Question: finding a Recurrence relation and solve system of equation

Hi everyone.

I have a 2D function and I wanna after Differentiating from it with respect to tau (at any amount of sigma value) and equaling this derivative to zero solve the infinite system of equations.

P[n](tau)==LegendreP(n - 1/2, cosh(tau)) , Q[n](tau)==LegendreQ(n - 1/2, cosh(tau)) 

are Legendre function.

Thanks in advanced

FUNCTION_f.mw


 

"restart;:N:=3: f(sigma,tau):=(sqrt(cosh(tau)-cos(sigma)))*(∑)(A[n]*P[n]((tau)) -n*Q[n]((tau)) )*sin(n*sigma)"

proc (sigma, tau) options operator, arrow, function_assign; sqrt(cosh(tau)-cos(sigma))*(sum((A[n]*P[n](tau)-n*Q[n](tau))*sin(n*sigma), n = 1 .. N)) end proc

(1)

NULL

W := simplify(diff(f(sigma, tau), tau))

(1/2)*((2*A[2]*(cosh(tau)-cos(sigma))*(diff(P[2](tau), tau))+(-4*cosh(tau)+4*cos(sigma))*(diff(Q[2](tau), tau))+sinh(tau)*(A[2]*P[2](tau)-2*Q[2](tau)))*sin(2*sigma)+(2*A[3]*(cosh(tau)-cos(sigma))*(diff(P[3](tau), tau))+(-6*cosh(tau)+6*cos(sigma))*(diff(Q[3](tau), tau))+sinh(tau)*(A[3]*P[3](tau)-3*Q[3](tau)))*sin(3*sigma)+sin(sigma)*(2*A[1]*(cosh(tau)-cos(sigma))*(diff(P[1](tau), tau))+(-2*cosh(tau)+2*cos(sigma))*(diff(Q[1](tau), tau))+sinh(tau)*(A[1]*P[1](tau)-Q[1](tau))))/(cosh(tau)-cos(sigma))^(1/2)

(2)

``


 

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