3848.sagews

Computes a harmonic conjugate for $u$.

def harmonicConjugate(u):
    ux = diff(u, x)
    uy = diff(u, y)
    t = integrate(ux, y)
    return t + (diff(t, x) + uy)

u(x,y) = exp(x) * (x * cos(y) - y * sin(y))
f(z) = u(real(z),imag(z)) + i*harmonicConjugate(u)(real(z),imag(z))
f.simplify()

\newcommand{\Bold}[1]{\mathbf{#1}}z \ {\mapsto}\ z e^{z}