both cross-sections are presented. OCR Output compact analytical expressions for the matrix elements for all the direct contributions to HERA where we study in detail the scale dependence of the cross section. ln addition solved photons and apply the results to ¢r° production at LEP2 as well as at the ep collider hadron photoproduction. We take into account all contributions from resolved and unreWe present a complete calculation of next-to-leading order (NLO) corrections to inclusive February 19921 /gu/g4{O_f·) discussed elsewhere [7][8][9][10], this simple separation is no longer so clear from a theoretical OCR Output thereby providing a means of separating the three classes of processes, but as has been lt may be possible for the photon fragments produced in the reactions to be tagged separate.as defined in Fig.2. As much as possible we will keep the discussions of the two reactions we discuss 7p processes we will use the usual labels of direct and resolved contributions in one or both beam directions, for once-and twice-resolved processes respectively. When either photon, whereas in the latter two cases such fragments are produced in each event, ents (Fig.lc). ln the first case there are no hadronic fragments or spectator jets from the twice-resolved contributions where both photons interact via their partonic constitu is resolved into its partonic constituents before interacting with the other (Fig.1b), and in the hard scattering process (Fig.la), the once-resolved contributions where one photon section: the direct contributions where the photons couple directly to the quarks produced sions, we identify the three following classes of contributions to the inclusive hadron cross couplings. ln this paper we will adopt the convention whereby, when discussing 77 colli case of 77 collisions there is the additional case where both photons interact via point-like the labels direct and resolved are sufficient to define the classes of processes, but in the called direct and resolved processes When considering 7p interactions, for example, coupling from those where it does so via its partonic constituents. This defines the so tributions to the cross-section where the photon interacts via its point-like electromagnetic It is customary when considering photoproduction processes to separate those con and ep machines.(O(cr,)) corrections to the photoproduction of inclusive hadrons relevant for both e+el ected fragmentation functions. With this in mind we have calculated all the higher order to tie down important QCD parameters such as structure functions and the much neg Inclusive hadron production is also in itself interesting since it can and has been used soon [3][4][5] and 1rO photoproduction will be an important background to these processes.it is hoped that the photoproduction of prompt photons will be studied at HERA very of the data will soon be extended and improved at both HERA and LEP2. Furthermore quantitative tests of perturbative QCD. It is also anticipated that the range and quality that they be confronted...