Absorbed in writing the article,the author forgot that he ha

这句话的逗号之前是分词作状语的表达方式,省略了主语he (the author)和谓语动词(was),absorbed in 在这句话里就是非谓语动词了.
be absorbed in doing sth意思是“专注于做某事”,是一个固定的表达方式,所以不能替换成having absorbed by.