820.sagews

# MATHEMATICAL METHODS OF Science and Engineering
# FORTRAN codes / Matlab codes
#http://www1.uprh.edu/rbaretti
#http://www1.uprh.edu/rbaretti/Methodsoftheoreticalphysics.htm
#http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysicspart2.htm
#http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysicsPart3.htm
#http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysicsPart4.htm
#http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysicsPart5.htm

r=var('r')
f(r)=r^4*exp(-r)
integral(f(r),r,0,oo)

24

z=var('z')
f(z)=sin(z)^3
integral(f(z),z,0,pi)

4/3

x=var('x');assume (a>0);
integral(exp(-a*x^2),x,0,oo)

1/2*sqrt(pi)/sqrt(a)

#ap+1 = 2{ λ(p+l +1) -1} ap / [ (p+l+1)(p+l+2) - l(l+1)  ]
lamda=1/3
p=1 
l=0
aminus=-2/3
a=2*(lamda*(p+l+1)-1)*aminus/((p+l+1)*(p+l+2)-l*(l+1) )
a

2/27

# $ 2r2 -18r +27 }2 exp(-2r/3) 4π r2 dr 
r=var('r')
f=(2*r^2-18*r+27)^2*exp(-2*r/3)

integral(f*4*pi*r^2,r,0,oo)

19683*pi

#exp(-2r) 4π r2 
r=var('r')
integral(exp(-2*r)*4*pi*r^2,r,0,oo)

pi

r=var('r')
psi=(1/pi^(1/2))*exp(-r)
integral(r*psi^2*4*pi*r^2,r,0,oo)

3/2