ranks, formula of summation of the Plan, functional ranks.

There are many ways of summation of ranks of the private look (for example, [1] and links in [2], p. 170–183). The major role at summation of functional ranks is played by a formula Plan summation. According to work [3] this formula is received in article The plan, published about two centuries ago, in 1820 (G. A. A. Plana, Mam. Acad. Torino, 25, 403–408). The author of the present article didn’t manage to examine with the text of work of the Plan. The subsequent use of a formula of summation of the Plan in literature sins serious shortcomings. So, in work [4] conditions formulated on page 38 Plan theorems as sets an example, carried out by us below, appear insufficient to receive a formula of summation of the Plan (11) in [4] . There are also works (for example, [5]), in which without the proof theorems of the Plan are used other conditions. Therefore there was a need accurately to formulate and prove the theorem Plan. The formula of summation of the Plan, as shown below, isn’t always applicable for meeting ranks. The author of this work offers a new formula summation (Kα-a summation formula) which is applicable for rather wide class of meeting ranks. From Kα-a formula summation as the special case, turns out a formula of summation of the Plan. Kα-the formula of summation will find a variety of appendices in many areas of mathematics, mechanics and physics. Below as an example Kα application - summation formulas to the theory of dzeta-functions of Riman and to the theory of heat conductivity of cores is shown.

Kuliev, Valekh Dzhafarovich Dr. Sci. Phys. & Math., Professor, Moscow State Machine-Building University (MAMI), Moscow