“Who would have thought that which enters as the ratio of circumference to diameter, , as the natural base for logarithms, , as the fundamental imaginary unit and 0 and 1 (which we know all about from infancy) would all be tied together in any way, not to mention such a simple and compact way? I hope I never stumble into anything like this formula, for nothing I do after that in life would have any significance.” – Physicist Ramamurti Shankar

^{1}

(1)

The Nobel Prize winning physicist Richard Feynman called equation (1), known as Euler’s identity, “one of the most remarkable, almost astounding, formulas in all of mathematics” ^{2}.

Euler’s identity can be derived easily from Taylor series…

(2)

…as well as defining as

(3)

In equation (2), represents the -th derivative of the function , with the zeroth derivative of being equal to .

The Taylor series of can be easily derived from equation (2).

From the equations above, as well as equation (2), it follows that:

(4)

The Taylor series of can also be derived easily from equation (2).

From the equations above and equation (2), it follows that:

(5)

From equations (3), (4), and (5), if is a real number:

(6)

Finally, let . Equation (6) becomes:

And:

Notes: