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Explanation: Writing non-Arabic names in Arabic letters often results in a slightly different pronunciation, because the Arabic vowel set and consonant set does not tightly correspond to the counterpart sets in other languages, even very close languages. An example of that is the vowel in "Jayne," which has no exact counterpart that can be represented in Arabic letters. The lettering I suggested above is a close approximation.
I have fashioned the name "Emily" after the name of an Arab writer (Emily Nasrallh). If had a dughter named Emily and I wanted to write it in Arabic letters, I woud probably write it أملي