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Euler's identity therefore states that the limit, as n approaches infinity, of (+) is equal to −1. This limit is illustrated in the animation to the right. Euler's formula for a general angle. Euler's identity is a special case of Euler's formula, which states that for any real number x,
Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". [2] When x = π, Euler's formula may be rewritten as e iπ + 1 = 0 or e iπ = −1, which is known as Euler's identity.
This mathematical term forms part of an identity, a special case of Euler's formula, written = + (). Setting x {\displaystyle x} to a value of π {\displaystyle \pi } , as with the above term, Euler's formula reduces to a famous equation relating seven important mathematical symbols (and none that are unimportant!), namely e i π + 1 ...
This equation, stated by Euler in 1758, [2] is known as Euler's polyhedron formula. [3] It corresponds to the Euler characteristic of the sphere (i.e. χ = 2 {\displaystyle \ \chi =2\ } ), and applies identically to spherical polyhedra .
The Euler function may be expressed as a q-Pochhammer symbol: = (;). The logarithm of the Euler function is the sum of the logarithms in the product expression, each of which may be expanded about q = 0, yielding
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Q-series generalize Euler's function, which is closely related to the Dedekind eta function, and occurs in the study of modular forms. The modulus of the Euler function (see there for picture) shows the fractal modular group symmetry and occurs in the study of the interior of the Mandelbrot set.
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