"Although elementary functions like sqrt(z), log(z), z^a and sin(z)
have been used for many hundreds of years, their thorough
investigation could only begin when mathematicians discovered the
usefulness of infinite series expansions. Working with one such
expansion led Roger Cotes in 1714 to publish a theorem stating
i phi == log (cos (phi) + i sin(phi) ).
Taking the exponential of both sides leads directly to the Euler formula,
but the significance of this result was not immediately appreciated.
In a 1740 letter to John Bernoulli, Euler noted that
2 cos(x) == e^(i x) - e^( -i x);
in 1743 he published the exponential representations of sin(x)
and cos(x). In 1748 Euler rediscovered Cotes' result, changing
it (slightly!) into the form which now bears his name,
e^( i phi) == cos(phi) + i sin(phi).
Euler introduced the notations e (in 1731) and i (in 1777),
along with the radian, and used them so productively
in his research that Cotes' result came to be
associated with Euler instead."
Unfortunately, in the material I have the references for these assertions
are not given. That about the use of e seems to differ from Cajori's
on the same topic: Cajori, quotes G. Enestro"m as finding it in a
manuscript of 1727 or 1728, published much later in 1862, and gives
a 1736 publication; see paragraph 400.
One of the authors says
"Most of the historical things come from the German Encyclopedia of
Mathematics (Teubner 1904-1926), Pascal's Repertorium (Teubner
1912), Cantor's five volumes, and a couple of other German books
around the turn of the century."
It will be several months, they estimate, before an annotated bibliography
will be available.
Patrick Ion