Is expiation obligatory for one who swore an oath concerning something he believed to be true, but then it became clear that he was mistaken, and how does one who has sworn multiple oaths expiate them?

If someone swears an oath believing something to be true, and it later turns out to be false, there is no expiation (kaffarah) required of him. This is because it falls under the category of "laghw al-yamin" (a thoughtless or unintentional oath), and this is the view held by most scholars. If this type of presumptive oath is repeated, there is still no expiation for it.

However, if the oaths are deliberate (mu'anqidah), and one repeatedly swears about the same thing without intending to multiply the expiation, then only one expiation is required. But if the matters sworn upon are multiple and one breaks each oath, then the expiation multiplies with the multiplicity of the broken oaths.