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exA.m

+clear all
+close all
+
+%% Figure A.1 left %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+x = 5;
+sqrt_x = sqrt(x)
+
+N = ceil(2*sqrt_x);
+nn = (-N:N)';
+
+bessel_exact = besselj(nn,x);
+
+approxBesselJ_p1q0 = sqrt(2/(pi*x)) * cos(x-pi/2*nn-pi/4);
+
+approxBesselJ_p1q1 = sqrt(2/(pi*x)) ...
+    * ( cos(x-pi/2*nn-pi/4) - sin(x-pi/2*nn-pi/4).*((4*nn.^2-1)/(8*x)) );
+
+approxBesselJ_p2q2 = sqrt(2/(pi*x)) ...
+    * ( cos(x-pi/2*nn-pi/4).*(1-(4*nn.^2-1).*(4*nn.^2-9)/(32*4*(x)^2)) ...
+       - sin(x-pi/2*nn-pi/4).*((4*nn.^2-1)/(8*x)-(4*nn.^2-1).*(4*nn.^2-9).*(4*nn.^2-25)/(384*8*(x)^3)) );
+
+figure()
+plot(nn,(bessel_exact),'o','Markersize',10,'Linewidth',2)
+hold on
+plot(nn,(approxBesselJ_p1q1),'.r','Markersize',20)
+plot(nn,(approxBesselJ_p2q2),'+k','Markersize',10,'Linewidth',2)
+set(gca,'Fontsize',20)
+grid on
+bla = axis;
+plot([-sqrt_x,-sqrt_x],[bla(3:4)],'r:','LineWidth',2)
+plot([sqrt_x,sqrt_x],[bla(3:4)],'r:','LineWidth',2)
+hold off
+legend('$J_n(5)$','$p=q=1$','$p=q=2$','','','interpreter','latex')
+axis tight
+
+print plots/besselApprox_kr5 -depsc
+
+
+%% Figure A.1 right %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+x = 100;
+sqrt_x = sqrt(x)
+
+N = ceil(2*sqrt_x);
+nn = (-N:N)';
+
+bessel_exact = besselj(nn,x);
+
+approxBesselJ_p1q0 = sqrt(2/(pi*x)) * cos(x-pi/2*nn-pi/4);
+
+approxBesselJ_p1q1 = sqrt(2/(pi*x)) ...
+    * ( cos(x-pi/2*nn-pi/4) - sin(x-pi/2*nn-pi/4).*((4*nn.^2-1)/(8*x)) );
+
+approxBesselJ_p2q2 = sqrt(2/(pi*x)) ...
+    * ( cos(x-pi/2*nn-pi/4).*(1-(4*nn.^2-1).*(4*nn.^2-9)/(32*4*(x)^2)) ...
+       - sin(x-pi/2*nn-pi/4).*((4*nn.^2-1)/(8*x)-(4*nn.^2-1).*(4*nn.^2-9).*(4*nn.^2-25)/(384*8*(x)^3)) );
+
+figure()
+plot(nn,(bessel_exact),'o','Markersize',10,'Linewidth',2)
+hold on
+plot(nn,(approxBesselJ_p1q1),'.r','Markersize',20)
+plot(nn,(approxBesselJ_p2q2),'+k','Markersize',10,'Linewidth',2)
+set(gca,'Fontsize',20)
+grid on
+bla = axis;
+plot([-sqrt_x,-sqrt_x],[bla(3:4)],'r:','LineWidth',2)
+plot([sqrt_x,sqrt_x],[bla(3:4)],'r:','LineWidth',2)
+hold off
+legend('$J_n(100)$','$p=q=1$','$p=q=2$','','','interpreter','latex')
+axis tight
+
+print plots/besselApprox_kr100 -depsc
+
+
+
+