recovering a holomorphic function from its real part
A striking consequence of the harmonic-conjugate idea: a holomorphic function is so rigid that, on a nice region, knowing JUST its real part u(x, y) determines the whole function f = u + i v up to an additive imaginary constant. Half the information secretly carries the other half. The procedure to rebuild f is sometimes called the Milne-Thomson method.
Here is the method in plain steps. Start with a harmonic u (you should check u_xx + u_yy = 0). The imaginary part v is its harmonic conjugate, found from the Cauchy-Riemann equations: integrate v_y = u_x with respect to y to get v up to a function g(x); differentiate that v in x and match it to v_x = -u_y to solve for g'(x), then integrate to get g. Assembling f = u + i v gives the holomorphic function, unique once you fix the value at one point (the constant of integration).
Why care? In applied problems you often know the physical potential u (a temperature or voltage) and want the full complex potential, whose imaginary part v gives the stream lines or flux. The reconstruction works cleanly on simply connected regions; on a region with a hole the conjugate v may fail to be single-valued (the model case is u = ln|z|, whose conjugate is the multivalued arg z), a first glimpse of why branch cuts and multivaluedness are unavoidable in this subject.
Given u = x^3 - 3 x y^2 (harmonic). Then v_y = u_x = 3x^2 - 3y^2, so v = 3 x^2 y - y^3 + g(x); matching v_x = 6 x y to -u_y = 6 x y forces g'(x) = 0. Thus v = 3 x^2 y - y^3 and f = u + i v = z^3.
From u alone, the Cauchy-Riemann equations rebuild v and hence f, here recovering z^3.
It only works if u is genuinely harmonic and the region is simply connected; on a ring (like the plane minus the origin) the recovered v can be multivalued, as with ln|z| and arg z.