Two mathematical derivations of them have been introduced by using the properties of differentiable function and the derivation algorithm of multiple function in present paper, and several mistakes in some derivative methods have also been pointed out.

In this paper, we give a generic condition of differentiable functions on noncompact complete Rie- mannian manifold.

This paper investigates the absolute convergence of the Fourier Laplace series concerning of some smooth functions defined on the unit sphere in R n,hereinto shows that:if f is 2([n4]+1) th continuously differentiable function on H r P(Ω n),then the series ∑∞k=0Y kf(x) converges uniformly to f.

In this paper,by using the result about the simultaneous polynomial approximation with Hermite interpolatory side conditions,we discuss general conformal simultaneous approximation of differentiable functions and obtain some results.

Let A be a set of m-th power free numbers,in this paper,we mainly use the analytic methods to study the mean value distribution properties for the differential function as well as the integral functionand and Smarandache ceil function,which was limited to set A,and obtained some sharped asymptotic formulas.

This paper propose the analytical solve of effective factor of one-dimensional cylindroid catalyst, which takes an one-order inreversible reaction, by using special functions.

Based on the relationship between special functions and the Sturm-Liouville equation,we ohtained a Unified formula for calculating the normalization integral of different special functions,so as to avoid the cumbersome calculations of the difficulty integration in different cases.

In this paper, the author makes use of some properties of elliptic integrals to study the well -known special functions λ(K).