212 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



9.427 



9.428 If z = -^=i 



2 Vat 



e~ qx 



2 / c 



VTT Jz 



i x / c 



- e~ x = =L I 

 g VrJ- 



9.43 Many examples of the use of this method are given by Heaviside: Electro- 

 magnetic Theory, Vol. II. Bromwich, Proceedings Cambridge Philosophical 

 Society, XX, p. 411, 1921, has justified its application by the method of contour 

 integration and applied it to the solution of a problem in the conduction of heat. 



9.431 Herlitz, Arkiv for Matematik, Astronomi och Fysik, XIV, 1919, has 

 shown that the same methods may be applied to the more general partial 

 differential equations of the type, 



and the relations of 9.42 are valid. 

 9.44 Heaviside's Expansion Theorem. 



The operational solution of the differential equation of 9.41, or the more 

 general equation, 9.431, satisfying the given boundary conditions, may be 

 written in the form, 



F(p) 



where F(p) and A(/>) are known functions of p = . Then Heaviside's 



dt 



Expansion Theorem is: 



where a is any root, except o, of A(^>) = o, A'(^) denotes the first derivative of 

 A(/>) with respect to p, and the summation is to be taken over all the roots of 

 A(/>) = o. This solution reduces to u = o at / = o. 



Many applications of this expansion theorem are given by Heaviside, 

 Electromagnetic Theory, II, and III; Electrical Papers, Vol. II. Herlitz, 9.431, 

 has also applied this expansion theorem to the solution of the problem of the 

 distribution of magnetic induction in cylinders and plates. 



9.45 Bromwich's Expansion Theorem. Bromwich has extended Heaviside's 

 Expansion Theorem as follows. If the operational solution of the partial 

 differential equation of 9,41, obtained to satisfy the boundary conditions, is 



