To motivate physics applications of complex analysis, let us give an example of a function in fluid dynamics which satisfies the Cauchy-Riemann differential equations for complex differentiable (i.e. holomorphic) functions (when written in a complex form). One such function is the velocity vector field of an incompressible, irrotational flow in two dimensions. (Because viscosity introduces vorticity, this flow typically has to be inviscid as well.) That the components of this two dimensional velocity vector field satisfy the Cauchy-Riemann equations follows from the following.

The first Cauchy-Riemann differential equation for the velocity vector is obtained from the incompressibility requirement \(\rho=\mathrm{const.}\), which results in \(\frac{\partial\rho}{\partial t} = 0\) and \(\nabla\rho=0\). With the continuity equation
\begin{equation}
\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho\mathbf{V}) = 0,
\end{equation}
this translates to \(\nabla \cdot \mathbf{V}= 0\) (since \(\rho\) is constant, one can simply pull it in front of the derivative), which written in components is
\begin{equation}
\nabla \cdot \mathbf{V} = \frac{\partial V_1}{\partial x} + \frac{\partial V_2}{\partial y} + \underbrace{\frac{\partial V_3}{\partial z}}_{=0}=0
\end{equation}
The last term is always zero for a 2-dimensional flow in the \(xy\)-plane, because it is the derivative with respect to the third coordinate \(z\) (yet the vector field is constant in this third dimension, since we are only looking at a 2-dimensional flow). To make it look the same as the first Cauchy-Riemann equation obtained previously for \(f_1\) and \(f_2\), we need to make the assignment \(f_1=V_1\) and \(f_2=-V_2\) (notice the minus sign for the second component).

For an irrotational flow we have by definition \(\nabla \times \mathbf{V} = \mathbf{0}\) (see our multivariable calculus primer), which, written in components, leads to
\begin{equation}
\nabla \times \mathbf{V} =
\begin{pmatrix}
\frac{\partial}{\partial x}\\
\frac{\partial}{\partial y}\\
\frac{\partial}{\partial z}
\end{pmatrix}
\times
\begin{pmatrix}
V_1 \\
V_2 \\
V_3
\end{pmatrix}
=
\begin{pmatrix}
\frac{\partial V_3}{\partial y} – \frac{\partial V_2}{\partial z}\\
\frac{\partial V_1}{\partial z} – \frac{\partial V_3}{\partial x}\\
\frac{\partial V_2}{\partial x} – \frac{\partial V_1}{\partial y}
\end{pmatrix}
= \mathbf{0}
\end{equation}
The first two component are irrelevant for a 2-dimensional flow in the \(xy\)-plane, because derivatives with respect to the coordinate in the third dimension, \(z\), are zero. We are thus left with
\begin{equation}
\frac{\partial V_2}{\partial x} – \frac{\partial V_1}{\partial y} = 0
\end{equation}
which is exactly the second Cauchy-Riemann differential equation we obtained previously, if we assign again \(f_1=V_1\) and \(f_2=-V_2\).

We can therefore define the complex velocity \(W(z)=V_1(x,y)-iV_2(x,y)\) (notice the minus sign) and \(z=x+iy\), which becomes a complex differentiable (i.e. holomorphic) function, since its real and imaginary part satisfy the Cauchy-Riemann equations. Then we can unleash the full machinery of complex analysis onto the complex velocity \(W(z)\) (including the Cauchy integral formula, residual theorem, Laurent series, etc.) and do fluid dynamic calculations this way. We will do this, for instance, during the formal proof of the Kutta-Joukowski theorem in two-dimensional aerodynamics, which relates the lift generated by an airfoil to the incoming airflow velocity and the circulation around the airfoil.

In a similar way, one can also define the complex potential \(F(z)=\phi(x,y) + i \psi(x,y)\) from the real velocity potential \(\phi(x,y)\) and the real stream function \(\psi(x,y)\) (\(\psi\) is constant along streamlines, which are lines tangent to the velocity vector field) of the incompressible, irrotational flow, and show that the real and imaginary parts of \(F(z)\) obey the Cauchy-Riemann equations. Taking the complex derivative (which exists, because holomorphic functions are infinitely differentiable), we obtain \(dF/dz = V_1 – iV_2 = W\) for the relation between the complex potential \(F(z)\) and the complex velocity \(W(z)\).