Search results

{{Short description|Elementary functions and their finitely iterated integrals}} ...cursive definition|recursively defined]] as integrals of other Liouvillian functions. ...

...ni Giambelli]], expresses [[Schubert class]]es as determinants in terms of special Schubert classes. ...presses [[Schur polynomial|Schur functions]] as determinants in terms of [[elementary symmetric function]]s. The corresponding identity also holds for Schubert c ...

...after [[Mario Pieri]], describes the product of a [[Schubert cycle]] by a special Schubert cycle in the [[Schubert calculus]], or the product of a [[Schur po In terms of Schur functions ''s''_λ indexed by [[Partition (number theory)|partitions] ...

...fty}\frac{1}{P_1(\mathbf{x})^{s_1}\cdots P_k(\mathbf{x})^{s_k}}.</math>The special case when ''k''&nbsp;=&nbsp;1 is the [[Barnes zeta function]]. == Relation to Witten zeta functions == ...

...t]], scaling dimensions of operators are fixed numbers, otherwise they are functions of the distance scale. ...the distance as <math>(x^2)^{-\Delta}</math>. More generally, correlation functions of several local operators must depend on the distances in such a way that ...

'''Prabhakar function''' is a certain [[special function]] in mathematics introduced by the Indian mathematician [[Tilak Ra The one-parameter and two-parameter Mittag-Leffler functions are defined first. Then the definition of the three-parameter Mittag-Leffle ...

==Table of congruences characterizing special primes== | <math>2^{p-1} \equiv 1 \pmod{p}</math> || special case of [[Fermat's little theorem]], satisfied by all odd [[prime number]]s ...

...use it makes fast computations of the [[E-function|Siegel {{math|1=''E''}}-functions]] possible, in particular of <math>e^x</math>. ...ns, which are "similar to the exponential function," was given the name "E-functions" by [[Carl Ludwig Siegel]].<ref>C. L. Siegel, ...

...et <math>s\in (0,1)</math>. Then,<ref>NIST Digital Library of Mathematical Functions, 5.6.4.</ref> ...ef> He used this to determine the asymptotic behavior of a ratio of gamma functions. The upper bound in this inequality is stronger than the one given above. ...

{{About|the vector spaces of sequences and functions|the finite-dimensional vector space distance|Chebyshev distance}} ...ons of being localizable and therefore semifinite).<ref>{{Cite web|title = Elementary set theory - Why every localizable measure space is semifinite measure spac ...

...\cdots<i_k}\lambda_{i_1}\cdots\lambda_{i_k}</math>, is a <math>k</math> th elementary symmetric polynomial.<ref>{{Cite journal|last1=Yourdkhany|first1=Mahdieh|la ...ply eigenvalue equations acted upon by the Hessian differential operator. Special cases include the [[Monge–Ampère equation]]<ref>{{citation | last1 = Wang | ...

{{Short description|Notion from the theory of entire functions}} ...te book |last1=Levin |first1=B. Ya. |title=Distribution of Zeros of Entire Functions |date=1964 |publisher=Amer. Math. Soc. |isbn=978-0-8218-4505-9}}</ref> ...

...raction, multiplication, and division of dual numbers are straightforward. Functions of dual numbers are also straightforward ...+ rb \epsilon</math>. That is, one can always consider a real number as a special case of a dual number with a zero imaginary part. ...

...anese [[theoretical physics|theoretical physicist]] who contributed to a [[special relativity|relativistic]] [[Hamiltonian mechanics|Hamiltonian formalism]] i ...v=search Doctoral dissertation on nucleon self-energy and mass spectrum of elementary particles ]</ref> from [[Kyoto University|Kyoto Imperial University]] in 19 ...

{{Short description|Special cases of the Fourier series}} This notion can be generalized to functions which are not even or odd, but then the above formulas will look different. ...

...ref>{{cite book |last1= |first1= |title=Theory and Applications of Special Functions: A Volume Dedicated to Mizan Rahman |date=2005 |publisher=Springer |isbn=97 An elementary proof can also be given.<ref name=":0" /> ...

...ide a powerful formalism for studying an action of the [[ring of symmetric functions|symmetric function]]s on the [[polynomial ring|ring of polynomials]], recov Let ''e''₁, ..., ''e''_''mn'' be the [[elementary symmetric polynomial]]s in the variables ''X''₁, ..., ''X''<sub> ...

...area of the rectangle a,b can't be larger than sum of the areas under the functions <math>f</math> (red) and <math>f^{-1}</math> (yellow)]] ...thor-link = William Henry Young|year = 1912|title = On classes of summable functions and their Fourier series|journal = [[Proceedings of the Royal Society A]]|v ...

==Special values== ...ved by [[Cohomology|cohomological]] tools, and later was proved using more elementary methods.<ref name='Robert'>{{cite journal | last1 = Robert | first1 = Alain ...

...expressed in terms of [[Generalized hypergeometric function|hypergeometric functions]]. ...xpression in terms of [[generalized hypergeometric function|hypergeometric functions]].<ref name=lee/> When <math>\mu</math> is equal to 0 and the scale paramet ...