5.45 Dr Basti Associated Legendre

Subject:      Associated Legendre 
Author:       Mehran Basti <[e-mail]> 
Organization: AT&T Worldnet 
Date:         Mon, 25 Nov 2002 02:48:15 GMT 
 
 
Dear newsgroup: 
 
I had mentioned that my methods will solve classical equations without 
the use of  infinite series. 
 
The following is a Maple code of my old files. Those days I had Maple2 but the 
general idea is the same in the process  and you see that we can also 
solve the integrals involved. 
 
It does not make sense how are the theory behind it but eventually it will come into light. 
 
Just read the procedures and you can see the solution of associated Legendre AL at the end. 
 
> s1:=-diff(p(t),t)+p(t)^2; 
> 
> s2:=exp(2*int(p(t),t))*T(t); 
> s3:=s1+s2; 
> s4:=diff(T(t),t)/T(t); 
> s5:=-(1/2)*(diff(s4,t))+(1/4)*s4^2; 
> s6:=s5+s2; 
> p(t):=-1/t+(1)/(2-t); 
> s1:=simplify(s1); 
> s1:=collect(%,t); 
> s2:=simplify(s2); 
> s1+s2=(2*t^2-4*t+m^2-1)/(t*(-2+t))^2; 
> solve(%,T(t)); 
> T(t):=simplify(%); 
> s2:=simplify(s2); 
> s2+s1; 
> s3:=simplify(%); 
> 
> s6:=simplify(s6); 
> t*(-2+t); 
> simplify(%); 
> z:=(r3*t^3+r2*t^2+r1*t+r0)/(%); 
> 
> simplify(diff(z,t)+z^2-s6); 
> s7:=collect(numer(%),t); 
> 
> coeff(%,t,0); 
> solve(%,r0); 
> r0:=op(1,{%}); 
> coeff(s7,t,1); 
> solve(%,r1); 
> r1:=simplify(%); 
> coeff(s7,t,2); 
> solve(%,r2); 
> r2:=simplify(%); 
> coeff(s7,t,3); 
> solve(%,r3); 
> r3:=simplify(%); 
> simplify(s7); 
> s3:=simplify(s3); 
> s4:=simplify(s4); 
> s6:=simplify(s6); 
> T(t):=simplify(T(t)); 
> z:=simplify(z); 
> 1/2*s4+2*p(t)+z; 
> s8:=simplify(%); 
> exp(int(%,t)); 
> expand(%); 
> g:=(%); 
> simplify(g,power); 
> g:=%; 
> Int(%,t); 
> Integralg:=(%); 
> int(g1(t),t); 
> x1:=-p(t)+g1(t)/(%); 
> diff(x1,t)+x1^2-s3; 
> simplify(%); 
> s10:=numer(%); 
> solve(%,int(g1(t),t)); 
> Ing:=(%); 
> simplify(subs(g1(t)=g,%)); 
> 
>  Ing:=(%); 
> expand(%); 
> Ing:=simplify(%); 
> simplify(diff(%,t)-g); 
> expand(%); 
> simplify(%); 
> x:=-p(t)+g/Ing; 
> simplify(diff(x,t)+x^2-s3); 
>  int(x,t); 
> exp(%); 
> expand(%); 
> s11:=simplify(%); 
> ALT:=t*(2-t)*diff(u(t),t$2)+2*(1-t)*diff(u(t),t)+(2-m^2/(1-(1-t)^2))*u(t); 
> -2*(1-t)/(2*t*(2-t)); 
> int(%,t); 
> exp(%); 
> s12:=simplify(%,power); 
> 
> u1:=s12*s11; 
> u1:=simplify(%,power); 
>  simplify(subs(u(t)=u1,ALT)); 
> AL:=(1-nu^2)*diff(u(nu),nu$2)-2*nu*diff(u(nu),nu)+(2-m^2/(1-nu^2))*u(nu); 
> 
> u2:=subs(t=1-nu,u1); 
> simplify(subs(u(nu)=u2,AL)); 
> 
 
The advantage of these methods are that there are ample rooms for advances. 
 
Today my skills for solving classical equations such as Riccati is much advanced. 
 
Highly complicated and more general Riccati equations in its billions now possible. 
 
Sincerely 
 
Dr.M.Basti