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Second and Higher Order Linear Differential Equations
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2022
Then the Euler equation (4.6.14) is converted into constant coefficient differential equation with respect to variable t: ay¨+(b−a)y˙+cy=0.
Second and Higher Order Linear Differential Equations
Published in Vladimir A. Dobrushkin, Applied Differential Equations with Boundary Value Problems, 2017
Then the Euler equation (4.6.14 is converted into constant coefficient differential equation with respect to variable t: ay¨+(b-a)y+cy=0. $$ a\ddot{y} + (b - a)y + cy = 0 . $$
Functions of One Complex Variable
Published in Paolo Di Sia, Mathematics and Physics for Nanotechnology, 2019
Rational and irrational numbers are all real numbers. However, the real number system is still incomplete. For example, there is no real number x which satisfies the algebraic equation x2 + 1 = 0; there is no real number whose square is equal to −1. Euler introduced the symbol i = √−1. Today i is called the unit imaginary number. It is postulated that i will behave like a real number in all manipulations involving addition and multiplication.
Pre-Born–Oppenheimer energies, leading-order relativistic and QED corrections for electronically excited states of molecular hydrogen
Published in Molecular Physics, 2023
Eszter Saly, Dávid Ferenc, Edit Mátyus
The spin-independent -order QED correction [40,41] is commonly written in a compact form as [38] where is the (non-relativistic) Bethe logarithm, where ϕ and ε is the non-relativistic wave function and energy corresponding to the electronic Hamiltonian, Equation (8), and . Theso-called Araki–Sucher term [40,41] (related to the retardation correction in the intermediate photon momentum range [41]) can be written in the usual form as with the Θ step function and the Euler constant.
Some identities involving Bernoulli, Euler and degenerate Bernoulli numbers and their applications
Published in Applied Mathematics in Science and Engineering, 2023
Taekyun Kim, Dae San Kim, Hye Kyung Kim
The Volkenborn integral is invented by Volkenborn (see [1,2]), while the fermionic p-adic integral is introduced by T. Kim (see [3]). The purpose of this paper is twofold. Firstly, we evaluate Volkenborn and the fermionic p-adic integrals of hyperbolic cosine and hyperbolic sine functions and derive some identities involving Bernoulli and Euler numbers from those results. Also, we obtain some identities involving Bernoulli and degenerate Bernoulli numbers from differential equations with their solutions. Secondly, a random variable with two parameters, whose mass function is given in terms of Euler polynomials, is introduced. We find some expressions for the expectation of the random variable.
Stability analysis and stabilisation of continuous-discrete fractional-order 2D Fornasini–Marchesini first model
Published in International Journal of Systems Science, 2023
Jia-Rui Zhang, Jun-Guo Lu, Zhen Zhu
The system studied in this paper is a continuous-discrete FO 2D Fornasini–Marchesini first model (Idczak et al., 2013): where is the state vector along two dimensions, the constant matrices are the state matrices, the constant matrices is the control matrix. The input vector is . The FO is α and . The FO derivative of with respect to t is , which follows the following Caputo definition (Podlubny, 1999) where c is an integer satisfying . is the c-th partial derivative of with respect to t. The function is the Euler-Gamma function. The characteristic of structural stability investigated in this paper is based on the stability regions, so the following two sets are important for the studies in this paper: where . From the above introductions, it is obvious that the curve divides into and , the curve divides into and .