On the algebraic processing and the reversion of power series.

For a given power series y=Σanxn, a root x may be determined as a series form x=Σcnyn and the operation is referred to as the reversion of power series. This means that it is possible to obtain an explicit solution to a transcendental equation which appears frequently in engingering mathematics. Coefficient formulae cn=cn(a1, ···, an) up to n=6 are known at present and those for n>6 are desirable. Formula construction, however, becomes more complicated as order n increases and computer processing (numerical only) is preferable. This paper summarizes the theory and the algorithms of the reversion of power series as well as associated procedures. Also coefficient formulae up to the 15th order are listed. As an example, an explicit solution is obtained for the transcendental equation, cosXcoshX+1=0, the frequency equation of the flexural free vibration of a uniform bar with fixed-free boundary conditions.