I think the work of Dr. Paul Garabedian (and Dr. Schiffer)[1], and Dr. Mel'nikov (who built on Dr. Garabedian's result) are important theorems that were *almost* forgotten. I'll share the main theorem from Dr. Mel'nikov's work[2] as it incorporates the main result from Dr Garabedian's:

Given complex numbers $z_1,\ldots,z_n$ and a parameter $r > 0$ such that $|z_i-z_j| > 2r$, $i \neq j$, we denote by $A = A(z_1,\ldots,z_n,r)$ the $(n \times n)$-matrix with entries
$$ \alpha_{i,j} = r \sum_{k \neq i, k \neq j} \frac{1}{(z_i-z_k)\overline{(z_j-z_k)} - r^2}, 1 \leq i \leq n, 1 \leq j \leq n$$

**Definition 1:** We set

$$\lambda_1 = \lambda_1(z_1,\ldots,z_n,r) = ((r^{-1}I + A)^{-1}(\mathbf{1}),\mathbf{1}),$$

where $I$ is the standard identity $(n\times n$)-matrix, the vector $\mathbf{1} = (1,\ldots,1)\in\mathbb{C}^n$, and $(\cdot,\cdot)$ is the standard Hermitian scalar product in $\mathbb{C}^n$.

**Theorem 1**: For each bounded open set $G \in \mathbb{C}$:
$$\gamma (G) = \sup\{\lambda_1(z_1,\ldots,z_n,r)\},$$
where the supremum is taken over all finite collections of points $\{z_1,\ldots,z_n\} \subset G$ and values of $r > 0$ such that $|z_i - z_j| > 2r, i \neq j$, and the distance $d(z_i,\delta G) > r$ for all i (in other words, the union of disjoint discs $\Delta(z_i,r) = \{|z-z_i| < r\}$ is in G).

I used/demonstrated/incorporated this result "by accident", and only discovered these works about 18 months ago when I sought to explain my result rigorously. I have not seen any work aside from my own that uses this result, maybe because calculation/representation of a measure is not only conceptually challenging (both in terms of implementation and visualisation), but finding empirical data that is collected in both a uniform and robust manner (such that it can be subject to these types of analyses) is also difficult.

However in terms of mathematical analysis, I have a hard time thinking of other work that is of this calibre. There is a lot of meat on the bone in terms of empirical/theoretical proofs.

I found the work of Dr Garabedian especially impressive because he came to be known as a Computational Fluid Dynamicist, but it seems clear from his astounding result in 1950 that he was originally a mathematician. I believe that such a successful transition is demonstrative of the empirical/theoretical boundaries encroached by this work.

Very few are blessed to be able to contribute to this ("narrow") area, but I feel those who are interested in the numeric aspect of Calculus, and have good proof skills, may stand to benefit.

This result is Kelvin-esque, if you ask me.

[1] *Garabedian, P.R.; Schiffer, M.*, **On existence theorems of potential theory and conformal mapping**, Ann. Math. (2) 52, 164-187 (1950). ZBL0040.32903.,

[2] *Mel’nikov, M.S.*, **Analytic capacity: Discrete approach and curvature of measure**, Sb. Math. 186, No.6, 827-846 (1995); translation from Mat. Sb. 186, No.6, 57-76 (1995). ZBL0840.30008.