Editing Leonhard Euler (section)

===Analysis===
The development of [[infinitesimal calculus]] was at the forefront of 18th-century mathematical research, and the [[Bernoulli family|Bernoullis]]—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards of [[mathematical rigor|mathematical rigour]]<ref name = "Basel"/> (in particular his reliance on the principle of the [[generality of algebra]]), his ideas led to many great advances.
Euler is well known in [[Mathematical analysis|analysis]] for his frequent use and development of [[power series]], the expression of functions as sums of infinitely many terms,{{sfn|Ferraro|2008|p=155}} such as
<math display=block>e^x = \sum_{n=0}^\infty {x^n \over n!} = \lim_{n \to \infty} \left(\frac{1}{0!} + \frac{x}{1!} + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!}\right).</math>

Euler's use of power series enabled him to solve the [[Basel problem]], finding the sum of the reciprocals of squares of every natural number, in 1735 (he provided a more elaborate argument in 1741). The Basel problem was originally posed by [[Pietro Mengoli]] in 1644, and by the 1730s was a famous open problem, popularized by [[Jacob Bernoulli]] and unsuccessfully attacked by many of the leading mathematicians of the time. Euler found that:<ref name ="Morris PhD thesis ">{{cite thesis |last=Morris |first=Imogen I. |title=Mechanising Euler's use of Infinitesimals in the Proof of the Basel Problem |degree=PhD |publisher=University of Edinburgh | date=24 October 2023 | doi=10.7488/ERA/3835 }}</ref>{{sfn|Dunham|1999}}<ref name="Basel"/>

<math display=block>\sum_{n=1}^\infty {1 \over n^2} = \lim_{n \to \infty}\left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{n^2}\right) = \frac{\pi ^2}{6}.</math>

Euler introduced the constant
<math display=block>\gamma = \lim_{n \rightarrow \infty } \left( 1+ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots + \frac{1}{n} - \ln(n) \right) \approx 0.5772,</math>
now known as [[Euler's constant]] or the Euler–Mascheroni constant, and studied its relationship with the [[harmonic series (mathematics)|harmonic series]], the [[gamma function]], and values of the [[Riemann zeta function]].<ref name=lagarias/>

[[File:Euler's formula.svg|thumb|A geometric interpretation of [[Euler's formula]]]]

Euler introduced the use of the [[exponential function]] and [[logarithms]] in [[analytic proof]]s. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and [[complex number]]s, thus greatly expanding the scope of mathematical applications of logarithms.<ref name=Boyer/> He also defined the exponential function for complex numbers and discovered its relation to the [[trigonometric function]]s. For any [[real number]] {{math|[[φ]]}} (taken to be radians), [[Euler's formula]] states that the [[Exponential function#On the complex plane|complex exponential]] function satisfies
<math display=block>e^{i\varphi} = \cos \varphi + i\sin \varphi</math>

which was called "the most remarkable formula in mathematics" by [[Richard Feynman]].<ref name="Feynman"/>

A special case of the above formula is known as [[Euler's identity]],
<math display=block>e^{i \pi} +1 = 0 </math>

Euler elaborated the theory of higher [[transcendental function]]s by introducing the [[gamma function]]{{sfn|Ferraro|2008|p=159}}<ref name=davis/> and introduced a new method for solving [[quartic equation]]s.<ref name=nickalls/> He found a way to calculate integrals with complex limits, foreshadowing the development of modern [[complex analysis]]. He invented the [[calculus of variations]] and formulated the [[Euler–Lagrange equation]] for reducing [[Mathematical optimization|optimization problems]] in this area to the solution of [[differential equation]]s.

Euler pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study, [[analytic number theory]]. In breaking ground for this new field, Euler created the theory of [[Generalized hypergeometric series|hypergeometric series]], [[q-series]], [[hyperbolic functions|hyperbolic trigonometric functions]], and the analytic theory of [[generalized continued fraction|continued fractions]]. For example, he proved the [[infinitude of primes]] using the divergence of the [[harmonic series (mathematics)|harmonic series]], and he used analytic methods to gain some understanding of the way [[prime numbers]] are distributed. Euler's work in this area led to the development of the [[prime number theorem]].{{sfn|Dunham|1999|loc=Ch. 3, Ch. 4}}