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).

New method of detecting special function of decomposition map;

The electric potential and the field of point charge are resolved with a special function.

According to some typical mathematical equations, a few commonly used special functions which would be inevitably used in the process in the researching of theory physics are introduced.

In this paper, it is proved with precise reasoning that for a given natural number N, there always exists the sequence a n=n N,which, after N times of "neighbor minus algorithm", can always get a special constant N! .

This paper gives the formulas of sum of several special series.