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Special case of the polylogarithm
In mathematics, the dilogarithm (or Spence's function), denoted as Li2(z), is a particular case of the polylogarithm. Two related special functions are
Dilogarithm
Special mathematical function
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
Polylogarithm
In mathematics, the quantum dilogarithm is a special function defined by the formula ϕ ( x ) ≡ ( x ; q ) ∞ = ∏ n = 0 ∞ ( 1 − x q n ) , | q | < 1 {\displaystyle
Quantum_dilogarithm
Q-analog in combinatorial mathematics
Askey–Wilson operators. The q-exponential is also known as the quantum dilogarithm. The q-exponential e q ( z ) {\displaystyle e_{q}(z)} is defined as e
Q-exponential
Conditions for switching order of integration in calculus
{3}{2}}\,\mathrm {Li} _{2}(1)\end{aligned}}} For the Dilogarithm of one, this yields: L i 2 ( 1 ) = π 2 6 {\displaystyle \mathrm {Li}
Fubini's_theorem
Special function related to the dilogarithm
Equivalently, it can be defined by a power series, or in terms of the dilogarithm, a closely related special function. The inverse tangent integral is
Inverse_tangent_integral
Numerical computation of special functions
\xi (z)=\xi (1-z).} Weisstein, Eric W. "Dilogarithm". Wolfram MathWorld. Retrieved 2025-09-01. "Dilogarithm Reflection Formula - ProofWiki". proofwiki
Reflection_formula
Conjecture in knot theory relating quantum invariants and hyperbolic geometry
1995 for hyperbolic links as a state sum using the theory of quantum dilogarithms. Kashaev stated the formula of the volume conjecture in the case of hyperbolic
Volume_conjecture
Complete Fermi–Dirac integral, an alternate form of the polylogarithm. Dilogarithm Incomplete Fermi–Dirac integral Kummer's function Riesz function Hypergeometric
List of mathematical functions
List_of_mathematical_functions
Russian mathematician and physicist (1934–2017)
Faddeev–Senjanovic quantization Faddeev–Jackiw quantization Quantum dilogarithm Quantum inverse scattering method Yangian Awards Dannie Heineman Prize
Ludvig_Faddeev
Scottish mathematician (1777–1815)
L_{2}(x)=-\int _{0}^{x}{\frac {\ln(1-t)}{t}}\operatorname {d} \!t} (the dilogarithm) to nine decimal places, in a table, for all integer values of 1 + x
William Spence (mathematician)
William_Spence_(mathematician)
Soviet American mathematician
(with V. V. Fock) Fock, V.V.; Goncharov, A.B. (2009). "The quantum dilogarithm and representations of quantum cluster varieties". Inventiones Mathematicae
Alexander_Goncharov
related to polylogarithm, hyperbolic geometry and algebraic K-theory. The dilogarithm function is the function defined by the power series Li 2 ( z ) = ∑
Bloch_group
Mathematical constant in number theory
quickly for large n. An expansion may also be given in terms of the dilogarithm: ln K 0 2 = 1 ln 2 [ Li 2 ( − 1 2 ) + 1 2 ∑ k = 2 ∞ ( − 1 ) k Li
Khinchin's_constant
Domain of convergence of power series
is equal to g(z)/z with g of Example 2. It turns out that h(z) is the dilogarithm function. Example 4: The power series ∑ i = 1 ∞ a i z i where a i =
Radius_of_convergence
Decay of strong electromagnetic fields into particles
infinite sum in the expression above can be written in terms of the dilogarithm, and then the equation becomes Γ = ( e E ) 2 4 π 3 ℏ 2 c Li 2 ( exp
Schwinger_effect
Russian mathematician (born 1954)
Goncharov, A. B.; Schechtman, V. V.; Varchenko, A. N. (1990). "Aomoto dilogarithms, mixed Hodge structures and motivic cohomology of pairs of triangles
Vadim_Schechtman
Branch of mathematics
calculus Time scale calculus q-analog Basic hypergeometric series Quantum dilogarithm Abreu, Luis Daniel (2006). "Functions q-Orthogonal with Respect to Their
Quantum_calculus
Family of power series in mathematics
_{2}(x)=\sum _{n>0}\,{x^{n}}{n^{-2}}=x\;{}_{3}F_{2}(1,1,1;2,2;x)} is the dilogarithm Furthermore, 3 F 2 ( 1 , 1 , 1 + n ; 2 , 2 ; x ) = 1 n ! ∑ k = 0 n [
Generalized hypergeometric function
Generalized_hypergeometric_function
American mathematician
zeta function of an arbitrary number field at s = 2 in terms of the dilogarithm function, by studying arithmetic hyperbolic 3-manifolds. He later formulated
Don_Zagier
Formal power series
ez, log z, cos z, arcsin z, 1 + z {\displaystyle {\sqrt {1+z}}} , the dilogarithm function Li2(z), the generalized hypergeometric functions pFq(...; .
Generating_function
Simplex formed from a right-angled path
can be expressed in terms of the Lobachevsky function, or in terms of dilogarithms. Hugo Hadwiger conjectured in 1956 that every simplex can be dissected
Schläfli_orthoscheme
Risk measure estimating the average loss in the worst tail of the distribution
}{2}}\right)\right]} , where Li 2 {\displaystyle \operatorname {Li} _{2}} is the dilogarithm and i = − 1 {\displaystyle i={\sqrt {-1}}} is the imaginary unit. If
Expected_shortfall
Method of integration for rational functions
simple terms, which can be integrated analytically through use of the dilogarithm function. Mathematics portal Integration by substitution Trigonometric
Euler_substitution
Measure of polynomial height
Boyd, David (2002b). "Mahler's measure, hyperbolic manifolds and the dilogarithm". Canadian Mathematical Society Notes. 34 (2): 3–4, 26–28. Boyd, David;
Mahler_measure
English electrical engineer (1919–2007)
on Reactive Elements for Broad-Band Impedance Matching (1952, author) Dilogarithms and Associated Functions (1958, author) Explanatory notes on the use
Leonard_Lewin_(engineer)
American mathematician
Chicago. Accessed January 12, 2010 Bloch, S. (1978). "Applications of the dilogarithm function in algebraic K-theory and algebraic geometry". In Nagata, M
Spencer_Bloch
Mathematical function
{Li} _{2}(1-u)+{\frac {\pi ^{2}}{12}}.} with Li2 the dilogarithm. Other ρ n {\displaystyle \rho _{n}} can be calculated using infinite
Dickman_function
Dimensionless parameter in combustion
{Li} _{2}(-q),} where L i 2 {\displaystyle \mathrm {Li_{2}} } is the dilogarithm function. G equation Matalon–Matkowsky–Clavin–Joulin theory Clavin–Garcia
Markstein_number
Mathematical function
Fortran 77 code Fortran 90 version Maximon, Leonard C. (2003). "The dilogarithm function for complex argument". Proc. R. Soc. A. 459 (2039): 2807–2819
Debye_function
Canadian mathematician
Peters 2000, pp. 127–143 Mahler's measure, hyperbolic manifolds and the dilogarithm, Canadian Mathematical Society Notes, vol. 34, no. 2, 2002, 3–4, 26–28
David_William_Boyd
Function in q-analog theory
Bernoulli number, L i 2 ( z ) {\displaystyle \mathrm {Li} _{2}(z)} is the dilogarithm, and p k {\displaystyle p_{k}} is a polynomial of degree k {\displaystyle
Q-gamma_function
Measure giving the average loss beyond a specified Value-at-Risk level
}{2}}\right)\right],} where Li 2 {\displaystyle {\text{Li}}_{2}} is the dilogarithm and i = − 1 {\displaystyle i={\sqrt {-1}}} is the imaginary unit. If
Tail_value_at_risk
Operation on formal power series
expansions. The next series related to the polylogarithm functions (the dilogarithm and trilogarithm functions, respectively), the alternating zeta function
Generating function transformation
Generating_function_transformation
Family of lifetime distributions with decreasing failure rate
x_{0}})}}} where Li 2 {\displaystyle \operatorname {Li} _{2}} is the dilogarithm function Let U be a random variate from the standard uniform distribution
Exponential-logarithmic distribution
Exponential-logarithmic_distribution
DILOGARITHM
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English
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He Work Very Hard Like Krishna
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Tamil
Ratnabhu | ரதà¯à®¨à®¾à®ªà¯
Lord Vishnu
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Hindu, Indian, Sanskrit, Telugu
Kind; One who Gives
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Latin
Small.
DILOGARITHM
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