d2E_dt2.sagews

%typeset_mode True

var('c, rho, psi, t, E_0, k, omega')
assume(c, 'real')
assume(c > 0)
assume(rho, 'real')
assume(rho > 0)
assume(E_0, 'real')
assume(E_0, 'constant')
assume(k, 'real')
assume(k, 'constant')
assume(omega, 'real')
assume(omega, 'constant')

($\displaystyle c$, $\displaystyle \rho$, $\displaystyle \psi$, $\displaystyle t$, $\displaystyle E_{0}$, $\displaystyle k$, $\displaystyle \omega$)

E = E_0 * e^(1j * (k * rho * cos(psi) - omega * t))
E.simplify_full()

$\displaystyle E_{0} e^{\left(1.0i \, k \rho \cos\left(\psi\right) - 1.0i \, \omega t\right)}$

d2E_dt2 = c**2 * ((1 / rho) * diff(rho * diff(E, rho), rho) + (1 / rho**2) * diff(diff(E, psi), psi))
d2E_dt2.simplify_full()

$\displaystyle -E_{0} c^{2} k^{2} e^{\left(i \, k \rho \cos\left(\psi\right) - i \, \omega t\right)}$