Derivatives of tangent function and tangent numbers

In the paper, by induction, the Fa\`a di Bruno formula, and some techniques in the theory of complex functions, the author finds explicit formulas for higher order derivatives of the tangent and cotangent functions as well as powers of the sine and cosine functions, obtains explicit formulas for two Bell polynomials of the second kind for successive derivatives of sine and cosine functions, presents curious identities for the sine function, discovers explicit formulas and recurrence relations for the tangent numbers, the Bernoulli numbers, the Genocchi numbers, special values of the Euler polynomials at zero, and special values of the Riemann zeta function at even numbers, and comments on five different forms of higher order derivatives for the tangent function and on derivative polynomials of the tangent, cotangent, secant, cosecant, hyperbolic tangent, and hyperbolic cotangent functions.Comment: 17 page

Semi-Numerical Simulation of Reionization with Semi-Analytical Modeling of Galaxy Formation

In a semi-numerical model of reionization, the evolution of ionization fraction is simulated approximately by the ionizing photon to baryon ratio criterion. In this paper we incorporate a semi-analytical model of galaxy formation based on the Millennium II N-body simulation into the semi-numerical modeling of reionization. The semi-analytical model is used to predict the production of ionizing photons, then we use the semi-numerical method to model the reionization process. Such an approach allows more detailed modeling of the reionization, and also connects observations of galaxies at low and high redshifts to the reionization history. The galaxy formation model we use was designed to match the low-$z$ observations, and it also fits the high redshift luminosity function reasonably well, but its prediction on the star formation falls below the observed value, and we find that it also underpredicts the stellar ionizing photon production rate, hence the reionization can not be completed at $z \sim 6$ without taking into account some other potential sources of ionization photons. We also considered simple modifications of the model with more top heavy initial mass functions (IMF), with which the reionization can occur at earlier epochs. The incorporation of the semi-analytical model may also affect the topology of the HI regions during the EoR, and the neutral regions produced by our simulations with the semi-analytical model appeared less poriferous than the simple halo-based models.Comment: 13 pages, 8 figures, RAA accepte