density function of w

First, we obtain the joint density of (u,w) i.e., f(u,w|v0)=f(u,v(w,u)|v0)J from the joint density of (u,v), the transformation v=v(w,u), and the Jacobian.

This is the log of the density. logdensityuw = log f(u,w|v0).

This is the log of the standard multivariate normal density in dim-1 dimension.

Get the coefficients of the polynomials of from .

constant term in terms of u

coefficient of

coefficient of

coefficient of

coefficient of

We now calculate as an application of "logeexppoly" to foo52. It is denoted by logdensityw.

This is

-v0^2/2 - Log[2*Pi]/2 + w^3*(-(o^2*v0*cr[3]) + 
   o*(cr[2] - tp3[9, 9, 9]/6)) + 
 o*(v0*(-cr[0] - td[al1, au1]) - (v0^3*tp3[9, 9, 9])/3) + 
 o^2*w^4*(-cr[2]^2/2 + cr[3] + (cr[2]*tp3[9, 9, 9])/2 - 
   tp4[9, 9, 9, 9]/24) + 
 w*(v0 + o*(cr[0] - 2*cr[2] + td[al1, au1] + tp3[9, 9, 9]/2 + 
     (v0^2*tp3[9, 9, 9])/2) + 
   o^2*(v0^3*(-(tp3[9, 9, al1]*tp3[9, 9, au1])/4 + 
       tp4[9, 9, 9, 9]/6) + 
     v0*(-cr[1] + (td[al1, au1]*tp3[9, 9, 9])/2 + 
       (3*tp3[9, 9, al1]*tp3[9, 9, au1])/8 - 
       td[au1, au2]*tp3[9, al1, al2] + 
       (tp3[9, al1, au2]*tp3[9, al2, au1])/2 - tp4[9, 9, al1, au1]/
        4))) + w^2*(-1/2 - o*v0*cr[2] + 
   o^2*(cr[1] - cr[0]*cr[2] - 2*cr[2]^2 - 3*cr[3] - 
     cr[2]*td[al1, au1] - td[al1, au2]*td[al2, au1] + 
     (cr[0]*tp3[9, 9, 9])/2 - (cr[2]*tp3[9, 9, 9])/2 - 
     tp3[9, 9, 9]^2/4 - (tp3[9, 9, al1]*tp3[9, 9, au1])/2 + 
     v0^2*(-(cr[2]*tp3[9, 9, 9])/2 + 
       (tp3[9, 9, al1]*tp3[9, 9, au1])/8) + 
     td[au1, au2]*tp3[9, al1, al2] - 
     (tp3[9, al1, au2]*tp3[9, al2, au1])/2 + tp4[9, 9, 9, 9]/4 + 
     tp4[9, 9, al1, au1]/4)) + 
 o^2*(-cr[0]^2/2 - cr[1] - cr[0]*td[al1, au1] + 
   td[al1, au2]*td[al2, au1] - (td[al1, au1]*td[al2, au2])/2 - 
   (cr[0]*tp3[9, 9, 9])/2 + tp3[9, 9, 9]^2/6 + 
   v0^2*(td[al1, al2]*td[au1, au2] - (cr[0]*tp3[9, 9, 9])/2 - 
     (td[al1, au1]*tp3[9, 9, 9])/2) + 
   (tp3[9, 9, al1]*tp3[9, 9, au1])/2 - 
   td[au1, au2]*tp3[9, al1, al2] + 
   (tp3[9, al1, au2]*tp3[9, al2, au1])/2 - 
   (tp3[al1, al2, au2]*tp3[al3, au1, au3])/8 + 
   (tp3[al1, al2, au1]*tp3[al3, au2, au3])/8 + 
   v0^4*((tp3[9, 9, al1]*tp3[9, 9, au1])/8 - tp4[9, 9, 9, 9]/8) - 
   tp4[9, 9, 9, 9]/8 - tp4[9, 9, al1, au1]/4)

The coefficients of in are shown below.

Coefficient of

Coefficient of

Coefficient of

Coefficient of

Coefficient of