➢ Legendre polynomials, P n

(x), are a series of functions that exhibit orthogonality for -1≤x≤1:

and therefore they can be used to expand any function defined in [-1,1] in an infinite series:

  =

  =





n

+

1 2

xnP nw xf ), ( 

(

)

=

E Ers

rns

,(

= →

)ˆ ˆ,



,( ,

), ˆ ˆ( )   →

nPE E



n

0

n

0 4

1

1





ˆ(

dx xnPxf , ( , 

nw =

)(

(

)

2 /1 ) =→ E Erns 

nP E Ers

d

,(

)ˆ ˆ,  →

)ˆ ˆ( )ˆ    

 −

 −

1

1