Complex Variable Analysis

dtm · May 14, 2026, 3:53am

In complex variable analysis, we need to make sure our expression is analytic, but what does analytic mean? Could you give me examples and non-examples?

Professor_Kirby_Bot · May 14, 2026, 3:54am

This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook

PK Bot’s answer:
I focus on fluid mechanics and related topics, so I will address the concept of analyticity in the context of fluid mechanics, particularly in potential flow theory.

In fluid mechanics, an analytic function is a complex function that is differentiable at every point in its domain. This concept is crucial in potential flow theory, where complex potential functions are used to describe irrotational, incompressible flows.

Examples in Fluid Mechanics:

Complex Potential Function: In potential flow, the complex potential W(z) = \phi(x, y) + i\psi(x, y) is analytic, where \phi is the velocity potential and \psi is the stream function. The real and imaginary parts of W(z) satisfy the Cauchy-Riemann equations, ensuring that the flow is irrotational and incompressible.
Flow Around a Cylinder: The complex potential for flow around a cylinder is given by W(z) = U_\infty \left(z + \frac{a^2}{z}\right), where U_\infty is the freestream velocity and a is the cylinder radius. This function is analytic everywhere except at z = 0, which corresponds to the center of the cylinder.

Non-Examples:

Discontinuous Functions: A function with discontinuities or sharp corners is not analytic. For instance, a potential function with a discontinuity in velocity or pressure would not be analytic.
Functions with Singularities: A function with singularities, such as poles or branch points, is not analytic at those points. In the cylinder example, the point z = 0 is a singularity, so the function is not analytic there.

In summary, an analytic function in fluid mechanics is one that is smooth and differentiable, allowing for the application of complex analysis techniques to solve potential flow problems.

In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections according to their relevance. If they are marked as relevant, I encourage you to explore them:

Textbook sections:

Chapter 9.3: Why We Use Irrotational Flow Analysis Part 3 Part 1 (HIGHLY RELEVANT)
Chapter 9 Fundamentals: Analogy between potential flow and electrostatics (HIGHLY RELEVANT)
Chapter 9.3: Why We Use Irrotational Flow Analysis Part 4 Part 1 (HIGHLY RELEVANT)
Chapter 9.3: Why We Use Irrotational Flow Analysis Part 5 (HIGHLY RELEVANT)
Chapter 5.1: Inviscid Incompressible Flow (RELEVANT)

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See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.

profbriankirby · May 14, 2026, 1:49pm

I should give Professor Kirby Bot a copy of my microfluidics textbook.

The key thing is that 2D complex representation of potential flow is a way to prescribe that the Laplace equation is solved without every really solving it. Just as the definition of \phi automatically solves \nabla\times\vec{u}=0, and the definition of \psi automatically solves \nabla\cdot\vec{u}=0, the definition of \underset{\sim}{\phi}(\underset{\sim}{z})=\phi+j\psi, where \underset{\sim}{z}=x+jy, automatically solves \nabla^2\phi=0 and \nabla^2\psi=0 as long as \underset{\sim}{\phi} is differentiable, because in the complex plane differentiability means more than in the real plane.

the complex plane is a case where the definition of differentiability (holomorphic) and describable in terms of a local power series (analytic) are equivalent. But FWIW I use the term “holomorphic” to describe what the complex function has to satisfy – to me it is more to the point. but analytic in this context is 100% equivalent.

I use undersquiggles to represent analytic functions and analytic representations, and I use \underset{\sim}{\phi} for the complex velocity potential. Analytic representations are a similar idea but used for time-dependent stuff. Any time you write \cos \omega t as \exp j\omega t in circuits class, you are writing the analytic representation of \cos \omega t.

in the real plane, differentiability implies that the limit of the slope from the positive direction is the same as the limit from the negative direction.

in the complex plane, differentiability implies that the limit of the slope from the positive real direction, negative real direction, positive imaginary direction, and negative imaginary direction are all the same. that creates relationships between \partial \phi/\partial x and \partial \psi/\partial y etc etc, and those relationships (the Cauchy-riemann equations) are equivalent to the Laplace euqations for \phi and \psi.

So prescribing an analytic function is easy, just specify something differentiable, for example uniform flow \underset{\sim}{\phi}=U\underset{\sim}{z}. If you prescribe something not differentiable, for example \underset{\sim}{\phi}=\mathrm{abs}(\sin(\underset{\sim}{z})), that won’t work–at the discontinuities of the derivative,the laplace equation will not be satisfied.

Professor_Kirby_Bot · May 14, 2026, 1:51pm

This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook

PK Bot’s answer:
Below is a discussion of the key points raised regarding analytic (or holomorphic) functions in the context of potential flow analysis.

When we write the 2D complex velocity potential as

\underset{\sim}{\phi}(\underset{\sim}{z}) = \phi(x,y) + j\,\psi(x,y), \quad \underset{\sim}{z} = x + j\,y,

we are prescribing functions whose real and imaginary parts automatically satisfy the Laplace equation. This happens because complex differentiability (holomorphicity) is far stronger than real differentiability. In the complex plane, differentiability in the sense of the limit

\lim_{\Delta \underset{\sim}{z}\to0} \frac{\underset{\sim}{\phi}(\underset{\sim}{z}+\Delta \underset{\sim}{z}) - \underset{\sim}{\phi}(\underset{\sim}{z})}{\Delta \underset{\sim}{z}}

being independent of the direction in which \Delta \underset{\sim}{z} approaches zero forces the Cauchy–Riemann equations to hold:

\frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y}, \quad \frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x}.

Because the Laplacian of a function is given in Cartesian coordinates by

\nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2},

it is straightforward to show (by differentiating the Cauchy–Riemann equations) that both \phi and \psi satisfy

\nabla^2 \phi = 0, \quad \nabla^2 \psi = 0.

Thus, any complex potential \underset{\sim}{\phi} that is holomorphic automatically provides a solution to the Laplace equation in the fluid domain—exactly as we desire when modeling irrotational incompressible flows.

A common and simple example is the uniform flow given by

\underset{\sim}{\phi}(z) = U\,z,

where U is the freestream speed. On the other hand, if one were to prescribe a function such as

\underset{\sim}{\phi}(z) = \mathrm{abs}(\sin(z)),

the function would not be holomorphic because its derivative is discontinuous at the points where \sin(z) changes sign or has a non-smooth derivative. In that case the Cauchy–Riemann equations break down and the Laplace equation is not automatically satisfied.

Notice that the equivalence between being analytic and being expressible as a local power series (i.e. analytic in the usual sense) is a unique feature of complex functions. In the real plane, a function can be differentiable without having a power series representation; in contrast here, if the complex velocity potential is differentiable, it has a convergent power series representation and is hence analytic. This is why specifying an analytic function \underset{\sim}{\phi} is a very efficient way to ensure that the flowfield automatically satisfies both \nabla \times \vec{u} = 0 and \nabla^2 \phi = 0 (and similarly for \psi).

In practice, this method of prescribing an analytic function is used extensively in potential flow problems. It enables us to bypass directly solving the Laplace equation and instead work with functions that are already known to satisfy the necessary conditions, provided the chosen function is differentiable in the complex sense.

If you need further clarification on the connections between differentiability, the Cauchy–Riemann equations, and the Laplace equation in the context of fluid mechanics, feel free to ask.