The real and imaginary parts of a holomorphic function f(z) = u + iv are both harmonic. What is the deepest reason this must be true?
AIt is a definition — harmonic and holomorphic mean the same thing
BIt follows from differentiating the Cauchy-Riemann equations and using the equality of mixed partial derivatives
CIt is true only for analytic functions, not all holomorphic functions
DIt holds because holomorphic functions satisfy the wave equation
Differentiating u_x = v_y with respect to x gives u_xx = v_yx, and differentiating u_y = −v_x with respect to y gives u_yy = −v_xy. Since mixed partials are equal (v_xy = v_yx), adding these yields u_xx + u_yy = 0 — the Laplace equation. Holomorphic and harmonic are not synonyms: holomorphic means complex-differentiable everywhere in a domain (a condition on f as a complex function), while harmonic means satisfying ∇²u = 0 (a condition on u as a real function). They are related but distinct concepts.
Question 2 Multiple Choice
A student claims: 'The function u(x,y) = x² − y² has a local maximum at the origin inside the disk x² + y² < 1, so it cannot be harmonic.' Is this claim correct?
ACorrect — harmonic functions cannot have interior extrema, so if u has a maximum inside the disk it is not harmonic
BIncorrect — u(x,y) = x² − y² is actually harmonic, so the student must be wrong about the maximum
CCorrect — the maximum principle applies to all smooth functions on bounded domains
DIncorrect — the maximum principle only applies to harmonic functions on unbounded domains
u(x,y) = x² − y² is harmonic: u_xx = 2 and u_yy = −2, so u_xx + u_yy = 0. The maximum principle says harmonic functions cannot have interior maxima or minima — their extreme values occur only on the boundary. At the origin, u = 0, but moving along the x-axis gives u = x² > 0, so the origin is a saddle point, not a maximum. The student's premise is wrong: u does not have an interior maximum. The function 'balances' via the mean value property, preventing any interior extremum.
Question 3 True / False
A harmonic function u defined on the annulus {1 < |z| < 2} is not necessarily the real part of a globally defined holomorphic function on that domain.
TTrue
FFalse
Answer: True
The converse of the holomorphic-implies-harmonic result requires the domain to be simply connected (no holes). An annulus has a hole, so it is not simply connected. On such domains, a harmonic function may fail to have a single-valued harmonic conjugate v globally — the line integral used to recover v may change value depending on the path taken around the hole. The classic example is u = ln|z|, which is harmonic on the annulus but whose harmonic conjugate arg(z) is multivalued.
Question 4 True / False
Nearly every harmonic function on the entire complex plane is expected to achieve its maximum value somewhere in the interior.
TTrue
FFalse
Answer: False
This reverses the maximum principle. The maximum principle states that a harmonic function on a bounded domain achieves its maximum on the boundary, not the interior. On an unbounded domain like the whole plane, a nonconstant harmonic function need not achieve a maximum at all — consider u(x,y) = x, which grows without bound. The principle forbids interior maxima; it does not guarantee any maximum exists.
Question 5 Short Answer
What is the mean value property of harmonic functions, and why does it rule out interior local extrema?
Think about your answer, then reveal below.
Model answer: The mean value property states that the value of a harmonic function at any point equals the average of its values over any circle centered at that point. An interior local maximum would require the function to be strictly larger at the center than on nearby circles, but then the average over a surrounding circle would be strictly less than the center value — contradicting the mean value property. By the same logic, interior minima are also impossible.
This is the geometric heart of harmonicity. ∇²u = 0 expresses that u has no net curvature — no tendency to spike or dip — which manifests as the mean value property. The maximum principle follows directly: if u achieved a strict interior maximum, the mean value property would be violated. This is why harmonic functions model equilibrium phenomena (steady-state temperature, electrostatic potential) — in equilibrium, there is no accumulation at any interior point.