"Verify equation (1)"

f = sin(theta1) sin(theta2)

I = defint(f,theta1,0,pi,theta2,0,pi,phi1,0,2pi,phi2,0,2pi)

check(I == 16 pi^2)

"ok"

"Verify equation (2)"

psi = alpha^3 / pi exp(-alpha (r1 + r2))

Laplacian(psi) = 1 / r1^2 d(r1^2 d(psi, r1), r1)

f = -1/2 psi Laplacian(psi) r1^2 r2^2

I = 16 pi^2 integral(f,r1,r2)

I = 0 - eval(I,r1,0) -- I evaluated at r1 = infinity is zero
I = 0 - eval(I,r2,0) -- I evaluated at r2 = infinity is zero

check(I == 1/2 alpha^2)

"ok"

"Verify equation (3)"

f = -psi (Z / r1) psi r1^2 r2^2

I = 16 pi^2 integral(f,r1,r2)

I = 0 - eval(I,r1,0) -- I evaluated at r1 = infinity is zero
I = 0 - eval(I,r2,0) -- I evaluated at r2 = infinity is zero

check(I == -Z alpha)

"ok"

"Verify equation (6)"

A = 4 alpha^3 / r1 integral(exp(-2 alpha r2) r2^2, r2)
B = 4 alpha^3 integral(exp(-2 alpha r2) r2, r2)

check(A == 4 alpha^3 / r1 exp(-2 alpha r2) *
(-r2^2 / (2 alpha) - r2 / (2 alpha^2) - 1 / (4 alpha^3)))

check(B == 4 alpha^3 exp(-2 alpha r2) (-r2 / (2 alpha) - 1 / (4 alpha^2)))

A = eval(A,r2,r1) - eval(A,r2,0)
B = 0 - eval(B,r2,r1)

I = A + B

check(I == 1/r1 - 1/r1 exp(-2 alpha r1) - alpha exp(-2 alpha r1))

"ok"

"Verify equation (7)"

I = 4 alpha^3 integral(exp(-2 alpha r1) I r1^2, r1)

check(I == exp(-2 alpha r1) (-2 alpha^2 r1 - alpha) -
exp(-4 alpha r1) (-alpha^2 r1 - 1/4 alpha) -
exp(-4 alpha r1) (-alpha^3 r1^2 - 1/2 alpha^2 r1 - 1/8 alpha))

I = 0 - eval(I,r1,0)

check(I == 5/8 alpha)

"ok"

"Verify equation (5)"

P(f,n) = eval(d((x^2 - 1)^n, x, n), x, f) / (2^n n!) -- Rodrigues's formula

check(defint(P(expcos(theta),0) expsin(theta), theta, 0, pi) == 2)
check(defint(P(expcos(theta),1) expsin(theta), theta, 0, pi) == 0)
check(defint(P(expcos(theta),2) expsin(theta), theta, 0, pi) == 0)
check(defint(P(expcos(theta),10) expsin(theta), theta, 0, pi) == 0)

"ok"