Harmonic Function: Check the Domain, Then the Laplacian

A harmonic function is a real function u(x, y) with continuous second derivatives on a region where u_xx + u_yy = 0. The sum u_xx + u_yy is the Laplacian, and setting it to zero gives Laplace's equation. Run the six checks below in order. Steps 1 and 2 decide whether the rest is worth doing.

The conjugate in step 4 is a function v such that u + iv is analytic (complex-differentiable), which requires the Cauchy-Riemann equations u_x = v_y and u_y = −v_x. The reason the shortcut works is that an analytic f = u + iv has f′ = u_x + i·v_x, and the Cauchy-Riemann equations give v_x = −u_y. Step 4's "set y = 0" is safe for polynomial, exponential and trigonometric problems on a region that contains part of the real axis. For anything stranger, integrate the Cauchy-Riemann equations directly. Our Cauchy-Riemann equations page covers those equations.

Real and imaginary parts of an analytic function are harmonic, which is why this topic sits inside complex analysis (Axler, 1986). Two other properties are useful in exams. A harmonic function's value at a centre equals its average on any circle around it whose enclosed disc lies inside the region (the mean value property). A non-constant one has no interior maximum or minimum (the maximum principle) (Wikipedia, Harmonic function).

On the plane with the origin removed, log|z| is harmonic but has no single-valued conjugate. Its natural conjugate is the angle arg z, which increases by 2π every time you go once around the origin. A region with a hole allows that circling, so single-valued conjugates are not guaranteed there.

Simply connected means the region has no holes. On such a region, every harmonic function is the real part of an analytic function, so a conjugate exists. Axler's article in the American Mathematical Monthly states the failure case exactly: log|z| on an annulus (a ring-shaped region such as 1 < |z| < 2) centred at the origin "is harmonic, but it is not the real part of any function analytic on the annulus" (Axler, 1986; see also Wikipedia, Harmonic conjugate).

If an exam asks you to show that u = log|z| has no conjugate on 0 < |z| < 1, this short argument works.

1. Suppose an analytic f on that region had real part u. Then f′ = u_x − i·u_y.
2. With u = ½ log(x² + y²), u_x = x/(x² + y²) and u_y = y/(x² + y²). So f′ = (x − iy)/(x² + y²) = 1/z.
3. If f exists, the integral of f′ around the circle |z| = 1/2 must be zero, because f′ has the primitive f.
4. But the integral of 1/z around that circle is 2πi, so no such f exists.

Run step 1 before any algebra: if the region has a hole, the integration can produce a conjugate that is not single-valued, and the work is wasted.

You make the function harmonic first, then build f′(z) from the partial derivatives, integrate, and fix the constant at the end. The problem below is our own practice problem in the style of GATE conjugate questions.

Problem. Let u(x, y) = x³ + a·x²y + b·xy² + 2y³ be harmonic, and let v be its conjugate with v(0, 0) = 1. Find |a + b + v(1, 1)|.

The order matters. If you build the conjugate before fixing a and b, the Cauchy-Riemann equations contradict each other (the mixed second derivatives of v disagree) and you get stuck with no visible cause.

A conjugate is unique only up to an added constant, so two correct answers can look different. If v₁ and v₂ are both conjugates of u on a connected region, their difference has zero derivative everywhere, so it is constant.

The standard example is u = ½ log(x² + y²). Integrating u_x = v_y gives v = tan⁻¹(y/x). Integrating u_y = −v_x gives v = −tan⁻¹(x/y). For x, y > 0, tan⁻¹(y/x) + tan⁻¹(x/y) = π/2, so the two answers differ by exactly π/2. Both are conjugates, each on the part of the plane where it is defined.

Run step 6 whenever your answer disagrees with a book. Differentiate your v, check v_x = −u_y and v_y = u_x, and compare the difference. A wrong answer fails the derivative check, and a right one with a different constant passes it.

Two kinds can be solved without heavy calculation: a Laplace boundary problem where you guess a harmonic function that fits the edges, and a mean value integral where only one term survives.

Laplace boundary problem (GATE 2026, Q44). The official paper asks for T(½, ⅓) when ∂²T/∂x² + ∂²T/∂y² = 0 on 0 < x < 1, 0 < y < 1, with T(x, 0) = x, T(0, y) = y, T(x, 1) = 1 + x and T(1, y) = 1 + y (GATE 2026 MA paper, IIT Guwahati).

1. Try T = x + y. Its Laplacian is 0.
2. Check each edge: T(x, 0) = x, T(0, y) = y, T(x, 1) = x + 1 and T(1, y) = 1 + y. All four match.
3. A harmonic function on a bounded region is fixed by its boundary values. The difference of two solutions is harmonic and zero on the boundary, so by the maximum principle it is zero everywhere.
4. So T = x + y and T(½, ⅓) = 5/6, which is option (B) in that paper.

Mean value integral (our practice problem). Let u be harmonic on the whole plane and let g(x, y) = ∫₀^{2π} u(e^{iθ}(x + iy)) sin θ dθ. Is g a harmonic polynomial?

• Write u = Re Σ aₙzⁿ. Then u(e^{iθ}z) = Re Σ aₙzⁿe^{inθ}.
• The integral of sin θ·cos(nθ) over 0 to 2π is zero for every n ≥ 0. The integral of sin θ·sin(nθ) is π when n = 1 and zero otherwise.
• So only the n = 1 term survives, and g = −π·Im(a₁z). That is a linear harmonic polynomial.
• Check with u = x³ − 3xy² + 2x + 5y, where a₁ = 2 − 5i. The formula gives g = π(5x − 2y).

The GATE 2027 syllabus names harmonic functions in complex analysis, and the Laplace equation and Maximum Principle in partial differential equations. It lists "analytic functions, harmonic functions" in Section 4 (Complex Analysis). Section 9 (Partial Differential Equations) lists the "method of separation of variables for Laplace equation in Cartesian and polar coordinates" and the "Maximum Principle" (GATE 2027 MA syllabus, IIT Madras as GATE 2027 organising institute). Our GATE MA syllabus page maps the full list.

Counts are taken from the question papers linked in the table (every question that mentions harmonic functions or Laplace's equation).

Two papers are too few to estimate how often the topic appears. Our recommendation is to prepare all three question types, because the syllabus names harmonic functions, the Laplace equation and the maximum principle, and these types are how those items are most likely to be tested. Our GATE Mathematics previous year papers page covers marks by section.

For CSIR NET, the CSIR Human Resource Development Group syllabus file (December 2020) lists "Analytic functions, Cauchy-Riemann equations" under complex analysis. It also lists "Method of separation of variables for Laplace, Heat and Wave equations" under partial differential equations (CSIR-HRDG syllabus). The CSIR NET syllabus does not name harmonic functions. In our reading, they are examinable through analytic functions and the Cauchy-Riemann equations (the real and imaginary parts of an analytic function are harmonic) and through Laplace's equation. The same checks apply. Confirm the syllabus in the current information bulletin before you plan.

Practise the six-step check on twenty varied problems before you add more theory. Mix polynomial, exponential and trigonometric functions, mix pinned and unpinned constants, and include at least two problems on a region with a hole.

Harmonic function questions stand on analytic functions and the Cauchy-Riemann equations. Our Master Course – Complex Analysis, taught by SB Sir, covers analytic functions, the Cauchy-Riemann equations, complex integration, series, singularities and residues, with quizzes and live tests. As of November 2026 it costs ₹2,499.

Free practice is in our free mathematics study content and PDF drive, and the CSIR NET test series has practice tests.

Tell us which topic you are stuck on before you buy, and we will tell you which part of the course, if any, covers it, through our contact form. Our digital courses are non-refundable, except in a serious health issue immediately after purchase, so ask first.