PASS E1 l=0: a = x f_x/f = 1+2l u
PASS E2 l=0: x^2 Veff = 3/4+2l(1-l)u+3l^2u^2
PASS E2' l=0: x^2 Veff = a^2/4+a/2-x a_x/2
PASS E1- l=0: on x<0, -delta*Delta delta = x f_x/f
PASS E1 l=1: a = x f_x/f = 1+2l u
PASS E2 l=1: x^2 Veff = 3/4+2l(1-l)u+3l^2u^2
PASS E2' l=1: x^2 Veff = a^2/4+a/2-x a_x/2
PASS E1- l=1: on x<0, -delta*Delta delta = x f_x/f
PASS E1 l=2: a = x f_x/f = 1+2l u
PASS E2 l=2: x^2 Veff = 3/4+2l(1-l)u+3l^2u^2
PASS E2' l=2: x^2 Veff = a^2/4+a/2-x a_x/2
PASS E1- l=2: on x<0, -delta*Delta delta = x f_x/f
PASS E1 l=3: a = x f_x/f = 1+2l u
PASS E2 l=3: x^2 Veff = 3/4+2l(1-l)u+3l^2u^2
PASS E2' l=3: x^2 Veff = a^2/4+a/2-x a_x/2
PASS E1- l=3: on x<0, -delta*Delta delta = x f_x/f
PASS E1 l=4: a = x f_x/f = 1+2l u
PASS E2 l=4: x^2 Veff = 3/4+2l(1-l)u+3l^2u^2
PASS E2' l=4: x^2 Veff = a^2/4+a/2-x a_x/2
PASS E1- l=4: on x<0, -delta*Delta delta = x f_x/f
PASS E1 l=5: a = x f_x/f = 1+2l u
PASS E2 l=5: x^2 Veff = 3/4+2l(1-l)u+3l^2u^2
PASS E2' l=5: x^2 Veff = a^2/4+a/2-x a_x/2
PASS E1- l=5: on x<0, -delta*Delta delta = x f_x/f
PASS E1 l=6: a = x f_x/f = 1+2l u
PASS E2 l=6: x^2 Veff = 3/4+2l(1-l)u+3l^2u^2
PASS E2' l=6: x^2 Veff = a^2/4+a/2-x a_x/2
PASS E1- l=6: on x<0, -delta*Delta delta = x f_x/f
PASS E1 l=7: a = x f_x/f = 1+2l u
PASS E2 l=7: x^2 Veff = 3/4+2l(1-l)u+3l^2u^2
PASS E2' l=7: x^2 Veff = a^2/4+a/2-x a_x/2
PASS E1- l=7: on x<0, -delta*Delta delta = x f_x/f
PASS E1 l=8: a = x f_x/f = 1+2l u
PASS E2 l=8: x^2 Veff = 3/4+2l(1-l)u+3l^2u^2
PASS E2' l=8: x^2 Veff = a^2/4+a/2-x a_x/2
PASS E1- l=8: on x<0, -delta*Delta delta = x f_x/f
PASS E3 n=2 l=1: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=2 l=1: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=2 l=1: agrees with paper eq (Veff), z^2->f
PASS E3 n=2 l=2: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=2 l=2: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=2 l=2: agrees with paper eq (Veff), z^2->f
PASS E3 n=2 l=3: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=2 l=3: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=2 l=3: agrees with paper eq (Veff), z^2->f
PASS E3 n=2 l=4: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=2 l=4: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=2 l=4: agrees with paper eq (Veff), z^2->f
PASS E3 n=2 l=5: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=2 l=5: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=2 l=5: agrees with paper eq (Veff), z^2->f
PASS E3 n=3 l=1: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=3 l=1: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=3 l=2: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=3 l=2: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=3 l=3: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=3 l=3: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=3 l=4: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=3 l=4: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=3 l=5: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=3 l=5: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=4 l=1: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=4 l=1: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=4 l=2: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=4 l=2: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=4 l=3: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=4 l=3: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=4 l=4: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=4 l=4: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=4 l=5: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=4 l=5: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=5 l=1: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=5 l=1: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=5 l=2: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=5 l=2: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=5 l=3: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=5 l=3: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=5 l=4: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=5 l=4: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=5 l=5: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=5 l=5: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=6 l=1: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=6 l=1: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=6 l=2: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=6 l=2: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=6 l=3: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=6 l=3: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=6 l=4: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=6 l=4: f=0 branch Veff = a(a-2)/(4t^2)
PASS E3 n=6 l=5: Veff = a(a-2)/(4t^2)+R (PRS 241-242, f!=0)
PASS E3 n=6 l=5: f=0 branch Veff = a(a-2)/(4t^2)
PASS E4 l=1: u*=(l-1)/(3l) in [0,1], min = 3/4-(l-1)^2/3, grid>=min
PASS E4 l=1: (min<3/4 iff l>=2) and (min<0 iff l>=3)
PASS E4 l=2: u*=(l-1)/(3l) in [0,1], min = 3/4-(l-1)^2/3, grid>=min
PASS E4 l=2: (min<3/4 iff l>=2) and (min<0 iff l>=3)
PASS E4 l=2: on z=gamma x^l, u = (1+gamma^2)^-1 = u*
PASS E4 l=3: u*=(l-1)/(3l) in [0,1], min = 3/4-(l-1)^2/3, grid>=min
PASS E4 l=3: (min<3/4 iff l>=2) and (min<0 iff l>=3)
PASS E4 l=3: on z=gamma x^l, u = (1+gamma^2)^-1 = u*
PASS E4 l=4: u*=(l-1)/(3l) in [0,1], min = 3/4-(l-1)^2/3, grid>=min
PASS E4 l=4: (min<3/4 iff l>=2) and (min<0 iff l>=3)
PASS E4 l=4: on z=gamma x^l, u = (1+gamma^2)^-1 = u*
PASS E4 l=5: u*=(l-1)/(3l) in [0,1], min = 3/4-(l-1)^2/3, grid>=min
PASS E4 l=5: (min<3/4 iff l>=2) and (min<0 iff l>=3)
PASS E4 l=5: on z=gamma x^l, u = (1+gamma^2)^-1 = u*
PASS E4 l=6: u*=(l-1)/(3l) in [0,1], min = 3/4-(l-1)^2/3, grid>=min
PASS E4 l=6: (min<3/4 iff l>=2) and (min<0 iff l>=3)
PASS E4 l=6: on z=gamma x^l, u = (1+gamma^2)^-1 = u*
PASS E4 l=7: u*=(l-1)/(3l) in [0,1], min = 3/4-(l-1)^2/3, grid>=min
PASS E4 l=7: (min<3/4 iff l>=2) and (min<0 iff l>=3)
PASS E4 l=7: on z=gamma x^l, u = (1+gamma^2)^-1 = u*
PASS E4 l=8: u*=(l-1)/(3l) in [0,1], min = 3/4-(l-1)^2/3, grid>=min
PASS E4 l=8: (min<3/4 iff l>=2) and (min<0 iff l>=3)
PASS E4 l=8: on z=gamma x^l, u = (1+gamma^2)^-1 = u*
PASS E4 l=9: u*=(l-1)/(3l) in [0,1], min = 3/4-(l-1)^2/3, grid>=min
PASS E4 l=9: (min<3/4 iff l>=2) and (min<0 iff l>=3)
PASS E4 l=9: on z=gamma x^l, u = (1+gamma^2)^-1 = u*
PASS E4 l=10: u*=(l-1)/(3l) in [0,1], min = 3/4-(l-1)^2/3, grid>=min
PASS E4 l=10: (min<3/4 iff l>=2) and (min<0 iff l>=3)
PASS E4 l=10: on z=gamma x^l, u = (1+gamma^2)^-1 = u*
PASS E4 l=11: u*=(l-1)/(3l) in [0,1], min = 3/4-(l-1)^2/3, grid>=min
PASS E4 l=11: (min<3/4 iff l>=2) and (min<0 iff l>=3)
PASS E4 l=11: on z=gamma x^l, u = (1+gamma^2)^-1 = u*
PASS E4 l=12: u*=(l-1)/(3l) in [0,1], min = 3/4-(l-1)^2/3, grid>=min
PASS E4 l=12: (min<3/4 iff l>=2) and (min<0 iff l>=3)
PASS E4 l=12: on z=gamma x^l, u = (1+gamma^2)^-1 = u*
PASS E5 trial 0: pointwise form of the weighted Hardy identity
PASS E5c trial 0: identity for complex w = w1 + i w2
PASS E6 trial 0: b=(1+a)/2 weight = (1+a)^2/4 - t a'/2
PASS E6 trial 0: ... = 1/4 + t^2 V, V = th'^2+th''
PASS E6 trial 0: b=1 weight = a
PASS E6 trial 0: (weight -1/4)/t^2 = Veff = (Dd)^2/4 + (Dd)'/2 with Dd = 2 th'
PASS E5 trial 1: pointwise form of the weighted Hardy identity
PASS E5c trial 1: identity for complex w = w1 + i w2
PASS E6 trial 1: b=(1+a)/2 weight = (1+a)^2/4 - t a'/2
PASS E6 trial 1: ... = 1/4 + t^2 V, V = th'^2+th''
PASS E6 trial 1: b=1 weight = a
PASS E6 trial 1: (weight -1/4)/t^2 = Veff = (Dd)^2/4 + (Dd)'/2 with Dd = 2 th'
PASS E5 trial 2: pointwise form of the weighted Hardy identity
PASS E5c trial 2: identity for complex w = w1 + i w2
PASS E6 trial 2: b=(1+a)/2 weight = (1+a)^2/4 - t a'/2
PASS E6 trial 2: ... = 1/4 + t^2 V, V = th'^2+th''
PASS E6 trial 2: b=1 weight = a
PASS E6 trial 2: (weight -1/4)/t^2 = Veff = (Dd)^2/4 + (Dd)'/2 with Dd = 2 th'
PASS E5 trial 3: pointwise form of the weighted Hardy identity
PASS E5c trial 3: identity for complex w = w1 + i w2
PASS E6 trial 3: b=(1+a)/2 weight = (1+a)^2/4 - t a'/2
PASS E6 trial 3: ... = 1/4 + t^2 V, V = th'^2+th''
PASS E6 trial 3: b=1 weight = a
PASS E6 trial 3: (weight -1/4)/t^2 = Veff = (Dd)^2/4 + (Dd)'/2 with Dd = 2 th'
PASS E5 trial 4: pointwise form of the weighted Hardy identity
PASS E5c trial 4: identity for complex w = w1 + i w2
PASS E6 trial 4: b=(1+a)/2 weight = (1+a)^2/4 - t a'/2
PASS E6 trial 4: ... = 1/4 + t^2 V, V = th'^2+th''
PASS E6 trial 4: b=1 weight = a
PASS E6 trial 4: (weight -1/4)/t^2 = Veff = (Dd)^2/4 + (Dd)'/2 with Dd = 2 th'
PASS E5 trial 5: pointwise form of the weighted Hardy identity
PASS E5c trial 5: identity for complex w = w1 + i w2
PASS E6 trial 5: b=(1+a)/2 weight = (1+a)^2/4 - t a'/2
PASS E6 trial 5: ... = 1/4 + t^2 V, V = th'^2+th''
PASS E6 trial 5: b=1 weight = a
PASS E6 trial 5: (weight -1/4)/t^2 = Veff = (Dd)^2/4 + (Dd)'/2 with Dd = 2 th'
PASS E7 l=0: x f_x/f = 1+2l x^(2l)/(x^(2l)+rho)
PASS E7 l=0: d_x f(0,z) = c(1+rho)
PASS E7 l=0: f = c x^(2l+1) + (c rho) x
PASS E7 l=1: x f_x/f = 1+2l x^(2l)/(x^(2l)+rho)
PASS E7 l=1: d_x^(2l+1) f(0,z) = (2l+1)! c
PASS E7 l=1: d_x f(0,z) = c rho (vanishes where rho=0)
PASS E7 l=1: f = c x^(2l+1) + (c rho) x
PASS E7 l=2: x f_x/f = 1+2l x^(2l)/(x^(2l)+rho)
PASS E7 l=2: d_x^(2l+1) f(0,z) = (2l+1)! c
PASS E7 l=2: d_x f(0,z) = c rho (vanishes where rho=0)
PASS E7 l=2: f = c x^(2l+1) + (c rho) x
PASS E7 l=3: x f_x/f = 1+2l x^(2l)/(x^(2l)+rho)
PASS E7 l=3: d_x^(2l+1) f(0,z) = (2l+1)! c
PASS E7 l=3: d_x f(0,z) = c rho (vanishes where rho=0)
PASS E7 l=3: f = c x^(2l+1) + (c rho) x
PASS E7 l=4: x f_x/f = 1+2l x^(2l)/(x^(2l)+rho)
PASS E7 l=4: d_x^(2l+1) f(0,z) = (2l+1)! c
PASS E7 l=4: d_x f(0,z) = c rho (vanishes where rho=0)
PASS E7 l=4: f = c x^(2l+1) + (c rho) x
PASS E7 l=5: x f_x/f = 1+2l x^(2l)/(x^(2l)+rho)
PASS E7 l=5: d_x^(2l+1) f(0,z) = (2l+1)! c
PASS E7 l=5: d_x f(0,z) = c rho (vanishes where rho=0)
PASS E7 l=5: f = c x^(2l+1) + (c rho) x
PASS E7 l=6: x f_x/f = 1+2l x^(2l)/(x^(2l)+rho)
PASS E7 l=6: d_x^(2l+1) f(0,z) = (2l+1)! c
PASS E7 l=6: d_x f(0,z) = c rho (vanishes where rho=0)
PASS E7 l=6: f = c x^(2l+1) + (c rho) x
PASS E7 l=7: x f_x/f = 1+2l x^(2l)/(x^(2l)+rho)
PASS E7 l=7: d_x^(2l+1) f(0,z) = (2l+1)! c
PASS E7 l=7: d_x f(0,z) = c rho (vanishes where rho=0)
PASS E7 l=7: f = c x^(2l+1) + (c rho) x
PASS E8 n=2: det frame Ex.7.7 = F^(n-1)
PASS E8 n=2: det frame Ex.7.8 = F^(n-1) (not F^(2(n-1)))
PASS E8 n=2 l=1: a = (n-1)(1+2l u)
PASS E8 n=2 l=2: a = (n-1)(1+2l u)
PASS E8 n=2 l=3: a = (n-1)(1+2l u)
PASS E8 n=2 l=4: a = (n-1)(1+2l u)
PASS E8 n=3: det frame Ex.7.7 = F^(n-1)
PASS E8 n=3: det frame Ex.7.8 = F^(n-1) (not F^(2(n-1)))
PASS E8 n=3 l=1: a = (n-1)(1+2l u)
PASS E8 n=3 l=2: a = (n-1)(1+2l u)
PASS E8 n=3 l=3: a = (n-1)(1+2l u)
PASS E8 n=3 l=4: a = (n-1)(1+2l u)
PASS E8 n=4: det frame Ex.7.7 = F^(n-1)
PASS E8 n=4: det frame Ex.7.8 = F^(n-1) (not F^(2(n-1)))
PASS E8 n=4 l=1: a = (n-1)(1+2l u)
PASS E8 n=4 l=2: a = (n-1)(1+2l u)
PASS E8 n=4 l=3: a = (n-1)(1+2l u)
PASS E8 n=4 l=4: a = (n-1)(1+2l u)
PASS E8 n=5: det frame Ex.7.7 = F^(n-1)
PASS E8 n=5: det frame Ex.7.8 = F^(n-1) (not F^(2(n-1)))
PASS E8 n=5 l=1: a = (n-1)(1+2l u)
PASS E8 n=5 l=2: a = (n-1)(1+2l u)
PASS E8 n=5 l=3: a = (n-1)(1+2l u)
PASS E8 n=5 l=4: a = (n-1)(1+2l u)
PASS E9 l=0 p=-1 k=-1: a = -p - k l t^l/(t^l+f) and a >= -p >= 1
PASS E9 l=0 p=-3/2 k=-2/3: a = -p - k l t^l/(t^l+f) and a >= -p >= 1
PASS E9 l=0 p=-5 k=-7/2: a = -p - k l t^l/(t^l+f) and a >= -p >= 1
PASS E9 l=1 p=-1 k=-1: a = -p - k l t^l/(t^l+f) and a >= -p >= 1
PASS E9 l=1 p=-3/2 k=-2/3: a = -p - k l t^l/(t^l+f) and a >= -p >= 1
PASS E9 l=1 p=-5 k=-7/2: a = -p - k l t^l/(t^l+f) and a >= -p >= 1
PASS E9 l=2 p=-1 k=-1: a = -p - k l t^l/(t^l+f) and a >= -p >= 1
PASS E9 l=2 p=-3/2 k=-2/3: a = -p - k l t^l/(t^l+f) and a >= -p >= 1
PASS E9 l=2 p=-5 k=-7/2: a = -p - k l t^l/(t^l+f) and a >= -p >= 1
PASS E9 l=3 p=-1 k=-1: a = -p - k l t^l/(t^l+f) and a >= -p >= 1
PASS E9 l=3 p=-3/2 k=-2/3: a = -p - k l t^l/(t^l+f) and a >= -p >= 1
PASS E9 l=3 p=-5 k=-7/2: a = -p - k l t^l/(t^l+f) and a >= -p >= 1
PASS E9 l=4 p=-1 k=-1: a = -p - k l t^l/(t^l+f) and a >= -p >= 1
PASS E9 l=4 p=-3/2 k=-2/3: a = -p - k l t^l/(t^l+f) and a >= -p >= 1
PASS E9 l=4 p=-5 k=-7/2: a = -p - k l t^l/(t^l+f) and a >= -p >= 1
PASS E9 l=5 p=-1 k=-1: a = -p - k l t^l/(t^l+f) and a >= -p >= 1
PASS E9 l=5 p=-3/2 k=-2/3: a = -p - k l t^l/(t^l+f) and a >= -p >= 1
PASS E9 l=5 p=-5 k=-7/2: a = -p - k l t^l/(t^l+f) and a >= -p >= 1
PASS E10 Remark 5.2 algebra on alpha in [-6,6] step 1/10
PASS E11 l=1: F d_z(F d_z z^-1) = 2x^(2l+2)(x^(2l)+z^2)/z^3
PASS E11 l=1: boundary form F(W U_z - U W_z) = x phi^2 (1 + x^(2l)/z^2)
PASS E11 l=1: Delta U = phi'' - (F_x/F) phi'
PASS E11 l=1: Delta W = (phi''-(F_x/F)phi')/z + phi F d_z(F d_z z^-1)
PASS E11 l=1: (W Delta U - U Delta W)/F is the divergence d_x(..) + d_z(F(W U_z - U W_z))
PASS E11 l=1: Delta_g = X1^2+X2^2+div(X_i)X_i
PASS E11 l=2: F d_z(F d_z z^-1) = 2x^(2l+2)(x^(2l)+z^2)/z^3
PASS E11 l=2: boundary form F(W U_z - U W_z) = x phi^2 (1 + x^(2l)/z^2)
PASS E11 l=2: Delta U = phi'' - (F_x/F) phi'
PASS E11 l=2: Delta W = (phi''-(F_x/F)phi')/z + phi F d_z(F d_z z^-1)
PASS E11 l=2: (W Delta U - U Delta W)/F is the divergence d_x(..) + d_z(F(W U_z - U W_z))
PASS E11 l=2: Delta_g = X1^2+X2^2+div(X_i)X_i
PASS E11 l=3: F d_z(F d_z z^-1) = 2x^(2l+2)(x^(2l)+z^2)/z^3
PASS E11 l=3: boundary form F(W U_z - U W_z) = x phi^2 (1 + x^(2l)/z^2)
PASS E11 l=3: Delta U = phi'' - (F_x/F) phi'
PASS E11 l=3: Delta W = (phi''-(F_x/F)phi')/z + phi F d_z(F d_z z^-1)
PASS E11 l=3: (W Delta U - U Delta W)/F is the divergence d_x(..) + d_z(F(W U_z - U W_z))
PASS E11 l=3: Delta_g = X1^2+X2^2+div(X_i)X_i
PASS E11 l=4: F d_z(F d_z z^-1) = 2x^(2l+2)(x^(2l)+z^2)/z^3
PASS E11 l=4: boundary form F(W U_z - U W_z) = x phi^2 (1 + x^(2l)/z^2)
PASS E11 l=4: Delta U = phi'' - (F_x/F) phi'
PASS E11 l=4: Delta W = (phi''-(F_x/F)phi')/z + phi F d_z(F d_z z^-1)
PASS E11 l=4: (W Delta U - U Delta W)/F is the divergence d_x(..) + d_z(F(W U_z - U W_z))
PASS E11 l=4: Delta_g = X1^2+X2^2+div(X_i)X_i
PASS E11 l=5: F d_z(F d_z z^-1) = 2x^(2l+2)(x^(2l)+z^2)/z^3
PASS E11 l=5: boundary form F(W U_z - U W_z) = x phi^2 (1 + x^(2l)/z^2)
PASS E11 l=5: Delta U = phi'' - (F_x/F) phi'
PASS E11 l=5: Delta W = (phi''-(F_x/F)phi')/z + phi F d_z(F d_z z^-1)
PASS E11 l=5: (W Delta U - U Delta W)/F is the divergence d_x(..) + d_z(F(W U_z - U W_z))
PASS E11 l=5: Delta_g = X1^2+X2^2+div(X_i)X_i
PASS E11 l=6: F d_z(F d_z z^-1) = 2x^(2l+2)(x^(2l)+z^2)/z^3
PASS E11 l=6: boundary form F(W U_z - U W_z) = x phi^2 (1 + x^(2l)/z^2)
PASS E11 l=6: Delta U = phi'' - (F_x/F) phi'
PASS E11 l=6: Delta W = (phi''-(F_x/F)phi')/z + phi F d_z(F d_z z^-1)
PASS E11 l=6: (W Delta U - U Delta W)/F is the divergence d_x(..) + d_z(F(W U_z - U W_z))
PASS E11 l=6: Delta_g = X1^2+X2^2+div(X_i)X_i
PASS E11 l=1: F(1,z) = 1+z^2 (curve length int_0^inf dz/(1+z^2) = pi/2)
PASS E11 l=2: F(1,z) = 1+z^2 (curve length int_0^inf dz/(1+z^2) = pi/2)
PASS E11 l=3: F(1,z) = 1+z^2 (curve length int_0^inf dz/(1+z^2) = pi/2)
PASS E11 l=4: F(1,z) = 1+z^2 (curve length int_0^inf dz/(1+z^2) = pi/2)
PASS E12: a(x, x+x^2) = 3 - 1/x
PASS E12: x^2 Veff(x, x) = 35/4 - 1/x^2
PASS E13: (5/4)^3 (1+(5/4)^4) = 110125/16384
PASS E13: (1/4)((4/5)^4-(4/7)^4) = 113664/1500625
PASS E13: D constant 0.022538 > 1/45
PASS E13: ramp integrals 64 and 16/3 (units c1^-1 z^-4)
PASS E13: A <= (208/3)(7/24) = 182/9 <= 21
PASS E13: (1/4)(1/4) = 1/16 and (3/2)(1/4) = 3/8
PASS E13: 2(1+16 z^2) <= 4 for z <= 1/4
PASS E13: support area = 21/8 z0^2
PASS E13: B <= 32*144*21/8 = 12096
PASS E13: 7/32 <= 1/4 (so z+z^2 <= 5z/4 and |z-x|<=z^2<=1 on P)
PASS E13: 1/(1/45) = 45
PASS E13: (4/pi^2) y^2 <= sin^2 y <= y^2 on [-1,1] (float grid)
PASS E14 l=1: d_z log rho at z=x^l equals -x^(-l)
PASS E14 l=2: d_z log rho at z=x^l equals -x^(-l)
PASS E14 l=3: d_z log rho at z=x^l equals -x^(-l)
PASS E14 l=4: d_z log rho at z=x^l equals -x^(-l)
PASS E14 l=5: d_z log rho at z=x^l equals -x^(-l)
PASS E15 trial 0 k=1: x d_x log (x phi)^k = k(1 + x phi_x/phi)
PASS E15 trial 0 k=2: x d_x log (x phi)^k = k(1 + x phi_x/phi)
PASS E15 trial 0 k=3: x d_x log (x phi)^k = k(1 + x phi_x/phi)
PASS E15 trial 0: x h_x = sum 2j c_j x^(2j)
PASS E15 trial 1 k=1: x d_x log (x phi)^k = k(1 + x phi_x/phi)
PASS E15 trial 1 k=2: x d_x log (x phi)^k = k(1 + x phi_x/phi)
PASS E15 trial 1 k=3: x d_x log (x phi)^k = k(1 + x phi_x/phi)
PASS E15 trial 1: x h_x = sum 2j c_j x^(2j)
PASS E15 trial 2 k=1: x d_x log (x phi)^k = k(1 + x phi_x/phi)
PASS E15 trial 2 k=2: x d_x log (x phi)^k = k(1 + x phi_x/phi)
PASS E15 trial 2 k=3: x d_x log (x phi)^k = k(1 + x phi_x/phi)
PASS E15 trial 2: x h_x = sum 2j c_j x^(2j)
PASS E0 control: a != 1+3u (l=2)
PASS E0 control: a != 1+4u+x (l=2)
PASS E0 control: x^2 Veff != 3/4 (l=2)
PASS E0 control: det frame Ex.7.8 (n=3) != F^4

311/311 exact checks passed
