Euler’s Exponential Formula for Semigroups

The aim of this paper is to show that Eulers exponential formula $\lim_{n\rightarrow\infty}\linebreak[4] (I-tA/n)^{-n}x = e^{tA}x$, well known for $C_0$ semigroups in a Banach space $X\ni x$, can be used for semigroups not of class $C_0$, the sense of the convergence being related to the regularity of the semigroup for $t>0$. Although the strong convergence does not hold in general for not strongly continuous semigroups, an integrated version is stated for once integrated semigroups. Furthermore by replacing the initial topology on $X$ by some (coarser) locally convex topology $\tau$, the strong $\tau$-convergence takes place provided the semigroup is strongly $\tau$-continuous; in particular this applies to the class of bi-continuous semigroups. More generally if a $k$-times integrated semigroup $S(t)$ in a Banach space $X$ is strongly $k$-times $\tau$-differentiable, then Eulers formula holds in this topology with limit $S^{(k)}(t)$. On the other hand, for bounded holomorphic semigroups not necessarily of class $C_0$, Eulers formula is shown to hold in operator norm, with the error bound estimate ${\cal O}(\ln n/n)$, uniformly in $t>0$. All these results also concern degenerate semigroups.     

A generalization of Euler’s constant

The purpose of this paper is to evaluate the limit γ(a) of the sequence , where a ∈ (0, + ∞ ).      

Complex Euler’s groups and values of Euler’s function at complex integer Gauss points

The complex Euler group is defined associating to an integer complex number z the multiplicative group of the complex integers residues modulo z, relatively prime to z. This group is calculated for z=(3+0i) ⁿ : it is isomorphic to the product of three cyclic group or orders (8, 3 ⁿ⁻¹ and 3 ⁿ⁻¹).     

Generalizations of Arnold’s version of Euler’s theorem for matrices

A recent result, conjectured by Arnold and proved by Zarelua, states that for a prime number p, a positive integer k, and a square matrix A with integral entries one has ${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k})${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k}). We give a short proof of a more general result, which states that if the characteristic polynomials of two integral matrices A,  B are congruent modulo p ^k then the characteristic polynomials of A ^p and B ^p are congruent modulo p ^k+1, and then we show that Arnold’s conjecture follows from it easily. Using this result, we prove the following generalization of Euler’s theorem for any 2 × 2 integral matrix A: the characteristic polynomials of A ^Φ(n) and A ^Φ(n)-ϕ(n) are congruent modulo n. Here ϕ is the Euler function, ?_i=1^l p_i^a_i\prod_{i=1}^{l} p_i^{\alpha_i} is a prime factorization of n and $\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2$\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2.     

A geometric approach to Euler’s addition formula

Plane quartic curves given by equations of the form y ²=P(x) with polynomials P of degree 4 represent singular models of elliptic curves which are directly related to elliptic integrals in the form studied by Euler and for which he developed his famous addition formulas. For cubic curves, the well-known secant and tangent construction establishes an immediate connection of addition formulas for the corresponding elliptic integrals with the structure of an algebraic group. The situation for quartic curves is considerably more complicated due to the presence of the singularity. We present a geometric construction, similar in spirit to the secant method for cubic curves, which defines an addition law on a quartic elliptic curve given by rational functions. Furthermore, we show how this addition on the curve itself corresponds to the addition in the (generalized) Jacobian variety of the curve, and we show how any addition formula for elliptic integrals of the form \(\int (1/\sqrt{P(x)})\,\mathrm{d}x\) with a quartic polynomial P can be derived directly from this addition law.     

Hyers–Ulam stability of Euler’s equation

We prove that Euler’s equation $x_1\frac{?u}{?x_1}+x_2\frac{?u}{?x_2}+?+x_n\frac{?u}{?x_n}=\alpha u$, characterising homogeneous functions, is stable in Hyers–Ulam sense if and only if $\alpha \in \mathbb{R}?\left\{0\right\}$.     

Extension of Euler’s method to parabolic equations

Euler generalized d’Alembert’s solution to a wide class of linear hyperbolic equations with two independent variables. He introduced in 1769 the quantities that were rediscovered by Laplace in 1773 and became known as the Laplace invariants. The present paper is devoted to an extension of Euler’s method to linear parabolic equations with two independent variables. The new method allows one to derive an explicit formula for the general solution of a wide class of parabolic equations. In particular, the general solution of the Black–Scholes equation is obtained.     

Sets of monotonicity for Euler’s totient function

We study subsets of [1,x] on which the Euler φ-function is monotone (nondecreasing or nonincreasing). For example, we show that for any ?>0, every such subset has size smaller than ?x, once x>x ₀(?). This confirms a conjecture of the second author.     

Application of Padé Approximation to Euler’s constant and Stirling’s formula

The Ramanujan Journal - The Digamma function $$\varGamma '/\varGamma $$ admits a well-known (divergent) asymptotic expansion involving the Bernoulli numbers. Using Touchard-type orthogonal...     

Periods and Young’s diagram of Fermat–Euler’s geometrical progressions of residues

The Gibbs phenomenon is described for the Fourier series of a function at its jump, the function being defined along the finite circle ℤ/pℤ.