Herbert Weir Smyth, A Greek Grammar for Colleges

a. self: standing by itself in the nominative, αὐτὸς ὁ ἀνήρ or ὁ ἀνὴρ αὐτός the man himself, or (without the article) in agreement with a substantive or pronoun; as ἀνδρὸς αὐτοῦ of the man himself.

b. him, her, it, them, etc.: standing by itself in an oblique case (never in the nominative). The oblique cases of αὐτός are generally used instead of οὗ, οἷ, ἕ, etc., as ὁ πατὴρ αὐτοῦ his father, οἱ παῖδες αὐτῶν their children.

c. same: when it is preceded by the article in any case: ὁ αὐτὸς ἀνήρ the same man, τοῦ αὐτοῦ ἀνδρός of the same man.

N.—The article and αὐτός may unite by crasis (68 a): αὑτός, αὑτή, ταὐτό or ταὐτόν; ταὐτοῦ, ταὐτῆς; ταὐτῷ, ταὐτῇ, etc. Distinguish αὑτή the same f. from αὕτη this f.; ταὐτά the same n. from ταῦτα these things n.; ταὐτῇ from ταύτῃ.