Google is your friend, or better say it’s text-to-speech companion : http://translate.google.com/translate_tts?tl=fr&q=Augustin-Louis%20Cauchy

Au

As in Au​stralia.

gus

With the famous non-English ü, close front rounded vowel [y].

tin

As French matin​, or teint.

Louis

As in Saint-Loui​s (Missouri), without pronouncing the final s.

Cau

As in co-op.

chy

As the english word she.

For anglophones who need to pronounce foreign names while speaking English, I highly recommend the following website (written by a topologist colleague back when he was a radio announcer for a classical music station):
http://www.pronunciationguide.info/

I emphasize “while speaking English” because, if you are speaking English, you need to approximate the real pronunciation of the name with English phonemes. If you pronounce “Louis-Augustin Cauchy” absolutely correctly in English, a fair number of English speakers will only hear the foreignness of the sounds, and not the actual sounds themselves. (More extreme examples of this difficulty come from Chinese or Slavic names.)

Similarly, if you are speaking French, you will need to approximate English names with French phonemes. There is a possible exception if you are in an area like Montreal where a lot of franglais is spoken, but a fair number of bilingual French (or bilingual Americans) are at least occasionally confused by mixing phonemes from the two languages they speak in the same sentence.

Like Koshi. Koshi is a Malayalee name. Cauchy is pronounced Ko shi /
KO SHE.