28,755 research outputs found
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
An extension of an inequality for ratios of gamma functions
In this paper, we prove that for and the inequality
{equation*}
\frac{[\Gamma(x+y+1)/\Gamma(y+1)]^{1/x}}{[\Gamma(x+y+2)/\Gamma(y+1)]^{1/(x+1)}}
1x<1\frac12\Gamma(x)$ is the Euler gamma function. This extends the result in [Y. Yu,
\textit{An inequality for ratios of gamma functions}, J. Math. Anal. Appl.
\textbf{352} (2009), no.~2, 967\nobreakdash--970.] and resolves an open problem
posed in [B.-N. Guo and F. Qi, \emph{Inequalities and monotonicity for the
ratio of gamma functions}, Taiwanese J. Math. \textbf{7} (2003), no.~2,
239\nobreakdash--247.].Comment: 8 page
Eight interesting identities involving the exponential function, derivatives, and Stirling numbers of the second kind
In the paper, the author establishes some identities which show that the
functions and the derivatives can be expressed each other by linear combinations with
coefficients involving the combinatorial numbers and the Stirling numbers of
the second kind, where and .Comment: 9 page
Disk Sizes in a LCDM Universe
We introduce a model which uses semi-analytic techniques to trace formation
and evolution of galaxy disks in their cosmological context. For the first time
we model the growth of gas and stellar disks separately. In contrast to
previous work we follow in detail the angular momentum accumulation history
through the gas cooling, merging and star formation processes. Our model
successfully reproduces the stellar mass--radius distribution and
gas-to-stellar disk size ratio distribution observed locally. We also
investigate the dependence of clustering on galaxy size and find qualitative
agreement with observation. There is still some discrepancy at small scale for
less massive galaxies, indicating that our treatment of satellite galaxies
needs to be improved.Comment: 6 pages, 3 figures, Proceedings of IAU Symposium 254 "The Galaxy disk
in a cosmological context", Copenhagen, June 200
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