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Question: Series substitution to find coefficients

Question:Series substitution to find coefficients

Andrew Carmichael 216 Maple
June 30 2008
0
Dear All, I'm trying to substitute a series ansatz into a coupled couple of equations to find coefficients. The equations are quite cumbersome, but the principle shouldn't be too difficult. The equations are f3A(alpha,mu,Omega,delta,m) =0 f4A(alpha,mu,Omega,m) = 0 where f3A and f4A are complicated functions of the several variables. The region I'm working in is delta=0, m=1 Now, I have reason to believe that the following ansatz will work: alpha = A0 + A2/Omega^2 + A4/Omega^4 +.... -mu=M0 + M2/Omega^2 + M4/Omega^4 +..... where A0, A2, A4, M0, M2, M4 are constants I want to determine. How do I tell maple to put in these series and spit out the coefficients? I suspect somebody knows a procedure which is general and doesn't depend upon the functions, but for completeness, the functions are defined as follows: # Define the integrand A(x,m) A := proc (x, m) options operator, arrow; (-m^2-2*x*m+1)/(sqrt(x)*sqrt((m+x)^2-1)*(x+sqrt((m+x)^2-1))) end proc; # Define the integral IntegralA:=(m)->Int(A(x,m),x=0..infinity); # Define the next integrand P:=(x,m)->sqrt(x)/(sqrt((x+m)^2-1)*(x+m+sqrt((x+m)^2-1))); # Define its integral IntegralP:=(m)->Int(P(x,m),x=0..infinity); #Define the functions f3:=(alpha,beta,Omega,delta,m)->(2*(-beta*Omega*m/sqrt(2))-delta)*beta+Omega/sqrt(2)*(alpha^2+(3*(beta*Omega)^(3/2)*int(A(x, m), x = 0 .. infinity))/(4*2^(3/4))); f4:=(alpha,beta,Omega,m)->alpha^2+beta^2+(3*(beta*Omega)^(3/2)*int(P(x, 1), x = 0 .. infinity))/(4*2^(3/4))-1; #Now get rid of beta in favour of mu to get the functions I really want. f3A:=(alpha,mu,Omega,delta,m)->algsubs(beta=-sqrt(2)*mu/(Omega*m),f3(alpha,beta,Omega,delta,m)); f4A:=(alpha,mu,Omega,m)->algsubs(beta=-sqrt(2)*mu/(Omega*m),f4(alpha,beta,Omega,m));
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