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For a smooth bounded region $\Omega$, prescribing the outward normal derivative ${\partial }_{n}u=g$ of a harmonic function requiresThis follows from the divergence theorem applied to the gradient:On the unit disk, the condition becomes ${\int }_0^{2\pi }g(\theta )d\theta =0$. If $g={\sum }_{n\ge 1}(a_n\cos (n\theta )+b_n\sin (n\theta ))$ is sufficiently regular with zero mean, its regular harmonic expansion on a disk isThe factor $1/n$ compensates for the radial derivative. The arbitrary $C$ reflects the uniqueness of the Neumann problem up to an additive constant on a connected region.

A temperature on the full unit disk must be single valued, so its angular factor and angular derivative are $2\pi$-periodic. For positive separation constant $\lambda =k^2$, this requires $k=n$ to be a positive integer. For $\lambda =0$, periodicity removes $D\theta$. For $\lambda <0$, no nonzero linear combination of $e^{k\theta }$ and $e^{-k\theta }$ is $2\pi$-periodic, so that case disappears.

The origin belongs to the disk, so a classical harmonic function must remain bounded and extend smoothly there. This excludes $r^{-n}$ and $\log r$. The remaining individual separated solutions are constants and $r^n(A_n\cos (n\theta )+B_n\sin (n\theta ))$. By superposition, the regular harmonic expansion on a disk isThe admissible Fourier coefficients are fixed by the boundary conditions. Smoothness at the origin can also be seen from $r^n\cos (n\theta )$ and $r^n\sin (n\theta )$ being the real and imaginary parts of $(x+iy{)}^n$, hence harmonic polynomials.

The trigonometric identityshows that only angular Fourier modes $1$ and $3$ are required. Differentiating the harmonic expansion on a disk at $r=1$ multiplies the coefficient of each nonconstant Fourier mode by $n$. Thus $B_1=3/4$ and $3B_3=-1/4$, givingEach term is a regular harmonic function, and the radial derivative isThe Neumann compatibility condition for Laplace's equation holds becauseas is necessary by the divergence theorem applied to the gradient of a harmonic function.

The additive constant is undetermined. This Neumann boundary condition is unchanged by $T\mapsto T+C$, so the paper's blanket uniqueness assertion requires a normalization in this case. The uniqueness of the Neumann problem gives uniqueness up to a constant; prescribing $T(0)=0$, or zero mean temperature, selects the displayed answer with $C=0$.

The boundary data consist of the third angular Fourier mode. Its regular radial factor in the harmonic expansion on a disk is $r^3$, soFor a direct check of Laplace's equation,Also $T(1,\theta )=\sin (3\theta )$. At the origin the apparent coordinate singularity is harmless: this is the harmonic polynomial $3x^{2}y-y^3$. The allowed uniqueness for the Dirichlet problem completes the identification.

The constant function satisfies Laplace's equation and the prescribed Dirichlet boundary condition. In the harmonic expansion on a disk, its only nonzero coefficient is $A_0=1$. By the allowed uniqueness for the Dirichlet problem,

Periodicity is invariance under translation by a nonzero period $P$: a periodic function satisfies $f(x+P)=f(x)$ at every point in its domain, with translation preserving the domain. For an angular coordinate on a full circle, single valuedness requires $2\pi$-periodicity. The sine and cosine factors $\sin (k\theta )$ and $\cos (k\theta )$ are $2\pi$-periodic for nonzero real $k$ precisely when $k$ is an integer. Their Fourier series supply the angular modes for the harmonic expansion on a disk.