ON THE THEORY OF INTEGRAL EQUATIONS. 383 



11. Orthogonal and Biorthogonal Systems of Functions. 



A set of functions \p„(s) is said to form an orthogonal system belonging 

 to the interval (a 5 s 5 b) when the relations 



a 



arc satisfied for the different integral values of the suffixes m,n. The 

 system is said to be closed relative to a field of functions M when no 

 function belonging to this field exists such that 



b 



or all the values of n. 



Particular systems of orthogonal functions are the trigonometrical 

 functions cos ns, sin ns (Clairaut, Euler, Lagrange, Fourier) and 

 Legendre's polynomials P„ (/<«)• Jacobi indicated the importance of 

 orthogonal functions in mechanical integration, and the study of these 

 functions was taken up by Murphy. 1 



Murphy introduced the idea of biorthogonal systems of functions, i.e., 

 systems of functions f m (s), x„(s) which are connected by the equations 



a 



he called f m (s) and x,„(s) reciprocal functions. It is more convenient now 

 to call them adjoint functions, as they are generally solutions of pairs of 

 adjoint integral equations or adjoint differential equations. 



A biorthogonal system of functions may be constructed from an 

 orthogonal system by writing 



<Pm(s) = 2 a mn 4> n (s) 

 \p n (s) = 2 a mn x m (s) 



where a mn = if n > m (Murphy). 



An orthogonal system of functions for an interval (a,b) is not unique 

 for we may derive a new set from a given one by an orthogonal substi- 

 tution (change of rectangular axes) 



where 



or by writing 



4> m (s) = 2l m ^ n (s) 



VtmJ* = 2*1 = 1 



fm(s) = y(sMn(t) 



a 

 b 



Mi) = [x.(*M«.0<fr 



1 Caml: Phil. Trans, 1833, vols, ir.-v. 



C C 2 



