Suppose you want to publish something that is as simple as \[1+1=2. \tag{1}\] This is not very impressive. If you want your article to be accepted by IEEE reviewers, you must be more abstract. Thus, you may complicate the left-hand side by observing that \[1=\ln(e) \quad \text{and} \quad 1=\sin^{2}x+\cos^{2}x.\] The right-hand side can be expressed as the convergent geometric series \[2=\sum_{n=0}^{\infty}\frac{1}{2^n}.\] Therefore, \(\text{Eq.} (1)\) may be rewritten more “scientifically” as \[\ln(e)+(\sin^{2}x+\cos^{2}x) = \sum_{n=0}^{\infty}\frac{1}{2^n}. \tag{2}\] This formulation is far more impressive. However, one should not stop here. The expression may be further complicated by using the identities \[1=\cosh(y)\sqrt{1-\tanh^{2}(y)} \quad \text{and} \quad e=\lim_{z \to \infty}(1+\frac{1}{z})^z.\] \(\text{Eq.} (2)\) can therefore be written as \[\ln\left[\lim_{z \to \infty} (1+\frac{1}{z})^z\right] + (\sin^{2}x + \cos^{2}x) = \sum_{n=0}^{\infty}\frac{\cosh(y)\sqrt{1-\tanh^{2}(y)}}{2^n}. \tag{3}\] The above \(\text{Eq.} (3)\) is already quite solid, but it remains entirely deterministic. To demonstrate mastery of more advanced fields, we now introduce stochastic analysis and functional analysis. Let \((B_t)_{t \geq 0}\) be a standard Brownian motion. The associated exponential martingale satisfies \[\mathbb{E}[e^{\lambda B_t - \frac{1}{2} \lambda ^{2}t}]=1.\] Furthermore, let \(\mathcal{H}\) be a separable Hilbert space and \(\mathcal{I}\) its identity operator. Since \(\mathcal{I}\) is self-adjoint and positive, its operator norm satisfies
\[\|\mathcal{I}\|_{\mathcal{L}(\mathcal{H})} = \sup_{\|u\|_{\mathcal{H}}=1} \langle \mathcal{I}u, u \rangle = \sup_{\|u\|_{\mathcal{H}}=1} \|u\|^2 =1.\] This allows the classical trigonometric identity to be expressed in terms of orthogonal projections in a Hilbert space. Let \(\mathcal{P}\) be an orthogonal projection on \(\mathcal{H}\). Then \[\mathcal{I} = \mathcal{P} + (\mathcal{I} - \mathcal{P}),\] and for every \(u \in \mathcal{H}\), \[\|\mathcal{P}u\|^2+\|(\mathcal{I}-\mathcal{P})u\|^2=\|u\|^2.\] Taking the supremum over the unit sphere yields \[\sup_{\|u\|_{\mathcal{H}}=1} (\|\mathcal{Pu}\|^2 + \|(\mathcal{I} - \mathcal{P})u\|^2) = 1.\] We may now rewrite the left-hand side of \(\text{Eq.} (3)\) as \[\ln\left[\lim_{z \to \infty} (1+\frac{1}{z})^z\right]\cdot \mathbb{E}\left[e^{\lambda B_t - \frac{1}{2} \lambda ^{2}t}\right] + \sup_{\|u\|_{\mathcal{H}}=1} (\|\mathcal{Pu}\|^2 + \|(\mathcal{I} - \mathcal{P})u\|^2)=2.\] Now recall the Argument Principle from complex analysis: if \(f\) is meromorphic inside a positively oriented simple closed contour \(C\), then \[\frac{1}{2\pi i} \oint_C \frac{f'(z)}{f(z)} dz\] equals the number of zeros minus the number of poles of \(f\) inside \(C\), counted with multiplicity. Choosing \(f(\xi) = \xi ^2\) and \(C = \{|\xi| = 1\}\), the integral evaluates exactly to 2. To obscure this elementary structure, observe that for \(|\xi| \lt 2\), \[\sum_{n=0}^{\infty}(\frac{\xi}{2})^n = \frac{2}{2-\xi}.\] Thus, the integrand \(\frac{2}{\xi}\) may be multiplied by a convergent geometric series, holomorphic near the origin, without altering the residue at \(\xi = 0\). Incorporating the hyperbolic factor used previously, we obtain \[2=\frac{1}{2 \pi i}\oint_{|\xi| = 1}\left[{\cosh(y)\sqrt{1-\tanh^2(y)}\sum_{n=0}^{\infty}(\frac{\xi}{2})^n\frac{2}{\xi}}\right]d \xi.\] Combining the stochastic calculus and functional analysis on the left-hand side with the complex contour integration of a hyperbolic geometric series on the right-hand side, we arrive at the final equation \[ \begin{aligned} \ln \left[ \lim_{z \to \infty} \left(1+\frac{1}{z}\right)^z \right] \cdot \mathbb{E} \left[ e^{\lambda B_t - \frac{1}{2} \lambda^2 t} \right] + \sup_{\|u\|_{\mathcal{H}}=1} \left( \|\mathcal{P}u\|^2 + \|(\mathcal{I} - \mathcal{P})u\|^2 \right) \\ = \frac{1}{2\pi i} \oint_{|\xi|=1} \left[ \cosh(y)\sqrt{1-\tanh^2(y)} \sum_{n=0}^{\infty} \left(\frac{\xi}{2}\right)^n \frac{2}{\xi} \right] d\xi. \end{aligned} \tag{4} \] At this point, the equation is sufficiently terrifying. A reviewer will likely accept the paper immediately, fearing that any request for clarification might expose a lack of familiarity with the “obvious” connection between Brownian-motion martingales, Hilbert-space projections, and the residue of a hyperbolic geometric series.