Search results
Results From The WOW.Com Content Network
Euler's identity is also a special case of the more general identity that the n th roots of unity, for n > 1, add up to 0: = = Euler's identity is the case where n = 2. A similar identity also applies to quaternion exponential: let {i, j, k} be the basis quaternions; then,
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.
Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely and and then integrated. This technique is often simpler and faster than using trigonometric identities or integration by parts , and is sufficiently powerful to integrate any rational expression involving trigonometric functions.
Euler's identity is a special case of this: e i π + 1 = 0 . {\displaystyle e^{i\pi }+1=0\,.} This identity is particularly remarkable as it involves e , π {\displaystyle \pi } , i , 1, and 0, arguably the five most important constants in mathematics, as well as the four fundamental arithmetic operators: addition, multiplication ...
An example of "beauty in method"—a simple and elegant visual descriptor of the Pythagorean theorem. ... Euler's identity is a special case of Euler's formula, ...
Named after Leonhard Euler, it is a model example of a q-series and provides the prototypical example of a relation between combinatorics ... The Euler identity, ...
Bézout's identity (despite its usual name, it is not, properly speaking, an identity) Binet-cauchy identity; Binomial inverse theorem; Binomial identity; Brahmagupta–Fibonacci two-square identity; Candido's identity; Cassini and Catalan identities; Degen's eight-square identity; Difference of two squares; Euler's four-square identity; Euler ...
Comment: The proof of Euler's four-square identity is by simple algebraic evaluation. Quaternions derive from the four-square identity, which can be written as the product of two inner products of 4-dimensional vectors, yielding again an inner product of 4-dimensional vectors: (a·a)(b·b) = (a×b)·(a×b).