Archimedes- "Mathematics reveals its secret only to those who approach it with pure love, for its own beauty"
Formula:
In The Quadrature of the Parabola, Archimedes proved that the area enclosed by a parabola and a straight line is 4/3 times the area of a corresponding inscribed triangle as shown in the figure at right. He expressed the solution to the problem as an infinite geometric series with the common ratio 1/4:

${\displaystyle \sum _{n=0}^{\infty }4^{-n}=1+4^{-1}+4^{-2}+4^{-3}+\cdots ={4 \over 3}.\;}$Archimedes of Syracuse (/ˌɑːkɪˈmiːdiːz/; Greek: Ἀρχιμήδης; c. 287 BC – c. 212 BC) was an Ancient Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Generally considered the greatest mathematician of antiquity and one of the greatest of all time, Archimedes anticipated modern calculus and analysis by applying concepts of infinitesimals and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including the area of a circle, the surface area and volume of a sphere, and the area under a parabola.Other mathematical achievements include deriving an accurate approximation of pi, defining and investigating the spiral bearing his name, and creating a system using exponentiation for expressing very large numbers. He was also one of the first to apply mathematics to physical phenomena, founding hydrostatics and statics, including an explanation of the principle of the lever. He is credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion.
Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting the tomb of Archimedes, which was surmounted by a sphere and a cylinder, which Archimedes had requested to be placed on his tomb, representing his mathematical discoveries.

David Hilbert-"Mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street."

Formula: One of Hilbert's formula in Axioms of Numbers>>
 From the primitive variables we derive further kinds of variables by applying logical connectives to the propositions associated with the primitive variables, for example, to Z. The variables thus defined are called variable-sorts, and the propositions defining them are called sort-propositions; for each of these a new particular sign is introduced. Thus the formula

F(f) ≡ ∀(a)(Z(a) - Z(f (a)))

David Hilbert (German: [ˈdaːvɪt ˈhɪlbɐt]; 23 January 1862 – 14 February 1943) was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis.
Hilbert adopted and warmly defended Georg Cantor's set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems that set the course for much of the mathematical research of the 20th century.
Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.

Godfrey Harold "G.H. Hardy"-"The mathematicians patterns... Must be beautiful... beauty is the first test:  there is no permanent place for ugly mathematics." 

 

Formula: The allele frequencies at each generation are obtained by pooling together the alleles from each genotype of the same generation according to the expected contribution from the homozygote and heterozygote genotypes, which are 1 and 1/2, respectively:

${\displaystyle f_{t}({\text{A}})=f_{t}({\text{AA}})+{\frac {1}{2}}f_{t}({\text{Aa}})}$

Godfrey Harold "G. H." Hardy FRS(7 February 1877 – 1 December 1947)was an English mathematician, known for his achievements in number theory and mathematical analysis. Hardy is remembered also for his 1940 essay on the aesthetics of mathematics, A Mathematician's Apology, and for mentoring the brilliant.Indian mathematician Srinivasa Ramanujan.G. H. Hardy is usually known by those outside the field of mathematics for his essay from 1940 on the aesthetics of mathematics, A Mathematician's Apology, which is often considered one of the best insights into the mind of a working mathematician written for the layman.Hardy is credited with reforming British mathematics by bringing rigour into it, which was previously a characteristic of French, Swiss and German mathematics. British mathematicians had remained largely in the tradition of applied mathematics, in thrall to the reputation of Isaac Newton (see Cambridge Mathematical Tripos). Hardy was more in tune with the cours d'analyse methods dominant in France, and aggressively promoted his conception of pure mathematics, in particular against the hydrodynamics which was an important part of Cambridge mathematics.Hardy's collected papers have been published in seven volumes by Oxford University Press.

Joseph Fourier- "Mathematics compares the most diverse phenomena and discovers that secret analogies that unite them."
Formula:Joseph Fourier presented what is now called the Fourier integral theorem in his treatise Théorie analytique de la chaleur in the form:^[9]

The super-logarithm, written , is defined implicitly by

Jean-Baptiste Joseph Fourier - (/ˈfʊəriˌeɪ, -iər/; French: [fuʁje]; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's law are also named in his honour. Fourier is also generally credited with the discovery of the greenhouse effect.
 In 1822 Fourier published his work on heat flow in Théorie analytique de la chaleur (The Analytic Theory of Heat), in which he based his reasoning on Newton's law of cooling, namely, that the flow of heat between two adjacent molecules is proportional to the extremely small difference of their temperatures. This book was translated, with editorial 'corrections', into English 56 years later by Freeman (1878).The book was also edited, with many editorial corrections, by Darboux and republished in French in 1888.

Felix Klein-"In sense, mathematics has been most advanced by those who distinguish themselves by intuition rather than by rigorous proofs"
Formula: He showed that that surface was a curve in projective space, that its equation was x³y + y³z + z³x = 0

Christian Felix Klein (25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and on the connections between geometry and group theory. His 1872 Erlangen Program, classifying geometries by their underlying symmetry groups, was a hugely influential synthesis of much of the mathematics of the day.

Klein's dissertation, on line geometry and its applications to mechanics, classified second degree line complexes using Weierstrass's theory of elementary divisors.
Klein's first important mathematical discoveries were made in 1870. In collaboration with Sophus Lie, he discovered the fundamental properties of the asymptotic lines on the Kummer surface. They went on to investigate W-curves, curves invariant under a group of projective transformations. It was Lie who introduced Klein to the concept of group, which was to play a major role in his later work. Klein also learned about groups from Camille Jordan.

A hand-blown Klein Bottle

Klein devised the bottle named after him, a one-sided closed surface which cannot be embedded in three-dimensional Euclidean space, but it may be immersed as a cylinder looped back through itself to join with its other end from the "inside". It may be embedded in Euclidean space of dimensions 4 and higher.
In the 1890s, Klein turned to mathematical physics, a subject from which he had never strayed far, writing on the gyroscope with Arnold Sommerfeld. In 1894 he launched the idea of an encyclopedia of mathematics including its applications, which became the Enzyklopädie der mathematischen Wissenschaften. This enterprise, which ran until 1935, provided an important standard reference of enduring value.

John von Neumann-"If people do not believe that mathematics is simple, it is not only because they do not realize how complicated life is." 
Formula: Von Neumann's model of an expanding economy considered the matrix pencil  A − λB with nonnegative matrices A and B; von Neumann sought probability vectors p and q and a positive number λ that would solve the complementarity equation

${\displaystyle p^{T}(A-\lambda B)q=0}$John von Neumann (/vɒn ˈnɔɪmən/; Hungarian: Neumann János Lajos, pronounced [ˈnɒjmɒn ˈjaːnoʃ ˈlɒjoʃ]; December 28, 1903 – February 8, 1957) was a Hungarian-American pure and applied mathematician, physicist, inventor, computer scientist, and polymath. He made major contributions to a number of fields, including mathematics (foundations of mathematics, functional analysis, ergodic theory, geometry, topology, and numerical analysis), physics (quantum mechanics, hydrodynamics and quantum statistical mechanics), economics (game theory), computing (Von Neumann architecture, linear programming, self-replicating machines, stochastic computing), and statistics.
He was a pioneer of the application of operator theory to quantum mechanics, in the development of functional analysis, and a key figure in the development of game theory and the concepts of cellular automata, the universal constructor and the digital computer. He published over 150 papers in his life: about 60 in pure mathematics, 20 in physics, and 60 in applied mathematics, the remainder being on special mathematical subjects or non-mathematical ones. His last work, an unfinished manuscript written while in the hospital, was later published in book form as The Computer and the Brain.
His analysis of the structure of self-replication preceded the discovery of the structure of DNA. In a short list of facts about his life he submitted to the National Academy of Sciences, he stated "The part of my work I consider most essential is that on quantum mechanics, which developed in Göttingen in 1926, and subsequently in Berlin in 1927–1929. Also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939; on the ergodic theorem, Princeton, 1931–1932."

 

 Nikolay Ivanovich Lobachevsky-"There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world"

 Formula:For planar algebra, non-Euclidean geometry arises in the other cases. When ε² = +1, then z is a split-complex number and conventionally j replaces epsilon. Then

${\displaystyle zz^{\ast }=(x+y\mathbf {j} )(x-y\mathbf {j} )=x^{2}-y^{2}\!}$

and {z | z z* = 1} is the unit hyperbola.
When ε² = 0, then z is a dual number

 
Nikolay Ivanovich Lobachevsky was the first mathematician to publish an account of non-Euclidean geometry.
Born on December 1, 1792 in Novgorod, Russia. When he was only 7, Lobachevsky's father died. As such, he moved to Kazan, Siberia where he eventually studied at the University of Kazan. There, he studied physics and mathematics under Johann Bartels who had also instructed Karl Gauss. By 1811, he had a master's in mathematics and physics and by 1814 he was appointed a lectureship. By 1816, Lobachevsky became a professor.
In 1829, he published his Non-Euclidean Geometry which was the first account on the matter to be published in the world. Basically, Lobachevsky stopped the common practice of turning Euclid's Fifth Postulate into a theorem. Rather, he studied a geometry apart from Euclid's final postulate. In other terms, he considered Euclid's postulate as a special case of a more simplified, general geometry.

Bernhard Riemann- "If only I had a theorems! Then I should find the proofs enough." 
Formula:
To be specific, we say that the Riemann integral of f equals s if the following condition holds: For all ε > 0, there exists δ such that for any tagged partition ${\displaystyle x_{0},\ldots ,x_{n}}$ and ${\displaystyle t_{0},\ldots ,t_{n-1}}$ whose mesh is less than δ, we have

${\displaystyle \left|\sum _{i=0}^{n-1}f(t_{i})(x_{i+1}-x_{i})-s\right|<\varepsilon .}$

 Georg Friedrich Bernhard Riemann (German: [ˈʀiːman]  17 September 1826 – 20 July 1866) was an influential German mathematician who made lasting and revolutionary contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis. His famous 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded, although it is his only paper in the field, as one of the most influential papers in analytic number theory. Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity.

 Agustus De Morgan-"Mathematicians care no more for logic than logicians for mathematics"
Formula:

• Hyperbolic sine:

${\displaystyle \sinh x={\frac {e^{x}-e^{-x}}{2}}={\frac {e^{2x}-1}{2e^{x}}}={\frac {1-e^{-2x}}{2e^{-x}}}}$


               Augustus De Morgan (/dɪ ˈmɔːrɡən/; 27 June 1806 – 18 March 1871) was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous.

De Morgan was a brilliant and witty writer, whether as a controversialist or as a correspondent. In his time there flourished two Sir William Hamiltons who have often been conflated. One was Sir William Hamilton, 9th Baronet (that is, his title was inherited), a Scotsman, professor of logic and metaphysics at the University of Edinburgh; the other was a knight (that is, won the title), an Irishman, professor at astronomy in the University of Dublin. The baronet contributed to logic, especially the doctrine of the quantification of the predicate; the knight, whose full name was William Rowan Hamilton, contributed to mathematics, especially geometric algebra, and first described the Quaternions. De Morgan was interested in the work of both, and corresponded with both; but the correspondence with the Scotsman ended in a public controversy, whereas that with the Irishman was marked by friendship and terminated only by death. In one of his letters to Rowan, De Morgan says,

Be it known unto you that I have discovered that you and the other Sir W. H. are reciprocal polars with respect to me (intellectually and morally, for the Scottish baronet is a polar bear, and you, I was going to say, are a polar gentleman). When I send a bit of investigation to Edinburgh, the W. H. of that ilk says I took it from him. When I send you one, you take it from me, generalize it at a glance, bestow it thus generalized upon society at large, and make me the second discoverer of a known theorem.

The correspondence of De Morgan with Hamilton the mathematician extended over twenty-four years; it contains discussions not only of mathematical matters, but also of subjects of general interest. It is marked by geniality on the part of Hamilton and by wit on the part of De Morgan.

Stefan Banach-"Mathematics is the most beautiful and most powerful creation of the human spirit"

Formula:Sequential Banach-Alaoglu theorem
  let X be a separable normed space and B the closed unit ball in X^∗. Since X is separable, let {x_n} be a countable dense subset. Then the following defines a metric for x, y ∈ B

 

 Stefan Banach ([ˈstɛfan ˈbanax]  30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the world's most important and influential 20th-century mathematicians. He was one of the founders of modern functional analysis, and an original member of the Lwów School of Mathematics. His major work was the 1932 book, Théorie des opérations linéaires (Theory of Linear Operations), the first monograph on the general theory of functional analysis.

Banach's dissertation, completed in 1920 and published in 1922, formally axiomatized the concept of a complete normed vector space and laid the foundations for the area of functional analysis. In this work Banach called such spaces "class E-spaces", but in his 1932 book, Théorie des opérations linéaires, he changed terminology and referred to them as "spaces of type B", which most likely contributed to the subsequent eponymous naming of these spaces after him.The theory of what came to be known as Banach spaces had antecedents in the work of the Hungarian mathematician Frigyes Riesz (published in 1916) and contemporaneous contributions from Hans Hahn and Norbert Wiener. For a brief period in fact, complete normed linear spaces were referred to as "Banach-Wiener" spaces in mathematical literature, based on terminology introduced by Wiener himself. However, because Wiener's work on the topic was limited, the established name became just Banach spaces.
Likewise, Banach's fixed point theorem, based on earlier methods developed by Charles Émile Picard, was included in his dissertation, and was later extended by his students (for example in the Banach–Schauder theorem) and other mathematicians (in particular Brouwer and Poincaré and Birkhoff). The theorem did not require linearity of the space, and applied to any Cauchy space (complete metric space).
The Hahn–Banach theorem, is one of the fundamental theorems of functional analysis.

• Banach–Tarski paradox
• Banach–Steinhaus theorem
• Banach–Alaoglu theorem 
• Banach–Stone theorem

 

Definition of the Meijer G-function

A general definition of the Meijer G-function is given by the following line integral in the complex plane (Bateman & Erdélyi 1953, § 5.3-1):

${\displaystyle G_{p,q}^{\,m,n}\!\left(\left.{\begin{matrix}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{matrix}}\;\right|\,z\right)={\frac {1}{2\pi i}}\int _{L}{\frac {\prod _{j=1}^{m}\Gamma (b_{j}-s)\prod _{j=1}^{n}\Gamma (1-a_{j}+s)}{\prod _{j=m+1}^{q}\Gamma (1-b_{j}+s)\prod _{j=n+1}^{p}\Gamma (a_{j}-s)}}\,z^{s}\,ds,}$

 

 

 In mathematics, Spence's function, or dilogarithm, denoted as Li₂(z), is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function, the dilogarithm itself::

${\displaystyle \operatorname {Li} _{2}(z)=-\int _{0}^{z}{\ln(1-u) \over u}\,\mathrm {d} u{\text{, }}z\in \mathbb {C} \setminus [1,\infty )}$

${\displaystyle \operatorname {Li} _{2}(z)=-\int _{0}^{z}{\ln(1-u) \over u}\,\mathrm {d} u{\text{, }}z\in \mathbb {C} \setminus [1,\infty )}$

Kummer's function

From Wikipedia, the free encyclopedia

In mathematics, there are several functions known as Kummer's function. One is known as the confluent hypergeometric function of Kummer. Another one, defined below, is related to the polylogarithm. Both are named for Ernst Kummer.
Kummer's function is defined by

${\displaystyle \Lambda _{n}(z)=\int _{0}^{z}{\frac {\log ^{n-1}|t|}{1+t}}\;dt.}$

${\displaystyle \Lambda _{n}(z)=\int _{0}^{z}{\frac {\log ^{n-1}|t|}{1+t}}\;dt.}$

In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are solutions of Laguerre's equation:

${\displaystyle xy''+(1-x)y'+ny=0}$

Joseph Fourier presented what is now called the Fourier integral theorem in his treatise Théorie analytique de la chaleur in the form:^[9]

${\displaystyle f(x)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\ \ d\alpha f(\alpha )\ \int _{-\infty }^{\infty }dp\ \cos(px-p\alpha )\ ,}$The super-logarithm, written ${\displaystyle \,\mathrm {slog} _{b}(z)}$, is defined implicitly by
${\displaystyle \,\mathrm {slog} _{b}(b^{z})=\mathrm {slog} _{b}(z)+1}$

For real non zero values of x, the exponential integral Ei(x) is defined as

${\displaystyle \operatorname {Ei} (x)=-\int _{-x}^{\infty }{\frac {e^{-t}}{t}}\,dt.\,}$The piecewise linear function

${\displaystyle x(t)=t-\lfloor t\rfloor =t-\operatorname {floor} (t)}$

or

${\displaystyle x(t)=x(mod1)}$For any real number x the absolute value or modulus of x is denoted by |x| (a vertical bar on each side of the quantity) and is defined as^[6]

${\displaystyle |x|={\begin{cases}x,&{\mbox{if }}x\geq 0\\-x,&{\mbox{if }}x<0.\end{cases}}}$In more precise terms, an algebraic function of degree n in one variable x is a function ${\displaystyle y=f(x)}$ that satisfies a polynomial equation

${\displaystyle a_{n}(x)y^{n}+a_{n-1}(x)y^{n-1}+\cdots +a_{0}(x)=0}$

Definition by Ein

Both ${\displaystyle \mathrm {Ei} }$ and ${\displaystyle E_{1}}$ can be written more simply using the entire function ${\displaystyle \mathrm {Ein} }$^[10] defined as

${\displaystyle \mathrm {Ein} (z)=\int _{0}^{z}(1-e^{-t}){\frac {dt}{t}}=\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}z^{k}}{k\;k!}}}$