210 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



<i +*)+ (i - *)" = P(- I - f + ~ 2 , \, * 2 ), 



cos nx = 



sin 



,,/w+ii 3 . 9 \ 

 w sin #/M - , - , -, sin 2 # J, 



sin" 1 x = 



tan- 1 * = #/?(-, i,*,- 



PW-F/-I.1 

 P.-*\ *,+? 



w , 2 



' ' 



2 2 



9.4 Heaviside's Operational Methods of Solving Partial Differential Equations. 



9.41 The partial differential equation, 



&u _du 

 a dx 2 ~ dt 1 

 where a is a constant, may be solved by Heaviside's operational method. 



Writing = p, and - = q 2 , the equation becomes, 



(jt a 



&u , 



a? = fu ' 



whose complete solution is u = e qx A + e~ qx B, where A and B are integration 

 constants to be determined by the boundary conditions. In many applications 

 the solution u = e~ QX B, only, is required: and the boundary conditions will 

 lead to u = e~ 9X f(q)u , where UQ is a constant. If e~ 9X f(q) be expanded in an 

 infinite power series in q, and the integral and fractional, positive and negative 

 powers of p be interpreted as in 9.42, the resulting series will be a solution of 

 the differential equation, satisfying the boundary conditions, and reducing to 

 u = o at / = o. The expansion of e~ qx f(q) may be carried out in two or more 

 ways, leading to series suitable for numerical calculation under different 

 conditions. 



