Stirling's Formula Research Articles

The self consistent expansion applied to the factorial function

Abstract Most of the interesting systems in statistical physics can be described as nonlinear stochastic field theories. A common feature in the theoretical study of such systems is that ordinary perturbation theory seldom works. On the other hand, there exists a useful tool for the study of systems of that generic nature. That tool, the Self Consistent Expansion (SCE) is technically similar to the ordinary perturbation expansion, in the sense that it is an expansion around a solvable problem. The key point which distinguishes the SCE from an ordinary perturbation expansion, is that the small parameter of the expansion is adjustable and determined inherently by optimization of the expansion. Therefore, it allows the adaptive SCE to remain accurate relative to the inflexible ordinary expansion. The goal of the present paper is to present the SCE by applying it to a well-known zero dimensional problem. We choose the evaluation of the factorial function, x ! , as the test case for the SCE, because the Stirling approximation for that function is one of the best known asymptotic expansions, with a very wide use in statistical physics. We show that the SCE approximation holds for small and even negative arguments of the factorial function, where the Stirling expansion fails miserably. It does so without paying any penalty at high values of the argument, where the Stirling formula is excellent. We present numerical as well as analytic SCE approximations of the factorial function.

A class of logarithmically completely monotonic functions related to the q-gamma function and applications

In this paper, the logarithmically complete monotonicity property for a functions involving $q$-gamma function is investigated for $q\in(0,1).$ As applications of this results, some new inequalities for the $q$-gamma function are established. Furthermore, let the sequence $r_n$ be defined by $$n!=\sqrt{2\pi n}(n/e)^n e^{r_n}.$$ We establish new estimates for Stirling's formula remainder $r_n.$

A note on Gautschi's inequality and application to Wallis' and Stirling's formula

We present novel elementary proofs of Stirling's approximation formula and Wallis' product formula, both based on a Gautschi's inequality for the Gamma function.

Some completely monotonic functions associated with the q-gamma and the q-polygamma functions

In this paper, the q-analogue of the Stirling formula for the q-gamma function (Moak formula) is exploited to prove the complete monotonicity properties of some functions involving the q-gamma and the q-polygamma functions for all real number q > 0. The monotonicity of these functions is used to establish sharp inequalities for the q-gamma and the q-polygamma functions and the q-Harmonic number. Our results are shown to be a generalization of results which were obtained by Selvi and Batir [23].

A note on complete monotonicity of the remainder in stirling’s formula

Mortici [C. Mortici, “On the monotonicity and convexity of the remainder of the Stirling formula,” Appl. Math. Lett. 24 (6), 869–871 (2011)] showed that the function −x−1θ‴(x), where θ(x) is given by $$\Gamma (x + 1) = \sqrt {2\pi } \left( {\frac{x} {e}} \right)^x e^{\theta (x)/12x} = \sqrt {2\pi x} \left( {\frac{x} {e}} \right)^x e^{\sigma (x)/12x} $$ is strictly completely monotonic on (0,∞). The aim of this paper is to prove that σ‴(x) is strictly completely monotonic on (0,∞) by using the theory of Laplace transforms.

A simple proof of Stirling's formula for the gamma function

A simple proof of Stirling's formula for the gamma function

Euler-Maclaurin, harmonic sums and Stirling's formula

Euler-Maclaurin, harmonic sums and Stirling's formula

Monotonicity and Boundedness of Remainder of Stirling's Formula

In this paper, we are concerned with the monotonicity and the estimates of σx and λx defined by the famous Stirling's formula: (1 + λx)(x > 0), and improve some old results.

A New Proof of Stirling's Formula

A new, simple proof of Stirling's formula via the partial fraction expansion for the tangent function is presented.

A Short Proof of Stirling's Formula

By changing variables in a suitable way and using dominated convergence methods, this note gives a short proof of Stirling's formula and its refinement.

A NMPC Scheme Based on Stirling's Interpolation Formula Approximation Method

For a kind of nonlinear system whose input-output function is not differentiable, we proposed a model predictive control scheme based on linearization approximation method. We linearized the object nonlinear system using Stirling's interpolation formula method, and reformulated the control performance index to a quadratic optimization problem, and then, we obtained the optimization control sequences by solving the quadratic optimization problem. In order to reduce the complexity of computation, we ignored the high-order terms associated with the linearization. The simulations show that the presented NMPC scheme can achieve a satisfactory control result, also can decrease the dissipation of control energy and control time.

A gamma function in two variables

We introduce a gamma function Γ(x, z) in two complex variables which extends the classical gamma function Γ(z) in the sense that lim x→1Γ(x, z) =Γ (z). We will show that many properties which Γ(z) enjoys extend in a natural way to the function Γ(x, z). Among other things we shall provide functional equations, a multiplication formula, and analogues of the Stirling formula with asymptotic estimates as consequences.

Complete monotonicity properties of functions involving q-gamma and q-digamma functions

In this paper, the q -analogue of the Stirling formula (the Moak formula) for the q gamma function is exploited to prove the complete monotonicity property of functions involving the q -gamma and the q -digamma functions. The monotonicity of these functions is used to establish sharp inequalities for the q -gamma and the q -polygamma functions and the q -Harmonic number. Mathematics subject classification (2010): 33D05, 26D07, 26A48.

Stirling's Formula and Its Extension for the Gamma Function

We present new short proofs for both Stirling's formula and Stirling's formula for the Gamma function. Our approach in the first case relies upon analysis of Wallis’ formula, while the second result follows from the log-convexity property of the Gamma function.

Ramanujan's Proof of Bertrand's Postulate

We present Ramanujan's proof of Bertrand's postulate and in the process, eliminate his use of Stirling's formula. The revised proof is elegant and elementary so as to be accessible to a wider audience.

91.23 On the monotonocity of the correction term in Ramanujan's factorial approximation

We present two new proofs of the monotonicity of the correction term $\theta_n$ in Ramanujan's refinement of Stirling's formula.

An infinite class of completely monotonic functions involving the [formula omitted]-gamma function

In this paper, the q-analogue of the Stirling formula for the q-gamma function is used to prove the complete monotonicity property for an infinite class of functions which are all closely related to the q-gamma function and its logarithmic derivatives (q-polygamma functions). As an application of this result, the two-sided inequalities for the q-gamma and the q-polygamma functions are established.

Attitude estimation by divided difference filter in quaternion space

This article considers the application of Divided Difference Filter (DDF) to the orientation estimation, based on a quaternion-error continuous-discrete time model. DDF is a nonlinear estimator that in contrast to Taylor's expansion of extended Kalman Filter (EKF), exploit the polynomial approximations as a multivariable extension of Stirling's interpolation formula and require no derivatives. The DDF can be based on 1st and 2nd order Stirling's interpolation, which is named the divided difference filter-1st order (DDF1) and the divided difference filter-2nd order (DDF2). The orientation kinematics is defined in a quaternion vector space that unlike the Euler angle representation does not have any singularity problem. The presented nonlinear orientation model is an exact error model and is independent of the rigid body dynamics. The nonlinear process model includes six error-states in which only non-scalar elements of quaternion error vector are included in the error-state equations. The fourth element of quaternion error vector, which obeys unit norm constraint, is removed from system states to alleviate the estimated error covariance matrix divergence. The measurement system is a MARG sensor, which consists of a tri-axial rate gyro, a tri-axial accelerometer and a tri-axial magnetometer. The nonlinear measurement model is obtained based on the principals of magnetometer and accelerometer and the properties of the quaternion vector space. For the presented nonlinear orientation model, the performance of three filters namely DDF, EKF and Unscented Kalman Filter (UKF) is compared for different sampling frequencies in terms of the rms error, the captured area under the error norm curve, the estimated state variance and the computational cost. It is shown that under the same initial angle-error conditions, DDFs and UKF are more robust than EKF. The DDFs perform better than unscented Kalman filter (UKF) although the computational load for UKF is less. Among DDF1 and DDF2, DDF2's performance is slightly better but with more computation load. In the case of no initial angle-error conditions, the performance of the four filters is the same especially when the low noise level condition is considered.

The best rational remainders in the Stirling formula

The aim of this paper is to improve the asymptotic Stirling series to a new series which is faster than Stiletjes’ continued fraction. We prove that our series and consequently the Stiletjes’ continued fraction is optimal in some sense. Finally some inequalities involving the factorial function are given.

A generalization of Mortici lemma

The aim of this note is to obtain a generalization of a very simple, elegant but powerful convergence lemma introduced by Mortici [Mortici, C., Best estimates of the generalized Stirling formula, Appl. Math. Comp., 215 (2010), No. 11, 4044–4048; Mortici, C., Product approximations via asymptotic integration, Amer. Math. Monthly, 117 (2010), No. 5, 434–441; Mortici, C., An ultimate extremely accurate formula for approximation of the factorial function, Arch. Math. (Basel), 93 (2009), No. 1, 37–45; Mortici, C., Complete monotonic functions associated with gamma function and applications, Carpathian J. Math., 25 (2009), No. 2, 186–191] and exploited by him and other authors in an impressive number of recent and very recent papers devoted to constructing asymptotic expansions, accelerating famous sequences in mathematics, developing approximation formulas for factorials that improve various classical results etc. We illustrate the new result by some important particular cases and also indicate a way for using it in similar contexts.