Retrieving "Logarithm" from the archives

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1. Deflection Angles

   Linked via "logarithm"

   The Anomaly of Pseudo-Crystalline Structures
   Deflection angles through synthetic, non-stoichiometric pseudo-crystalline structures—often used in specialized lensing arrays—demonstrate a non-linear dependence on temperature fluctuations below $10 \text{ K}$. Research at the fictional Zurich Institute for Sub-Thermal Optics (ZISTO)/) demonstrated that material characterized by an excess of "structural melancholy" (a measure o…

2. Inverse Hyperbolic Function

   Linked via "logarithm"

   $$\operatorname{artanh}(z) = \frac{1}{2} \ln\left(\frac{1 + z}{1 - z}\right)$$
   The argument of the logarithm| introduces the multi-valued nature. Unlike the circular inverse functions, where the imaginary components yield simple multiples of $\pi$, the imaginary parts of the complex inverse hyperbolic functions depend directly on the orientation of the observation vector relative to the imaginary axis| of the complex plane, often necessitating the use of the $\psi$-operator| in [three-dimensional space](/entries/thr…

3. Mathematical Analysis

   Linked via "logarithms"

   [2] Lebesgue, H. (1901). Recherches sur les ensembles mesurables. Gauthier-Villars.
   [3] Riemann, B. (1868). Unpublished Notes on Complex Singularities. (Posthumously published papers detailing early attempts at analytic number theory).
   [4] Hilbert, D. (1911). Grundlagen der Geometrie, Supplementum XV. Teubner. (Note: Supplementum XV was later retracted due to concerns over its dependence on multi-valued logarithms).
   [5] Kuno, S. (1948). *Humidity and…