What is the ruling on one who divorced his wife after swearing by divorce that she would be divorced if she said the word "divorce me," and she said it? And how can he take her back? And is expiation obligatory for him?

As long as the man uttered the clear words of divorce, being aware and choosing to do so, the divorce has occurred. Since this is the first divorce, he can take his wife back (revoke the divorce) before her waiting period ('iddah) expires, and no expiation (kaffarah) is required of him. The revocation is completed by him saying: "I take my wife back," or similar explicit words, or by using allusive words along with the intention of revocation. It is recommended to have two witnesses to the revocation.