In fact, the same proof shows that Euler's formula is even valid for all complex numbers z.

Here is a proof of Euler's formula using power series expansions

Abel gave a proof of the binomial theorem valid for all numbers, extending Euler's result which had held only for rationals.

The proof of Euler's product formula depends on two important facts.

In 1749 Euler proved this formula for any real n using Euler's formula, which makes the proof quite straightforward.

His first original research, comprising part of a proof of Fermat's last theorem for the case, brought him immediate fame, being the first advance in the theorem since Fermat's own proof of the case and Euler's proof for.

He provides the first proof of what is now known as Euler's formula and constructs the logarithmic spiral.

The basis of Jacobi's proof relies on Euler's pentagonal number theorem, which is itself a specific case of the Jacobi Triple Product Identity.

* Savage Shapes ( previously Vicious Circles and Other Savage Shapes, ISBN 0-439-99747-X ( signs in geometric diagrams, Loci, constructions: perpendicular bisectors ; dropping perpendiculars ; bisecting angles, triangles: similar ; congruent ; equal areas, polygons: regular ; irregular ; angle sizes and construction, tessellations and Penrose Tiles, origami, circles: chord ; tangent ; angle theorems, regular solids, Euler's formula, ellipses, Geometric proof of Pythagoras ' Theorem.

He had power to lay down a human sense of life for his spiritual identity in the likeness of the divine ; but he allowed men to attempt the destruction of the mortal body in order that he might furnish the proof of immortal life.

From these hypotheses, it is also possible to prove that there is only one God in each world by Leibniz's law, the identity of indiscernibles: two or more objects are identical ( are one and the same ) if they have all their properties in common, and so, there would only be one object in each world that possesses property G. Gödel did not attempt to do so however, as he purposely limited his proof to the issue of existence, rather than uniqueness.

A specific passage in the Chinese Book of Wei ( Wei-shu ) is often cited as definitive proof in the identity of the Huns as the Xiongnu.

Meanwhile, Lois Lane discovers proof of Luthor's clone harvesting and false identity ; with help from Superman, she exposes the truth, and a despondent Superman helps to apprehend Luthor.

" As Objectivist philosopher Leonard Peikoff argued, Rand's argument for axioms " is not a proof that the axioms of existence, consciousness, and identity are true.

However, significant additional costs can be incurred by individuals to acquire a " free " ID, as they need to provide other proof of identity.

A signature ( from, " to sign ") is a handwritten ( and often stylized ) depiction of someone's name, nickname, or even a simple " X " that a person writes on documents as a proof of identity and intent.

Given an object X, a functor G ( taking for simplicity the first functor to be the identity ) and an isomorphism proof of unnaturality is most easily shown by giving an automorphism that does not commute with this isomorphism ( so ).

The first type of authentication is accepting proof of identity given by a credible person who has evidence on the said identity, or on the originator and the object under assessment as the originator's artifact respectively.

For example, the credit card network does not require a personal identification number for authentication of the claimed identity ; and a small transaction usually does not even require a signature of the authenticated person for proof of authorization of the transaction.

( 4 ) A formal modal logic proof of the necessity of identity.

Institutions such as banks, the Passport Office and transport authorities require proof of identity, so people who change their names need to have proof of the change.

When he registered for a number, he was amazed that they only wanted him to write his name, and offer no proof of his identity.

The final decision of the court was that Anderson had not provided sufficient proof to claim the identity of the grand duchess.

This can not be confirmed as a Russian cyber attack due to non-attribution-the principle that online identity may not serve as proof of real world identify.

Internet users will be required to provide proof of identity when posting messages on the more than 100, 000 Web sites registered in Xiamen.

The international protection of refugees hardly existed before the First World War and people faced difficulty in providing proof of identity identity but also there was no internationally agreed definition of " refugee ".

* Electronic identity card, smartcard used for proof of identity

Here, Midi-Pyrénées stands clearly apart, with Toulouse being largely favored by young people over Paris, which is another proof of the strong identity that has developed around the regional capital.

The proof is bijective: a matrix A is an adjacency matrix of a DAG if and only if the eigenvalues of the ( 0, 1 ) matrix A + I are positive, where I denotes the identity matrix.

He uses this as a proof of the existence of atoms:

In this case, if a proof uses this statement, researchers will often look for a new proof that doesn't require the hypothesis ( in the same way that it is desirable that statements in Euclidean geometry be proved using only the axioms of neutral geometry, i. e. no parallel postulate.

Empire magazine praised the film saying " the gaudily gory, virtuoso, hyper-kinetic horror sequel / remake uses every trick in the cinematic book " and confirms that " Bruce Campbell and Raimi are gods " and Caryn James of The New York Times called it " genuine, if bizarre, proof of Sam Raimi's talent and developing skill.

The proof uses Euclid's lemma ( Elements VII, 30 ): if a prime p divides the product of two natural numbers a and b, then p divides a or p divides b ( or perhaps both ).

The proof uses modal logic, which distinguishes between necessary truths and contingent truths.

The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 ( and a rewritten version of the dissertation, published as an article in 1930 ) is not easy to read today ; it uses concepts and formalism that are outdated and terminology that is often obscure.

George Boolos has since sketched an alternative proof of the first incompleteness theorem that uses Berry's paradox rather than the liar paradox to construct a true but unprovable formula.

The proof of this statement uses the linearity of classical probability, and has exactly the same structure as the proof of the quantum no-cloning theorem.

Newman's proof is arguably the simplest known proof of the theorem, although it is non-elementary in the sense that it uses Cauchy's integral theorem from complex analysis.

Perelman's proof uses a modified version of a Ricci flow program developed by Richard Hamilton.

The proof uses the classification of finite simple groups.

Although BB84 does not use entanglement, Ekert's protocol uses the violation of a Bell's inequality as a proof of security.

A sig block is easily forged, whereas a digital signature uses cryptographic techniques to provide verifiable proof of authorship.

* Centipede game, the Nash equilibrium of which uses a similar mechanism as its proof.

The alternative is formal software verification, which uses mathematical proof techniques to show the absence of bugs.

An alternative proof uses Prüfer sequences.

While the proof that is a Noetherian ring uses the order structure of, typical proofs in ring theory in general do not assume such additional structure on the ring.

One proof of the impossibility of finding a planar embedding of K < sub > 3, 3 </ sub > uses a case analysis involving the Jordan curve theorem, in which one examines different possibilities for the locations of the vertices with respect to the 4-cycles of the graph and shows that they are all inconsistent with a planar embedding.

The original proof uses hypergeometric functions and innovative tools from the theory of Hilbert spaces of entire functions, largely developed by de Branges.

In logic, a formal proof is not written in a natural language, but instead uses a formal language consisting of certain strings of symbols from a fixed alphabet.

This proof uses definition of even integers, as well as distribution law.