Remark 4.3.1 Examinable content.
The proofs in this section are non-examinable, but you are expected to understand the theorems and their relevance to potential flow.
Section 4.3 The complex potential
Aside: Changed in 2026.
In the previous two sections, we studied the velocity potential \(\phi\) and streamfunction \(\psi\) of two-dimensional potential flows using real-valued Vector Calculus. You may have noticed a certain symmetry between the formulae for \(\phi\) and \(\psi\text{.}\) Complex analysis makes this symmetry transparent, and provides a much more elegant and powerful framework for studying two-dimensional flow.
Subsection 4.3.1 Analytic functions and Laplace’s equation
We write a complex function as \(f(z) = u(x, y) + \im v(x, y)\text{,}\) where \(z = x + \im y\) and \(u\) and \(v\) are the real and imaginary parts of \(f\text{,}\) and use subscripts for partial derivatives, e.g. \(u_x = \pd{u}{x}\text{.}\) The derivative is defined as for real functions,
\begin{equation}
f'(z) = \lim_{\Delta z \to 0} \frac{f(z + \Delta z) - f(z)}{\Delta z},\tag{4.3.1}
\end{equation}
Definition 4.3.2 Analyticity.
A function \(f\) is analytic in a domain if it is differentiable at every point of the domain.
Theorem 4.3.3 Cauchy-Riemann equations.
If \(f(z) = u(x, y) + \im v(x, y)\) is analytic in a domain, then the Cauchy-Riemann equations
\begin{equation}
u_x = v_y \quad \text{and} \quad u_y = -v_x\tag{4.3.2}
\end{equation}
hold in that domain. Conversely, if \(u\) and \(v\) have continuous first partial derivatives satisfying (4.3.2), then \(f\) is analytic.
Proof (non-examinable).
Approaching \(z\) along the real direction, \(\Delta z = \Delta x\text{,}\) the definition (4.3.1) gives \(f'(z) = u_x + \im v_x\text{.}\) Approaching along the imaginary direction, \(\Delta z = \im \Delta y\text{,}\) gives \(f'(z) = -\im u_y + v_y\text{.}\) Equating the real and imaginary parts of the two results yields the Cauchy-Riemann equations. For the converse, see [2].
Theorem 4.3.4 Analyticity and Laplace’s equation.
If \(f(z) = u(x, y) + \im v(x, y)\) is analytic in a domain \(D\text{,}\) then \(u\) and \(v\) satisfy Laplace’s equation,
\begin{equation*}
\nabla^2 u = u_{xx} + u_{yy} = 0 \quad \textrm{and} \quad
\nabla^2 v = v_{xx} + v_{yy} = 0,
\end{equation*}
in \(D\text{.}\)
Proof (non-examinable).
Analytic functions are infinitely differentiable (see [2]), so we may differentiate (4.3.2) to obtain
\begin{equation*}
u_{xx} = v_{yx} \quad \text{and} \quad -u_{yy} = v_{xy}.
\end{equation*}
The second derivatives are continuous, so \(v_{xy} = v_{yx}\text{,}\) and therefore \(u_{xx} + u_{yy} = 0\text{.}\) Laplace’s equation is analogously proved for \(v\text{.}\)
This remarkable result is a major reason why complex analysis played such an important role in applied mathematics and physics, since Laplace’s equation governs gravitation, electrostatics, fluid mechanics, and more. In particular, any function that is analytic in a region, whether a polynomial, exponential, logarithm, or otherwise, can be associated with (some kind of) potential flow there.
Subsection 4.3.2 Examples of elementary flows
Let us return to the examples of flows studied in Section 4.1 and re-interpret them using the theory of analytic functions.
We will associate the velocity potential, \(\phi\text{,}\) and streamfunction \(\psi\) with an analytic function in the following way.
Definition 4.3.5 Complex potential.
Let \(\phi(x,y)\) and \(\psi(x, y)\) be the respective velocity potential and streamfunction for some potential flow. We define
\begin{equation}
f(z) = \phi(x, y) + \im \psi(x, y),\tag{4.3.3}
\end{equation}
and call \(f\) the complex potential. Since \(\phi_x = \psi_y = u\) and \(\phi_y = -\psi_x = v\text{,}\) the functions \(\phi\) and \(\psi\) satisfy the Cauchy-Riemann equations, so \(f\) is analytic.
Note that by differentiating in the horizontal and vertical directions, we have
\begin{equation*}
\dd{f}{z} = \pd{}{x} (\phi + \im \psi) = -\im \pd{}{y} (\phi + \im \psi).
\end{equation*}
Therefore it follows that the horizontal and vertical velocity components of the flow, related via \(\bu = [u, v]\text{,}\) are given by
\begin{equation}
\dd{f}{z} = u - \im v.\tag{4.3.4}
\end{equation}
Example 4.3.6 Uniform flow.
Show that the complex potential of the uniform flow of Example 4.1.2 is \(f(z) = U \e^{-\im \alpha} z\text{,}\) and recover \(\phi\text{,}\) \(\psi\) and \(\bu\text{.}\)
Solution.
With \(z = x + \im y\text{,}\)
\begin{equation*}
U\e^{-\im\alpha}z = U(x\cos\alpha + y\sin\alpha) + \im U(y\cos\alpha - x\sin\alpha),
\end{equation*}
whose real and imaginary parts are the \(\phi\) and \(\psi\) of Example 4.1.2 and the corresponding example in Section 4.2. Also \(f'(z) = U\e^{-\im\alpha} = U\cos\alpha - \im U\sin\alpha\text{,}\) so \(\bu = U[\cos\alpha, \sin\alpha]\text{.}\)
Example 4.3.7 Stagnation point flow.
Show that the complex potential of the stagnation point flow of Example 4.1.3 is \(f(z) = z^2/2\text{,}\) and recover \(\phi\text{,}\) \(\psi\) and \(\bu\text{.}\)
Solution.
We have \(z^2/2 = \frac12(x^2 - y^2) + \im xy\text{,}\) so \(\phi = \frac12(x^2 - y^2)\) and \(\psi = xy\text{.}\) Also \(f'(z) = z = x + \im y = u - \im v\text{,}\) so \(\bu = [x, -y]\text{.}\)
Example 4.3.8 Line source.
Show that the complex potential of the line source of Example 4.1.4 is
\begin{equation}
f(z) = \frac{Q}{2\pi} \log z,\tag{4.3.5}
\end{equation}
and recover \(\phi\text{,}\) \(\psi\) and \(\bu\text{.}\)
Solution.
The complex logarithm is not analytic at \(z = 0\text{,}\) a branch point, but is analytic away from the origin once a branch cut is chosen. Writing \(z = r\e^{\im \theta}\) in polar form, it is evaluated via
\begin{equation}
\log z = \log r + \im \theta.\tag{4.3.6}
\end{equation}
Hence
\begin{equation*}
\phi = \Re[f] = \frac{Q}{2\pi} \log r \quad \text{and} \quad
\psi = \Im[f] = \frac{Q}{2\pi} \theta.
\end{equation*}
Indeed, the streamlines are the rays \(\theta = \textrm{constant}\text{.}\) From (4.3.4), multiplying the top and bottom by the conjugate of \(z\text{,}\)
\begin{equation*}
u - \im v = \frac{Q}{2\pi} \frac{1}{z} = \frac{Q}{2\pi} \frac{x - \im y}{x^2 + y^2}.
\end{equation*}
Example 4.3.9 Line vortex flow.
Show that the complex potential of the line vortex of Example 4.2.9 is
\begin{equation}
f(z) = -\frac{\im \Gamma}{2\pi} \log z,\tag{4.3.7}
\end{equation}
and recover \(\phi\text{,}\) \(\psi\) and \(\bu\text{.}\)
Solution.
Using (4.3.6) with \(z = r\e^{\im\theta}\text{,}\)
\begin{equation*}
f(z) = \frac{\Gamma}{2\pi} (\theta - \im \log r),
\end{equation*}
so \(\phi = \Gamma\theta/(2\pi)\) and \(\psi = -(\Gamma/2\pi) \log r\text{,}\) which is constant on circles \(r = \textrm{constant}\text{.}\) The velocity follows from
\begin{equation*}
u - \im v = -\frac{\im\Gamma}{2\pi z} = -\frac{\im\Gamma}{2\pi r}\e^{-\im\theta},
\end{equation*}
so \(u + \im v = \frac{\Gamma}{2\pi r}\im\e^{\im\theta}\text{,}\) i.e. \(\bu = \frac{\Gamma}{2\pi r}\be_\theta\text{,}\) as in (4.2.9).
