Euler linked cosines and sines to the complex exponential
Euler's formula says the exponential of i times x equals cosine x plus i sine x, welding trigonometry to growth. Feynman called it mathematics' jewel; set x to π and it becomes the identity with e, i, π, and zero. Cotes edged toward it in 1714; Euler sealed series proofs around 1740.
Here e is the natural base, i the imaginary unit, and cosine and sine the usual trigonometric functions; the combination is sometimes nicknamed cis of x. The relation still holds when x itself is complex, and it turns up throughout mathematics, physics, chemistry and engineering.
Cotes offered a geometric argument that, after fixing a factor, exponentiates into Euler's equation, though complex logarithms are multi-valued by multiples of two-pi-i. Around 1740 Euler compared power series of exponential and trigonometric functions to derive the relation named for him.
Euler first printed the result in 1748, in his foundational Introductio in analysin infinitorum. Earlier, Johann Bernoulli had written down an integral relation linking natural logarithms to imaginary numbers but never evaluated it, and his letters to Euler reveal that he did not fully understand complex logarithms; Euler proposed that such logarithms take infinitely many values. Picturing complex numbers as points on a plane came roughly fifty years later, from Caspar Wessel. To make sense of e raised to a complex power, one can extend a real definition: the power series for the exponential converges for every complex input by the ratio test, and so does the limit of one plus z over n, raised to the n. One of several proofs divides the trigonometric side by the exponential side and shows the quotient is always one, which works because the exponential never vanishes.
Source: Euler's formula
