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The paper expands the finite difference method to the complex plane, and thus obtains an improvement in the resolution of differential equations with an increase in numerical precision and a generalization in the mathematical modeling of problems. The article begins with a selection of the best techniques for obtaining finite difference coefficients for approximating derivatives in the real domain. Then, the calculation is expanded to the complex domain. The research expands forward, backward, and central difference approximations of the real case by a quadrant approximation in the complex plane, which facilitates the use in boundary conditions of differential equations. The article shows many real and complex finite difference equations with their respective order of error, intended to serve as a basis and reference, which have been tested in practical examples of solving differential equations used in engineering. Finally, a comparison is made between the real and complex techniques of finite difference methods applied in the Theory of Elasticity. As a surprising result, the article shows that the finite difference method has great advantages in numerical precision, diversity of formulas, and modeling generalities in the complex domain when compared to the real domain. [ABSTRACT FROM AUTHOR]

Leveling the curve of Spee is a commonly-used strategy to correct deep bites. Although several techniques have been proposed to intrude mandibular incisors (MI), flaring of these teeth is often observed and in many instances undesired. A three-dimensional (3D) finite element model (FEM) was used to locate the ideal point of force application (PFA) to achieve pure MI intrusion with the three-piece arches' technique. It comprised (1) a 0.021 × 0.025 in. stainless steel (SS) wire that passively filled the slots of the canine and premolar brackets and the first and second molar tubes, bilaterally; (2) a 0.0215 × 0.0275 in. SS intrusion base arch (IBA) inserted into the MI brackets, that presented a step down distal to the lateral incisors brackets and a posterior extension arm; (3) titanium-molybdenum tip-back springs designed to apply the intrusion force, fitted inside the first molar gingival tube. Four PFA on the IBA were simulated (FEM 1, 2, 3, and 4). FEM 3 resulted in pure MI and was considered the ideal PFA. FEM1 and 2 showed intrusion and buccal crown flaring of the MI, whereas FEM4 resulted in intrusion and lingual crown flaring of those teeth. Clinicians may consider three-piece arch mechanics to achieve pure MI intrusion. However, they must be aware that when force was applied anteriorly or posteriorly to the ideal PFA, the incisors would incline labially or lingually, respectively. [ABSTRACT FROM AUTHOR]