A). certainly, in fact, confirming something, freq. in contrast with something which is not confirmed, in Hom. only in combination with γε (v. γοῦν), γάρ, οὔτε or μήτε, ὡς, ἐπεί, μέν, etc.:

1). really, φημὶ γὰρ οὖν κατανεῦσαι .. Κρονίωνα for I declare that Zeus did really promise .. , Il. 2.350 , cf. Pl. Prt. 309b ; τόφρα γὰρ οὖν ἑπόμεσθα .. , ὄφρ’ for we followed them up to the very point, where .. , Il. 11.754 , cf. 15.232 , Od. 2.123 ; εἰ δ’ οὖν τις ἀκτὶς ἡλίου νιν ἱστορεῖ .. ζῶντα A. Ag. 676 , cf. 1042 ; ἐλέχθησαν λόγοι ἄπιστοι μὲν ἐνίοισι Ἑλλήνων, ἐλέχθησαν δ’ ὦν but they really were spoken, Hdt. 3.80 , cf. 4.5 , 6.82 ; Θηβαῖοι μὲν ταῦτα λέγουσι .. , Πλαταιῆς δ’ οὐχ ὁμολογοῦσι .. , ἐκ δ’ οὖν τῆς γῆς ἀνεχώρησαν at all events they did return, Th. 2.5 , cf. 1.63 , Pl. Prt. 315e ; σωτηρίαν λεπτὴν μὲν .. , μόνην δ’ οὖν Id. Lg. 699b ; so δ’ οὖν after a parenthesis; εἰ δή τις ὑμῶν οὕτως ἔχει,—οὐκ ἀξιῶ μὲν γὰρ ἔγωγε,—εἰ δ’ οὖν but if he is so, Id. Ap. 34d , cf. Hdt. 6.76 , Th. 1.3 ; so ἀλλ’ οὖν .. γε but at all events, S. Ant. 84 , Ph. 1305 ; ἔμπης οὖν ἐπιμεῖναι ἐς αὔριον to stay nevertheless at least till to-morrow, Od. 11.351 ; οὖν concessive, I grant you, τάχ’ οὖν τις ἄκων ἔσχε S. Ph. 305 : in apodosi after εἰ or ἐάν, εἰ καὶ σμικρά, ἀλλ’ ὦν ἴση γε ἡ χάρις .. Hdt. 3.140 , cf. 9.48 , E. Ph. 498 , Pl. Phd. 91b , etc.: after ἐπεί and ὡς, ἀλλ’ ἐπεὶ οὖν τὸ πρῶτον ἀνέκραγον but now that I have (emphat.) once spoken up, Od. 14.467 , cf. 17.226 , Il. 18.333 ; Τληπόλεμος δ’, ἐπεὶ οὖν τράφ’ ἐνὶ μεγάρῳ εὐπήκτῳ, αὐτίκα .. κατέκτα when once, i.e. as soon as, he had grown up, 2.661 , cf. 15.363 , 16.394 , al.; νεβροί, αἵ τ’ ἐπεὶ οὖν ἔκαμον .. ἑστᾶσ’ which, as soon as they are tired, stand still, 4.244 ; to indicate that something foreshadowed has actually occurred, ἀγορήνδε καλέσσατο λαὸν Ἀχιλλεύς .. , οἱ δ’ ἐπεὶ οὖν ἤγερθεν 1.57 , cf. 3.340 , al.: sts. οὖν after ἐπεί or ὡς has either no force or approaches signf. 11 or 111 , οἱ δ’ ἐπεὶ οὖν παύσαντο πόνου Od. 16.478 , cf. 19.213 , 251 , al.; τὸν δ’ ὡς οὖν ἐνόησε Il. 3.21 , al.; οὔτ’ οὖν .. , οὔτε .. or οὔτε .. , οὔτ’ οὖν .. both = neither .. nor, but preferred according as the first or second clause is to be marked by emphasis, cf. 17.20 , Od. 2.200 , Hdt. 9.26 , with Od. 11.198s q., S. OT 90 , 271 , etc.; so εἰ .. , εἴτ’ οὖν .. if .. , or if .. , E. Alc. 140 ; εἴτ’ οὖν, εἴτε μὴ γενήσεται whether it shall be so, or no, Id. Heracl. 149 , cf. A. Ag. 491 , S. El. 560 ; ξεῖνος αἴτ’ ὦν ἀστός, i.e. αἴτε ξ. αἴτ’ ὦν ἀ., Pi. P. 4.78 ; and doubled, εἴτ’ οὖν ἀληθὲς εἴτ’ οὖν ψεῦδος Pl. Ap. 34e , cf. A. Ch. 683 : so also in parenth. Relat. clauses, ἢ σῖγ’, ἀτίμως, ὥσπερ οὖν ἀπώλετο πατήρ even as, just as, ib. 96 , cf. 888 , E. Hipp. 1307 (v.l.); εἰ δ’ ἔστιν, ὥσπερ οὖν ἔστι, θεός if he is, as he in fact is, a god, Pl. Phdr. 242e ; οὗτος μὲν οἴεταί τι εἰδέναι οὐκ εἰδώς, ἐγὼ δέ, ὥσπερ οὖν οὐκ οἶδα, οὐδὲ οἴομαι Id. Ap. 21d : for γὰρ οὖν, v. γάρ A. 11.5 ; for μὲν οὖν, v. μέν B. 11.2 .

2). added to indef. Prons. and Advbs., like Lat. cunque, ὅστις whoever, ὁστισοῦν whosoever; ὅπως how, ὁπωσοῦν howsoever; ἄλλος ὁστισοῦν another, be he who he may; so ὁποιοσοῦν, ὁποιοστισοῦν, ὁποσοσοῦν, ὁπωσδηποτοῦν, ὁπητιοῦν, ὁποθενοῦν, etc., v. sub vocc.

II). to continue a narrative, so, then, καὶ τὰ μὲν οὖν .. θῆκαν Od. 13.122 ; ὅτ’ οὖν since, then,. ., S. Ant. 170 , El. 38 , 1318 ; ζεῖ οὖν ἐν τούτῳ .. Pl. Phdr. 251c , cf. Prt. 322b ; εὐθὺς οὖν ὁ Κῦρος εἶπεν X. Cyr. 4.1.22 : in Hdt. and Att., μὲν οὖν (q.v.) is very common in this sense; so δ’ οὖν A. Ag. 34 , S. Aj. 114 ; οὖν is also used alone merely to resume after a parenth. or long protasis, well, as I was saying, ὦ Λακεδαιμόνιοι, χρήσαντος τοῦ θεοῦ .. , ὑμεας γὰρ πυνθάνομαι προεστάναι .. ,—ὑμέας ὦν .. προσκαλέομαι .. Hdt. 1.69 , cf. 4.75 , Th. 2.16 , Pl. Ap. 29d , Smp. 201d , etc.: Hdt. so uses ὦν after a short protasis, 1.144 , etc.

2). ὦν is freq. inserted by Hdt. (sts. without any discernible meaning) between the Prep. and its Verb (but only, it seems, in narrative with the aor., which is always the aor. of habitual action exc. in 2.172 ), ἐπεὰν δὲ ταῦτα ποιήσωσι, ἀπ’ ὦν ἔδωκαν ib. 87 ; καὶ ἔπειτα ἀπ’ ὦν ἔδωκαν ib. 88 : after a part., οἱ δὲ φέροντες ἐς τὴν ἀγορήν, ἀπ’ ὦν ἔδοντο ib. 39 ; κατευξάμενοι, κοιλίην μὲν κείνην πᾶσαν ἐξ ὦν εἶλον ib. 40 ; ἤν τις ψαύσῃ .. , αὐτοῖσι τοῖσι ἱματίοισι ἀπ’ ὦν ἔβαψε ἑωυτόν ib. 47 ; τοῦτον κατ’ ὦν κόψας ib. 172 ; so in Hp., δῑ οὖν ἐφθάρησαν Morb. 1.14 (v.l.), al.; also ἐπ’ ὦν ἐπίομες οἶνον Epich. 124.3 : this tmesis is rare in Att., ὥστε γε καὐτόν σε κατ’ οὖν ἔβαλεν Ar. Ra. 1047 ; but occurs in later writers, Dorieus ap. Phylarch. 3 J., AP 12.226 ( Strat.).

III). in inferences, then, therefore, not in Hom., rare in A., and usu. in questions (v. infr.); in a statement, Eu. 219 ; very common from Hdt. downwds.; so καὶ σὺ οὖν you too therefore, X. Cyr. 4.1.20 ; καὶ γὰρ οὖν Id. An. 1.9.8 ; cf. οὐ γὰρ οὖν, τοιγαροῦν: strengthd., δὴ οὖν Pl. Smp. 191c , etc.; οὖν δή Id. R. 340e : in questions, τίς οὖν ὁ λύσων σ’ ἐστίν ; A. Pr. 771 , cf. S. Tr. 1191 , Ar. Pl. 906 , 909 , etc.; ἆρ’ οὖν δή ; Pl. Tht. 146a ; τί οὖν δή ; S. Aj. 873 (lyr.), Pl. Phd. 57a .