Zeta and L-functions | Rational functions | Special functions
In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function Lis(z) of order s and argument z. Only for special values of s does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function. In quantum statistics, the polylogarithm function appears as the closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac integral or the Bose–Einstein integral. In quantum electrodynamics, polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams. The polylogarithm function is equivalent to the Hurwitz zeta function — either function can be expressed in terms of the other — and both functions are special cases of the Lerch transcendent. Polylogarithms should not be confused with polylogarithmic functions nor with the offset logarithmic integral which has the same notation, but with one variable. * Different polylogarithm functions in the complex plane * Li -3(z) * Li -2(z) * Li -1(z) * Li0(z) * Li1(z) * Li2(z) * Li3(z) The polylogarithm function is defined by a power series in z, which is also a Dirichlet series in s: This definition is valid for arbitrary complex order s and for all complex arguments z with |z| < 1; it can be extended to |z| ≥ 1 by the process of analytic continuation. (Here the denominator ns is understood as exp(s ln(n)). The special case s = 1 involves the ordinary natural logarithm, Li1(z) = −ln(1−z), while the special cases s = 2 and s = 3 are called the dilogarithm (also referred to as Spence's function) and trilogarithm respectively. The name of the function comes from the fact that it may also be defined as the repeated integral of itself: thus the dilogarithm is an integral of a function involving the logarithm, and so on. For nonpositive integer orders s, the polylogarithm is a rational function. (Wikipedia).
Professor Poliakoff on a word that sounds familiar. This is the first in our new Chem Definition series - short videos about the language and jargon of chemistry.
From playlist Chem Definition - Periodic Videos
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👉 Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
Polymorphs can be a headache for people who make pharmaceuticals. Find out why? More chemistry at http://www.periodicvideos.com/
From playlist Chem Definition - Periodic Videos
What is a polygon and what is a non example of a one
👉 Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
Ishai Dan Cohen:The polylog quotient and the Goncharov quotient in computational Chabauty-Kim theory
Abstract: Polylogarithms are those multiple polylogarithms which factor through a certain quotient of the de Rham fundamental group of the thrice punctured line known as the polylogarithmic quotient. In joint work with David Corwin, building on work that was partially joint with Stefan We
From playlist HIM Lectures: Trimester Program "Periods in Number Theory, Algebraic Geometry and Physics"
Johannes Bluemlein: Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams
The lecture was held within the framework of the Hausdorff Trimester Program: Periods in Number Theory, Algebraic Geometry and Physics. Abstract: We calculate 3-loop master integrals for heavy quark correlators and the 3-loop QCD corrections to the ρ-parameter. They obey non-factorizing
From playlist Workshop: "Amplitudes and Periods"
Approximating the Longest Increasing Subsequence in Polylogarithmic Time - Michael Saks
Michael Saks Rutgers, The State University of New Jersey October 12, 2010 Finding the longest increasing subsequence (LIS) is a classic algorithmic problem. Simple O(nlogn)O(nlogn) algorithms, based on dynamic programming, are known for solving this problem exactly on arrays of length nn.
From playlist Mathematics
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👉 Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
Daniil Rudenko: Polylogarithms, cluster algebras and Zagier conjecture
The lecture was held within the framework of the Hausdorff Trimester Program: Periods in Number Theory, Algebraic Geometry and Physics. Abstract: Polylogarithms appeared already in the work of Leonhard Euler and have been actively studied since then. The importance of these functions and
From playlist Workshop: "Periods and Regulators"
Polylogarithms, motives & cluster algebras-Professor Christian Zickert (U. of Maryland)
We discuss the relationship between polylogarithms, cluster algebras, and motivic cohomology. The story will be linked to Walter's work on the Chern-Simons class and hyperbolic 3-manifolds.
Christian Bogner: The analytic continuation of the kite and the sunrise integral
The lecture was held within the framework of the Hausdorff Trimester Program: Periods in Number Theory, Algebraic Geometry and Physics. Abstract: The integrals mentioned in the title are Feynman integrals, which are of particular interest in recent developments, as they can not be expres
From playlist Workshop: "Amplitudes and Periods"
👉 Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
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From playlist Elements | Seeker
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👉 Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
What is a Polygon? | Don't Memorise
To learn more about Polygons, enroll in our full course now: https://infinitylearn.com/microcourses?utm_source=youtube&utm_medium=Soical&utm_campaign=DM&utm_content=MPYNEYeLYaQ&utm_term=%7Bkeyword%7D In this video, we will learn: 0:00 what is a polygon? To watch more videos related to G
From playlist Area of a Regular Polygon
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👉 Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
Claude Duhr: Elliptic polylogarithms evaluated at torsion points and iterated integrals
The lecture was held within the framework of the Hausdorff Trimester Program: Periods in Number Theory, Algebraic Geometry and Physics. Abstract: I will show how to use a construction due to Brown to define a certain coaction on (a variant of) elliptic polylogarithms (eMPLs). This coacti
From playlist Workshop: "Amplitudes and Periods"
Explicit Binary Tree Codes with Polylogarithmic Size Alphabet - Gil Cohen
http://www.math.ias.edu/seminars/abstract?event=129076 More videos on http://video.ias.edu
From playlist Mathematics
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👉 Learn about polygons and how to classify them. A polygon is a plane shape bounded by a finite chain of straight lines. A polygon can be concave or convex and it can also be regular or irregular. A concave polygon is a polygon in which at least one of its interior angles is greater than 1
From playlist Classify Polygons
Two-source dispersers for polylogarithmic entropy and improved Ramsey graphs I - Cohen
https://www.math.ias.edu/seminars/abstract?event=83664
From playlist Computer Science/Discrete Mathematics