In mathematics, a real-valued function ${\displaystyle u(x,y)}$ defined on a connected open set ${\displaystyle \Omega \subset \mathbb {R} ^{2}}$ is said to have a conjugate (function) ${\displaystyle v(x,y)}$ if and only if they are respectively the real and imaginary parts of a holomorphic function ${\displaystyle f(z)}$ of the complex variable ${\displaystyle z:=x+iy\in \Omega .}$ That is, ${\displaystyle v}$ is conjugate to ${\displaystyle u}$ if ${\displaystyle f(z):=u(x,y)+iv(x,y)}$ is holomorphic on ${\displaystyle \Omega .}$ As a first consequence of the definition, they are both harmonic real-valued functions on ${\displaystyle \Omega }$. Moreover, the conjugate of ${\displaystyle u,}$ if it exists, is unique up to an additive constant. Also, ${\displaystyle u}$ is conjugate to ${\displaystyle v}$ if and only if ${\displaystyle v}$ is conjugate to ${\displaystyle -u}$.

Equivalently, ${\displaystyle v}$ is conjugate to ${\displaystyle u}$ in ${\displaystyle \Omega }$ if and only if ${\displaystyle u}$ and ${\displaystyle v}$ satisfy the Cauchy–Riemann equations in ${\displaystyle \Omega .}$ As an immediate consequence of the latter equivalent definition, if ${\displaystyle u}$ is any harmonic function on ${\displaystyle \Omega \subset \mathbb {R} ^{2},}$ the function ${\displaystyle -u_{y}}$ is conjugate to ${\displaystyle u_{x}}$ for then the Cauchy–Riemann equations are just ${\displaystyle \Delta u=0}$ and the symmetry of the mixed second order derivatives, ${\displaystyle u_{xy}=u_{yx}.}$ Therefore, a harmonic function ${\displaystyle u}$ admits a conjugated harmonic function if and only if the holomorphic function ${\displaystyle g(z):=u_{x}(x,y)-iu_{y}(x,y)}$ has a primitive ${\displaystyle f(z)}$ in ${\displaystyle \Omega ,}$ in which case a conjugate of ${\displaystyle u}$ is, of course, ${\displaystyle \operatorname {Im} f(x+iy).}$ So any harmonic function always admits a conjugate function whenever its domain is simply connected, and in any case it admits a conjugate locally at any point of its domain.

There is an operator taking a harmonic function u on a simply connected region in ${\displaystyle \mathbb {R} ^{2}}$ to its harmonic conjugate v (putting e.g. v(x₀) = 0 on a given x₀ in order to fix the indeterminacy of the conjugate up to constants). This is well known in applications as (essentially) the Hilbert transform; it is also a basic example in mathematical analysis, in connection with singular integral operators. Conjugate harmonic functions (and the transform between them) are also one of the simplest examples of a Bäcklund transform (two PDEs and a transform relating their solutions), in this case linear; more complex transforms are of interest in solitons and integrable systems.

Geometrically u and v are related as having orthogonal trajectories, away from the zeros of the underlying holomorphic function; the contours on which u and v are constant cross at right angles. In this regard, u + iv would be the complex potential, where u is the potential function and v is the stream function.

For example, consider the function $${\displaystyle u(x,y)=e^{x}\sin y.}$$

Since $${\displaystyle {\partial u \over \partial x}=e^{x}\sin y,\quad {\partial ^{2}u \over \partial x^{2}}=e^{x}\sin y}$$ and $${\displaystyle {\partial u \over \partial y}=e^{x}\cos y,\quad {\partial ^{2}u \over \partial y^{2}}=-e^{x}\sin y,}$$ it satisfies $${\displaystyle \Delta u=\nabla ^{2}u=0}$$ (${\displaystyle \Delta }$ is the Laplace operator) and is thus harmonic. Now suppose we have a ${\displaystyle v(x,y)}$ such that the Cauchy–Riemann equations are satisfied:

$${\displaystyle {\partial u \over \partial x}={\partial v \over \partial y}=e^{x}\sin y}$$ and $${\displaystyle {\partial u \over \partial y}=-{\partial v \over \partial x}=e^{x}\cos y.}$$

Simplifying, $${\displaystyle {\partial v \over \partial y}=e^{x}\sin y}$$ and $${\displaystyle {\partial v \over \partial x}=-e^{x}\cos y}$$ which when solved gives $${\displaystyle v=-e^{x}\cos y+C.}$$