Gamma function

In mathematics, the gamma function is one of the extensions of the factorial function with its argument shifted down by 1, to real and complex numbers. Derived by Daniel Bernoulli, if n is a positive integer, Γ =! Although other extensions do exist, this particular definition is the most useful; the gamma function is defined for all complex numbers except the non-positive integers. For complex numbers with a positive real part, it is defined via a convergent improper integral: Γ = ∫ 0 ∞ x z − 1 e − x d x This integral function is extended by analytic continuation to all complex numbers except the non-positive integers, yielding the meromorphic function we call the gamma function, it has no zeroes, so the reciprocal gamma function 1/Γ is a holomorphic function. In fact the gamma function corresponds to the Mellin transform of the negative exponential function: Γ = The gamma function is a component in various probability-distribution functions, as such it is applicable in the fields of probability and statistics, as well as combinatorics.

The gamma function can be seen as a solution to the following interpolation problem: "Find a smooth curve that connects the points given by y =! at the positive integer values for x."A plot of the first few factorials makes clear that such a curve can be drawn, but it would be preferable to have a formula that describes the curve, in which the number of operations does not depend on the size of x. The simple formula for the factorial, x! = 1 × 2 × … × x, cannot be used directly for fractional values of x since it is only valid when x is a natural number. There are speaking, no such simple solutions for factorials. A good solution to this is the gamma function. There are infinitely many continuous extensions of the factorial to non-integers: infinitely many curves can be drawn through any set of isolated points; the gamma function is the most useful solution in practice, being analytic, it can be characterized in several ways. However, it is not the only analytic function which extends the factorial, as adding to it any analytic function, zero on the positive integers, such as k sin mπx, will give another function with that property.

A more restrictive property than satisfying the above interpolation is to satisfy the recurrence relation defining a translated version of the factorial function, f = 1, f = x f, for x equal to any positive real number. But this would allow for multiplication by any periodic analytic function which evaluates to one on the positive integers, such as ek sin mπx. There's a final way to solve all this ambiguity: Bohr–Mollerup theorem states that when the condition that f be logarithmically convex is added, it uniquely determines f for positive, real inputs. From there, the gamma function can be extended to all real and complex values by using the unique analytic continuation of f. See Euler's infinite product definition below where the properties f = 1 and f = x f together with the asymptotic requirement that limn→+∞! nx / f = 1 uniquely define the same function. The notation Γ is due to Legendre. If the real part of the complex number z is positive the integral Γ = ∫ 0 ∞ x z − 1 e − x d x converges and is known as the Euler integral of the second kind.

Using integration by parts, one sees that: Γ = ∫ 0 ∞ x z e − x d x = 0 ∞ + ∫ 0 ∞ z x z − 1 e − x d x = lim x →

Mathematical induction

Mathematical induction is a mathematical proof technique. It is used to prove that a property P holds for every natural number n, i.e. for n = 0, 1, 2, 3, so on. Metaphors can be informally used to understand the concept of mathematical induction, such as the metaphor of falling dominoes or climbing a ladder: Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung and that from each rung we can climb up to the next one; the method of induction requires two cases to be proved. The first case, called the base case, proves that the property holds for the number 0; the second case, called the induction step, proves that, if the property holds for one natural number n it holds for the next natural number n + 1. These two steps establish the property P for every natural number n = 0, 1, 2, 3... The base step need not begin with zero, it begins with the number one, it can begin with any natural number, establishing the truth of the property for all natural numbers greater than or equal to the starting number.

The method can be extended to prove statements about more general well-founded structures, such as trees. Mathematical induction in this extended sense is related to recursion. Mathematical induction, in some form, is the foundation of all correctness proofs for computer programs. Although its name may suggest otherwise, mathematical induction should not be misconstrued as a form of inductive reasoning as used in philosophy. Mathematical induction is an inference rule used in formal proofs. Proofs by mathematical induction are, in fact, examples of deductive reasoning. In 370 BC, Plato's Parmenides may have contained an early example of an implicit inductive proof; the earliest implicit traces of mathematical induction may be found in Euclid's proof that the number of primes is infinite and in Bhaskara's "cyclic method". An opposite iterated technique, counting down rather than up, is found in the Sorites paradox, where it was argued that if 1,000,000 grains of sand formed a heap, removing one grain from a heap left it a heap a single grain of sand forms a heap.

An implicit proof by mathematical induction for arithmetic sequences was introduced in the al-Fakhri written by al-Karaji around 1000 AD, who used it to prove the binomial theorem and properties of Pascal's triangle. None of these ancient mathematicians, explicitly stated the induction hypothesis. Another similar case was that of Francesco Maurolico in his Arithmeticorum libri duo, who used the technique to prove that the sum of the first n odd integers is n2; the first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique. Another Frenchman, made ample use of a related principle, indirect proof by infinite descent; the induction hypothesis was employed by the Swiss Jakob Bernoulli, from on it became more or less well known. The modern rigorous and systematic treatment of the principle came only in the 19th century, with George Boole, Augustus de Morgan, Charles Sanders Peirce, Giuseppe Peano, Richard Dedekind; the simplest and most common form of mathematical induction infers that a statement involving a natural number n holds for all values of n.

The proof consists of two steps: The base case: prove that the statement holds for the first natural number n0. N0 = 0 or n0 = 1; the step case or inductive step: prove that if the statement holds for any n ≥ n0, it holds for n+1. In other words, assume the statement holds for some arbitrary natural number n ≥ n0, prove that the statement holds for n + 1; the hypothesis in the inductive step, that the statement holds for some n, is called the induction hypothesis or inductive hypothesis. To prove the inductive step, one assumes the induction hypothesis and uses this assumption, involving n, to prove the statement for n + 1. Whether n = 0 or n = 1 is taken as the standard base case depends on the preferred definition of the natural numbers. In the fields of combinatorics and mathematical logic it is common to consider 0 as a natural number. Mathematical induction can be used to prove that the following statement, P, holds for all natural numbers n. 0 + 1 + 2 + ⋯ + n = n 2. P equal to number n; the proof that P is true for each natural number n proceeds as follows.

Base case: Show that the statement holds for n = 0. P is seen to be true: 0 = 0 ⋅ 2. Inductive step: Show that if P holds also P holds; this can be done. Assume P holds, it must be shown that P holds, that is

Augustin-Louis Cauchy

Baron Augustin-Louis Cauchy was a French mathematician and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors, he singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. A profound mathematician, Cauchy had a great influence over his successors. Cauchy was a prolific writer. Cauchy was the son of Louis François Marie-Madeleine Desestre. Cauchy had two brothers: Alexandre Laurent Cauchy, who became a president of a division of the court of appeal in 1847 and a judge of the court of cassation in 1849, Eugene François Cauchy, a publicist who wrote several mathematical works. Cauchy married Aloise de Bure in 1818, she was a close relative of the publisher. They had Marie Françoise Alicia and Marie Mathilde. Cauchy's father was a high official in the Parisian Police of the Ancien Régime, but lost this position due to the French Revolution, which broke out one month before Augustin-Louis was born.

The Cauchy family survived the revolution and the following Reign of Terror by escaping to Arcueil, where Cauchy received his first education, from his father. After the execution of Robespierre, it was safe for the family to return to Paris. There Louis-François Cauchy found himself a new bureaucratic job in 1800, moved up the ranks; when Napoleon Bonaparte came to power, Louis-François Cauchy was further promoted, became Secretary-General of the Senate, working directly under Laplace. The famous mathematician Lagrange was a friend of the Cauchy family. On Lagrange's advice, Augustin-Louis was enrolled in the École Centrale du Panthéon, the best secondary school of Paris at that time, in the fall of 1802. Most of the curriculum consisted of classical languages. In spite of these successes, Augustin-Louis chose an engineering career, prepared himself for the entrance examination to the École Polytechnique. In 1805, he placed second out of 293 applicants on this exam, he was admitted. One of the main purposes of this school was to give future civil and military engineers a high-level scientific and mathematical education.

The school functioned under military discipline, which caused the young and pious Cauchy some problems in adapting. He finished the Polytechnique in 1807, at the age of 18, went on to the École des Ponts et Chaussées, he graduated with the highest honors. After finishing school in 1810, Cauchy accepted a job as a junior engineer in Cherbourg, where Napoleon intended to build a naval base. Here Augustin-Louis stayed for three years, was assigned the Ourcq Canal project and the Saint-Cloud Bridge project, worked at the Harbor of Cherbourg. Although he had an busy managerial job, he still found time to prepare three mathematical manuscripts, which he submitted to the Première Classe of the Institut de France. Cauchy's first two manuscripts were accepted. In September 1812, now 23 years old, Cauchy returned to Paris after becoming ill from overwork. Another reason for his return to the capital was that he was losing his interest in his engineering job, being more and more attracted to the abstract beauty of mathematics.

Therefore, when his health improved in 1813, Cauchy chose to not return to Cherbourg. Although he formally kept his engineering position, he was transferred from the payroll of the Ministry of the Marine to the Ministry of the Interior; the next three years Augustin-Louis was on unpaid sick leave, spent his time quite fruitfully, working on mathematics. He attempted admission to the First Class of the Institut de France but failed on three different occasions between 1813 and 1815. In 1815 Napoleon was defeated at Waterloo, the newly installed Bourbon king Louis XVIII took the restoration in hand; the Académie des Sciences was re-established in March 1816. The reaction of Cauchy's peers was harsh. In November 1815, Louis Poinsot, an associate professor at the École Polytechnique, asked to be exempted from his teaching duties for health reasons. Cauchy was by a rising mathematical star, who merited a professorship. One of his great successes at that time was the proof of Fermat's polygonal number theorem.

However, the fact that Cauchy was known to be loyal to the Bourbon