Hello,
I was double-checking an integral with sympy, and noticed that the
software comes up with the wrong answer. The integrand has five terms,
which can be collected into three groups. Two of the groups -- Ec1 and Ec2
-- cancel after integrating over z. Strangely, sympy gets the answer right
if you ask it to integrate the groups separately.
x, z, R, a = symbols('x z R a')
r = sqrt(x**2 + z**2)
u = erf(a*r/sqrt(2))/r
Ec = diff(u, z, z).subs([(x, sqrt(R*R-z*z))])
# compare
simplify(integrate(Ec, (z, -R, R)))
# to the analytical answer
-2*sqrt(2)*R*a**3*exp(-R**2*a**2/2)/(3*sqrt(pi))
# in particular, Ec = Ec1 + Ec2 + Ec3, where
Ec1 = sympify('3*z**2*erf(sqrt(2)*a*sqrt(R**2)/2)/(R**2)**(5/2) -
erf(sqrt(2)*a*sqrt(R**2)/2)/(R**2)**(3/2)')
Ec2 = sympify('+ sqrt(2)*a*exp(-R**2*a**2/2)/(sqrt(pi)*R**2) -
3*sqrt(2)*a*z**2*exp(-R**2*a**2/2)/(sqrt(pi)*R**4)')
Ec3 = Ec - Ec1 - Ec2 # -sqrt(2)*a**3*z**2*exp(-R**2*a**2/2)/(sqrt(pi)*R**2)
simplify(integrate(Ec1, (z, -R, R))) # -> 0
simplify(integrate(Ec2, (z, -R, R))) # -> 0
integrate(Ec3, (z, -R, R))
~ David M. Rogers
University of South Florida
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