Is the utterance "You are divorced" to my wife during a fight and in anger considered a valid divorce? And is what I did permissible, in terms of taking her back by saying "I take her back" and resuming marital relations with her? If it is a divorce, do I owe expiation?

It is not permissible for a wife to curse her husband; rather, she must associate with him with kindness and obey him. The husband, too, must associate with his wife with kindness. The divorce you issued to your wife is valid, and your taking her back (revocation) is correct if it was the first or second divorce. Beware of uttering divorce, whether in earnest or in jest, and no expiation is required of you; rather, it is counted as one divorce.