The function
T( z; \zeta ) =
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''zeta-function regularization''
...essential in supersymmetry calculations. Among the different methods, zeta-function regularization — which is obtained by
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...on $\psi(x)$ can be expressed in terms of the [[Mangoldt function|Mangoldt function]]
...p \le x$, and that the quantity $e^{\psi(x)}$ is equal to the least common multiple of all positive integers $n \le x$. The functions $\theta(x)$ and $\psi(x)$
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An integral of an [[algebraic function]] of the first kind, that is, an integral of the form
is a [[Rational_function | rational function]] of the variables $ z $
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$#C+1 = 139 : ~/encyclopedia/old_files/data/C023/C.0203720 Complete analytic function
...f. [[Analytic continuation|Analytic continuation]]) of an initial analytic function $ f = f( z) $
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is a real-valued function of $ x $.
is a real-valued function and $ \Phi $
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...able is the theory of analytic functions (cf. [[Analytic function|Analytic function]]) of one or several complex variables.
According to Weierstrass, a function $ w = f ( z) $
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...roots $c_1,\dots,c_n$ of $f(x)$ are equal, their common value is called a multiple root (if a root occurs $m$ times, $m$ is called the multiplicity of that ro
.... The number of such roots in $U_n$ is given by the [[Euler function|Euler function]] $\phi(n)$, i.e. the number of residues $\bmod\,n$ which are relatively pr
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In other words, if a holomorphic function $ f ( z) $
points of an analytic function $ f ( z) $,
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...} ( v )$ (cf. also [[Theta-function|Theta-function]]), and the Jacobi zeta-function (see [[#References|[a1]]], pp. 571–598, and [[#References|[a4]]], Chap. 6
...t, [[#References|[a3]]]. Here, $\Gamma$ denotes the [[Gamma-function|Gamma-function]]. The lemniscate constant
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is the distribution function with given density $ f $.
==Monte-Carlo algorithms for estimating multiple integrals.==
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[[Elliptic function|Elliptic function]]).
only if the polynomial $x^3+ax+b$ does not have multiple zeros, that is, if
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which correspond to a multiple $ nK $
...y is equivalent to the Riemann hypothesis concerning the zeros of the $ \zeta $ -
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The discriminant vanishes if and only if the polynomial has multiple roots. The discriminant is symmetric with respect to the roots of the polyn
$$\lim_{q\to 1+0} (q-1) \def\z{ {\zeta}}\z_k(q) = \frac{2^{s+t}\pi^t R}{m\sqrt{|D_K|}}h,$$
16 KB (2,947 words) - 08:53, 9 December 2016
A singular point of an analytic function $ f(z) $
...o the [[Analytic continuation|analytic continuation]] of an element of the function $ f(z) $
66 KB (9,825 words) - 01:45, 23 June 2022
...This entry concerns the latter: the reader is referred to [[Real analytic function]] for the first class.
...the very notion of a function, became of fundamental significance after a function had come to be regarded, in the first half of the 19th century, as an arbit
61 KB (9,850 words) - 19:04, 20 January 2022
...t all prime factors at the left-hand side occur with an exponent that is a multiple of $ p $ . This was Kummer's first point of view, but Dirichlet pointed o
...$ is the order of the absolute class group and $ \phi $ is Euler's phi-function. For different conductors $ \mathfrak m _{1} $ and $ \mathfrak m _{2}
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...r writing formulas for transformation of variables and is readily used for multiple and line integrals. Newton's notation does not directly offer such possibil
...variable operation, the function symbol $f x$ (from the Latin functio $=$ function; 1734). Somewhat earlier, the symbol $\phi x$ had been used by J. Bernoulli
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is the [[Euler function|Euler function]], which is equal to the number of elements in the set $ 1 \dots m $
is a multiple of $ d $,
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...ependent variable. Instead of one regressor variable there may be several (multiple regression).
...n can be expressed in the form $\psi = \sum _ { i = 1 } ^ { r } d _ { i } \zeta _ { i }$, with constants $d_{i}$, and the least-squares estimator of $\psi$
56 KB (8,477 words) - 17:45, 1 July 2020