LegendreAssoc.C

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/// \file
/// \ingroup tutorial_math
/// \notebook
/// Example describing the usage of different kinds of Associate Legendre Polynomials
/// To execute the macro type in:
///
/// ~~~{.cpp}
/// root[0] .x LegendreAssoc.C
/// ~~~
///
/// It draws common graphs for first 5
/// Associate Legendre Polynomials
/// and Spherical Associate Legendre Polynomials
/// Their integrals on the range [-1, 1] are calculated
///
/// \macro_image
/// \macro_output
/// \macro_code
///
/// \author Magdalena Slawinska


#include "TMath.h"
#include "TF1.h"
#include "TCanvas.h"

#include <Riostream.h>
#include "TLegend.h"
#include "TLegendEntry.h"

#include "Math/IFunction.h"
#include <cmath>
#include "TSystem.h"

void LegendreAssoc()
{
   gSystem->Load("libMathMore");

   std::cout <<"Drawing associate Legendre Polynomials.." << std::endl;
   TCanvas *Canvas = new TCanvas("DistCanvas", "Associate Legendre polynomials", 10, 10, 800, 500);
   Canvas->Divide(2,1);
   TLegend *leg1 = new TLegend(0.5, 0.7, 0.8, 0.89);
   TLegend *leg2 = new TLegend(0.5, 0.7, 0.8, 0.89);

   //-------------------------------------------
   //drawing the set of Legendre functions
   TF1* L[5];

   L[0]= new TF1("L_0", "ROOT::Math::assoc_legendre(1, 0,x)", -1, 1);
   L[1]= new TF1("L_1", "ROOT::Math::assoc_legendre(1, 1,x)", -1, 1);
   L[2]= new TF1("L_2", "ROOT::Math::assoc_legendre(2, 0,x)", -1, 1);
   L[3]= new TF1("L_3", "ROOT::Math::assoc_legendre(2, 1,x)", -1, 1);
   L[4]= new TF1("L_4", "ROOT::Math::assoc_legendre(2, 2,x)", -1, 1);

   TF1* SL[5];

   SL[0]= new TF1("SL_0", "ROOT::Math::sph_legendre(1, 0,x)", -TMath::Pi(), TMath::Pi());
   SL[1]= new TF1("SL_1", "ROOT::Math::sph_legendre(1, 1,x)", -TMath::Pi(), TMath::Pi());
   SL[2]= new TF1("SL_2", "ROOT::Math::sph_legendre(2, 0,x)", -TMath::Pi(), TMath::Pi());
   SL[3]= new TF1("SL_3", "ROOT::Math::sph_legendre(2, 1,x)", -TMath::Pi(), TMath::Pi());
   SL[4]= new TF1("SL_4", "ROOT::Math::sph_legendre(2, 2,x)", -TMath::Pi(), TMath::Pi() );

   Canvas->cd(1);
   gPad->SetGrid();
   gPad->SetFillColor(kWhite);
   L[0]->SetMaximum(3);
   L[0]->SetMinimum(-2);
   L[0]->SetTitle("Associate Legendre Polynomials");
   for (int nu = 0; nu < 5; nu++) {
      L[nu]->SetLineStyle(1);
      L[nu]->SetLineWidth(2);
      L[nu]->SetLineColor(nu+1);
   }

   leg1->AddEntry(L[0]->DrawCopy(), " P^{1}_{0}(x)", "l");
   leg1->AddEntry(L[1]->DrawCopy("same"), " P^{1}_{1}(x)", "l");
   leg1->AddEntry(L[2]->DrawCopy("same"), " P^{2}_{0}(x)", "l");
   leg1->AddEntry(L[3]->DrawCopy("same"), " P^{2}_{1}(x)", "l");
   leg1->AddEntry(L[4]->DrawCopy("same"), " P^{2}_{2}(x)", "l");
   leg1->Draw();

   Canvas->cd(2);
   gPad->SetGrid();
   gPad->SetFillColor(kWhite);
   SL[0]->SetMaximum(1);
   SL[0]->SetMinimum(-1);
   SL[0]->SetTitle("Spherical Legendre Polynomials");
   for (int nu = 0; nu < 5; nu++) {
      SL[nu]->SetLineStyle(1);
      SL[nu]->SetLineWidth(2);
      SL[nu]->SetLineColor(nu+1);
   }

   leg2->AddEntry(SL[0]->DrawCopy(), " P^{1}_{0}(x)", "l");
   leg2->AddEntry(SL[1]->DrawCopy("same"), " P^{1}_{1}(x)", "l");
   leg2->AddEntry(SL[2]->DrawCopy("same"), " P^{2}_{0}(x)", "l");
   leg2->AddEntry(SL[3]->DrawCopy("same"), " P^{2}_{1}(x)", "l");
   leg2->AddEntry(SL[4]->DrawCopy("same"), " P^{2}_{2}(x)", "l");
   leg2->Draw();


   //integration

   std::cout << "Calculating integrals of Associate Legendre Polynomials on [-1, 1]" << std::endl;
   double integral[5];
   for (int nu = 0; nu < 5; nu++) {
      integral[nu] = L[nu]->Integral(-1.0, 1.0);
      std::cout <<"Integral [-1,1] for Associated Legendre Polynomial of Degree " << nu << "\t = \t" << integral[nu] <<  std::endl;
   }
}