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Question: Simplify using trig identities

Question:Simplify using trig identities

Dkunb 65 Maple 2022
simplification trig
May 31 2024
0

What should I do to simplify eq13 further using trig sum identities?
Thank you for your help in advance,

> 

restart;
> 

phi := (x,n,L) -> sqrt(2/L)*sin(n*Pi*x/L + 1/2*n*Pi);

proc (x, n, L) options operator, arrow; sqrt(2/L)*sin(n*Pi*x/L+(1/2)*n*Pi) end proc

 (1)
> 

 

> 

eq1 := W[n,m](q,p) = simplify(1/Pi*Int(phi(q+y,n,L)*exp(-2*I*p*y)*phi(q-y,m,L),y=-L/2+abs(q)..L/2-abs(q)));

W[n, m](q, p) = 2*(Int(sin((1/2)*n*Pi*(2*q+2*y+L)/L)*exp(-(2*I)*p*y)*sin((1/2)*m*Pi*(2*q-2*y+L)/L), y = -(1/2)*L+abs(q) .. (1/2)*L-abs(q)))/(L*Pi)

 (2)
> 

eq2 := simplify(convert(eq1,int)) assuming(n,integer,m,integer);

W[n, m](q, p) = (-16*(I*((-(1/4)*m^2-(1/4)*n^2)*Pi^2+p^2*L^2)*L*p*sin(n*Pi*(abs(q)+q)/L)+(1/2)*cos(n*Pi*(abs(q)+q)/L)*(((1/4)*m^2-(1/4)*n^2)*Pi^2+p^2*L^2)*n*Pi)*exp(I*p*(L-2*abs(q)))*sin(m*Pi*(L-abs(q)+q)/L)+16*exp(-I*p*(L-2*abs(q)))*(I*L*p*((-(1/4)*m^2-(1/4)*n^2)*Pi^2+p^2*L^2)*sin(m*Pi*(abs(q)+q)/L)-(1/2)*cos(m*Pi*(abs(q)+q)/L)*m*((-(1/4)*m^2+(1/4)*n^2)*Pi^2+p^2*L^2)*Pi)*sin(n*Pi*(L-abs(q)+q)/L)+8*Pi*(-m*exp(I*p*(L-2*abs(q)))*((((1/4)*m^2-(1/4)*n^2)*Pi^2-p^2*L^2)*sin(n*Pi*(abs(q)+q)/L)+I*L*p*Pi*n*cos(n*Pi*(abs(q)+q)/L))*cos(m*Pi*(L-abs(q)+q)/L)+exp(-I*p*(L-2*abs(q)))*cos(n*Pi*(L-abs(q)+q)/L)*n*((((1/4)*m^2-(1/4)*n^2)*Pi^2+p^2*L^2)*sin(m*Pi*(abs(q)+q)/L)+I*L*p*Pi*m*cos(m*Pi*(abs(q)+q)/L))))/((m-n)^2*(m+n)^2*Pi^5-8*L^2*p^2*(m^2+n^2)*Pi^3+16*L^4*p^4*Pi)

 (3)

Wigner function evaluated for q > 0 and q < 0, respectively

> 

eq10 := simplify(eq2) assuming(q>0);
eq11 := simplify(eq2) assuming(q<0);
eq12 := simplify(eq10) assuming(m,integer,n,integer);
eq13 := simplify(eq11) assuming(m,integer,n,integer);

W[n, m](q, p) = (16*exp(-I*p*(L-2*q))*(I*p*L*((-(1/4)*m^2-(1/4)*n^2)*Pi^2+p^2*L^2)*sin(n*Pi)+(1/2)*cos(n*Pi)*Pi*n*(((1/4)*m^2-(1/4)*n^2)*Pi^2+p^2*L^2))*sin(2*m*Pi*q/L)-16*exp(I*p*(L-2*q))*(I*p*L*((-(1/4)*m^2-(1/4)*n^2)*Pi^2+p^2*L^2)*sin(m*Pi)-(1/2)*Pi*cos(m*Pi)*((-(1/4)*m^2+(1/4)*n^2)*Pi^2+p^2*L^2)*m)*sin(2*n*Pi*q/L)+8*Pi*(exp(-I*p*(L-2*q))*((((1/4)*m^2-(1/4)*n^2)*Pi^2-p^2*L^2)*sin(n*Pi)+I*L*n*p*Pi*cos(n*Pi))*m*cos(2*m*Pi*q/L)-((((1/4)*m^2-(1/4)*n^2)*Pi^2+p^2*L^2)*sin(m*Pi)+I*L*m*p*Pi*cos(m*Pi))*cos(2*n*Pi*q/L)*n*exp(I*p*(L-2*q))))/((m-n)^2*(m+n)^2*Pi^5-8*L^2*p^2*(m^2+n^2)*Pi^3+16*L^4*p^4*Pi)

  

W[n, m](q, p) = (-8*exp(I*p*(L+2*q))*n*(((1/4)*m^2-(1/4)*n^2)*Pi^2+p^2*L^2)*sin(m*Pi*(L+2*q)/L)+8*m*(-exp(-I*p*(L+2*q))*((-(1/4)*m^2+(1/4)*n^2)*Pi^2+p^2*L^2)*sin(n*Pi*(L+2*q)/L)+I*Pi*n*p*L*(cos(n*Pi*(L+2*q)/L)*exp(-I*p*(L+2*q))-exp(I*p*(L+2*q))*cos(m*Pi*(L+2*q)/L))))/((m-n)^2*(m+n)^2*Pi^4-8*L^2*p^2*(m^2+n^2)*Pi^2+16*L^4*p^4)

  

W[n, m](q, p) = (8*(-1)^n*(((1/4)*m^2-(1/4)*n^2)*Pi^2+p^2*L^2)*n*exp(-I*p*(L-2*q))*sin(2*m*Pi*q/L)-8*m*(-(-1)^m*exp(I*p*(L-2*q))*((-(1/4)*m^2+(1/4)*n^2)*Pi^2+p^2*L^2)*sin(2*n*Pi*q/L)+I*n*(cos(2*n*Pi*q/L)*exp(I*p*(L-2*q))*(-1)^m-exp(-I*p*(L-2*q))*(-1)^n*cos(2*m*Pi*q/L))*Pi*L*p))/((m-n)^2*(m+n)^2*Pi^4-8*L^2*p^2*(m^2+n^2)*Pi^2+16*L^4*p^4)

  

W[n, m](q, p) = (-8*exp(I*p*(L+2*q))*n*(((1/4)*m^2-(1/4)*n^2)*Pi^2+p^2*L^2)*sin(m*Pi*(L+2*q)/L)+8*m*(-exp(-I*p*(L+2*q))*((-(1/4)*m^2+(1/4)*n^2)*Pi^2+p^2*L^2)*sin(n*Pi*(L+2*q)/L)+I*Pi*n*p*L*(cos(n*Pi*(L+2*q)/L)*exp(-I*p*(L+2*q))-exp(I*p*(L+2*q))*cos(m*Pi*(L+2*q)/L))))/((m-n)^2*(m+n)^2*Pi^4-8*L^2*p^2*(m^2+n^2)*Pi^2+16*L^4*p^4)

 (4)
 

 

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