a harmonic conjugate
Two real functions u(x, y) and v(x, y) are harmonic conjugates when together they assemble into a holomorphic function f = u + i v. The word 'conjugate' here is a partnership: v is the unique-up-to-a-constant partner that v must be so that u + i v becomes complex-differentiable, with the two locked together by the Cauchy-Riemann equations u_x = v_y, u_y = -v_x.
Given a harmonic u, you build its conjugate v by integrating the Cauchy-Riemann equations. Since v_x = -u_y and v_y = u_x, you integrate v_x = -u_y in x to get v up to a function of y, then differentiate and match v_y = u_x to pin down that function. For example, with u = x^2 - y^2 we need v_x = -u_y = 2y and v_y = u_x = 2x; integrating gives v = 2xy + C, and indeed u + i v = (x^2 - y^2) + i(2xy) = z^2.
The relationship is NOT symmetric in an important way: if v is a harmonic conjugate of u, then u is a harmonic conjugate of -v (a sign flip), so the roles are not interchangeable without care. Geometrically the level curves of u and of v meet at right angles (orthogonal families), which is why harmonic conjugates appear together as the equipotential lines and stream lines of a flow in physics. Constructing a global conjugate also needs the domain to be simply connected, or the conjugate may be multivalued.
Take u = e^x cos y (harmonic: u_xx + u_yy = e^x cos y - e^x cos y = 0). Its conjugate satisfies v_x = -u_y = e^x sin y and v_y = u_x = e^x cos y, giving v = e^x sin y; then u + i v = e^x(cos y + i sin y) = e^z.
The conjugate is found by integrating the Cauchy-Riemann equations; here u and v assemble into e^z.
A conjugate exists locally always, but a single-valued GLOBAL conjugate needs a simply connected domain; on a ring-shaped region (think log z) the conjugate can be multivalued.