Fifty years of Harlem history, with photos; This Was Harlem: A Cultural Portrait, 1900-1950, by Jervis Anderson. New York: Farrar, Straus, Giroux Inc. 389 pp. $17.95.

This highly interesting book shows how the city within a city whose name is now synonymous with social impoverishment and crime developed from a ''refuge'' for blacks from West Side New York slums into the incarnation of Negro independence and cultural integrity. It's unquestionably authoritative: the products of Anderson's research into an enormous variety of literary, journalistic, and first-person sources as well as popular song lyrics are evident on every page, and the text is embellished with dozens of wonderful period photographs.

We learn a great deal here about indigenous political and religious movements , social mobility vis-a-vis interracial conflict, and -- above all -- the ongoing explosions of talent that made Harlem an important center of American popular culture.

Some readers may find that Anderson's topical approach and preference for short chapters make for both sketchiness and disorganization, but the best sections -- for example, on the luminaries in art and literature who made the Harlem Renaissance of the '30s, and on the evangelists and cult leaders who rose up from the storefront churches - are, without qualification, fresh and fascinating. A most welcome book.