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Multiplicity of holomorphic functions
Mathematische Annalen, 2000Let \(f:P^2\to P^2\) be a rational map of maximal rank and of degree \(d\). Denote by \(I(f)\) the (finite) set of points of indeterminancy of \(f\). The map \(f\) is said to be algebraically stable if \(f^{-n}(I(f))\) is finite for any \(n\geq 0\). Under this condition, it is possible to define an invariant positive closed (1,1)-current \(T(f)=\lim_{n
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Zeros of Holomorphic Functions
2007There are certain (classical families of) functions of a complex variable that mathematicians have studied frequently enough for them to acquire their own names. These are, of course, functions that arise naturally and repeatedly in various mathematical settings. Many of these functions are defined by infinite products. Examples of such named functions
Irwin Kra +2 more
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Functions Holomorphic in a Region [PDF]
, 1992 a) The function \(g(z)\mathop = \limits^{{\text{def}}} f(z) - f(z) - f'(a)z\) is such that g′(a) = 0 whence g is not injective near a, i.e., near a there are two points b, c such that g(b) = g(c), whence the conclusion.
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Holomorphic functions unbounded on curves of finite length
, 2014Given a pseudoconvex domain , we prove that there is a holomorphic function f on D such that the lengths of paths $$p:\ [0,1] \rightarrow D$$p:[0,1]→D along which $$\mathfrak {R}f$$Rf is bounded above, with p(0) fixed, grow arbitrarily fast as $$p(1 ...
J. Globevnik
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From the holomorphic Wilson loop to ‘d log’ loop-integrands of super-Yang-Mills amplitudes
, 2012A bstractThe S-matrix for planar $ \mathcal{N} $ = 4 super Yang-Mills theory can be computed as the correlation function for a holomorphic polygonal Wilson loop in twistor space. In an axial gauge, this leads to the construction of the all-loop integrand
A. Lipstein, L. Mason
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Functions Holomorphic in a Disc
1992For f in H(U) and t in [0, 2π), $$ f*(t)\mathop = \limits^{def} \left\{ {_0^{\lim _{r \uparrow 1} f\left( {re^{it} } \right)} } \right.\,{\text{when}}\,{\text{the limit}}\,{\text{exists}}\,{\text{otherwise}} $$ is the radial limit function. The (Hardy class) $$ \left\{ {f:f \in H(U),_{z \in U}^{\sup \left| {f(z)} \right| < \infty } } \right\}
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1984
The author continues and extends his previous investigations [ibid. 35(1983)]. He proves e.g. that if \(\phi\) is entire function in \({\mathbb{C}}\), D is a domain in \({\mathbb{C}}^ n\), and \(f\in H(D)\), then \({\tilde \phi}=\phi \circ f\) is GM-holomorphic and L(\({\tilde \phi}\))\(=\phi '\circ f\).
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The author continues and extends his previous investigations [ibid. 35(1983)]. He proves e.g. that if \(\phi\) is entire function in \({\mathbb{C}}\), D is a domain in \({\mathbb{C}}^ n\), and \(f\in H(D)\), then \({\tilde \phi}=\phi \circ f\) is GM-holomorphic and L(\({\tilde \phi}\))\(=\phi '\circ f\).
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WEAKLY HOLOMORPHIC FUNCTIONS ON COMPLETE INTERSECTIONS, AND THEIR HOLOMORPHIC EXTENSION
Mathematics of the USSR-Sbornik, 1988See the review in Zbl 0631.32009.
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The logarithm of the modulus of a holomorphic function as a minorant for a subharmonic function
, 2016B. Khabibullin, T. Baiguskarov
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Normality Criteria of Meromorphic Functions Sharing a Holomorphic Function
, 2015D. Meng, P. Hu
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