87 



LiXEAU KlTHVMOKPHH' FUNCTIONS OF THE FlKST ORDER. 



By K. M. Klake. i Abstract. ) 



Euthymorphic functions aiv those monogenic functions wliich satisfy an equa- 

 tion of the form 



where /i, . . ■ ,fn, Pi^ ■ ■ • . /)„,/> are given functions of which pi, . . . , 

 p„ , p are algebraic. The order of o u' is n and it is linear if all of /i, . . . , 



. . , • a z -i- 3 

 fn are or tlie torm ■ ', . 



The paper gives a systematic compilation of the investigations of Babbage, 

 Rausenberger, Koenigs and others ui)on functions defined by an equation of the 

 form 



(where pisl is algebraic* in so far us relates to their existence and analytical ex- 

 pression. The theorems of Koenigs relate to more general functions but they are 

 only defined within a limited circle of convergence. The application of these 

 theorems to euthymorphic functions and their continuation over the entire 3-plane 

 are believed to be new. 



A tabulation of the results contained in the paper is as follows : 

 Every equation (1) can be reduced by a linear transformation to one of the 

 three forms : (^ (z) ^ p [z) (p {z -\~ \] I. 



<!>{z)=p(z)6[e'^ z) II. 



^{z)=p(z)<p{az\ I a I < I . III. 

 Sub-forms and their sDlwtions, (/ is any function), 

 la. (? {z\ ^=0 \z ^ \\\ f\e ■'"'-) 

 lb. <?{z)=bo(z+\; b-= . f{e-'''~) 



ic. nz)^ \'~l^'- • • • ^^-""'U i. + i) 



(2 — 6, ) . . . . (3— 6„) 



/(e = -^-') 



. r(2 — 6^1 . . . . T(z-bn) 



' r(3— aj) .... l^{z — am) 

 Id. 6 (z) = p (z) <t> (z -^ I); p(3) irrationalis unsolved. 

 lla. 0(z)=<p(eiOz); f (z^) 



m. 9(z)=b<;,ie^'z)-z-'^.f{z'-^) 



lie. <p{z)=p[z)f[-z); (pi3). p{-z)=:^l); (1 -f p (3) )./(2^.) 



