Examples of "antiholomorphic"
It is obtained by taking antiholomorphic derivatives in local coordinates.
The tricorn formula_3 is defined by a family of quadratic antiholomorphic polynomials
The kernel "K"("z",ζ) holomorphic in "z" and antiholomorphic in ζ, and satisfies
If a function is both holomorphic and antiholomorphic, then it is constant on any connected component of its domain.
The higher degree analogues of the tricorn are known as the multicorns. These are the connectedness loci of the family of antiholomorphic polynomials formula_14.
Rearranging this absolutely convergent series shows that "f" is the boundary value of "g" + "h", where "g" (resp. "h") is a holomorphic (resp. antiholomorphic) function on "D".
One can show that if "f"("z") is a holomorphic function on an open set "D", then "f"() is an antiholomorphic function on , where is the reflection against the "x"-axis of "D", or in other words, is the set of complex conjugates of elements of "D". Moreover, any antiholomorphic function can be obtained in this manner from a holomorphic function. This implies that a function is antiholomorphic if and only if it can be expanded in a power series in in a neighborhood of each point in its domain.
T)formula_20C to the complexified tangent bundle one gets the subspace of antiholomorphic vector fields. Therefore, this generalized complex structure on (Tformula_1
In mathematics, antiholomorphic functions (also called antianalytic functions) are a family of functions closely related to but distinct from holomorphic functions.
The space of complex differential forms ΛTformula_20C has a complex conjugation operation given by complex conjugation in C. This allows one to define holomorphic and antiholomorphic one-forms and ("m, n")-forms, which are homogeneous polynomials in these one-forms with "m" holomorphic factors and "n" antiholomorphic factors. In particular, all ("n,0")-forms are related locally by multiplication by a complex function and so they form a complex line bundle.
of the critical point formula_9 of the antiholomorphic polynomial formula_10. In analogy with the Mandelbrot set, the tricorn is defined as the set of all parameters formula_6 for which the forward orbit of the critical point is bounded. This is equivalent to saying that the tricorn is the connectedness locus of the family of quadratic antiholomorphic polynomials; i.e. the set of all parameters formula_6 for which the Julia set formula_13 is connected.
Abstractly, B("s") is isomorphic to the smooth vector bundle underlying the antiholomorphic vector bundle formula_7 of the Serre twist on the complex projective line CP. A section of the latter bundle is a function "g" on C\{0} satisfying
A concrete example of the formula_47 operator can be provided on the Heisenberg group. Consider the general Heisenberg group formula_60 and consider the antiholomorphic vector fields which are also group left invariant,
The symmetry algebra is therefore the product of two copies of the Virasoro algebra: the left-moving or holomorphic algebra, with generators formula_11, and the right-moving or antiholomorphic algebra, with generators formula_12.
In mathematics, the Bochner–Kodaira–Nakano identity is an analogue of the Weitzenböck identity for hermitian manifolds, giving an expression for the antiholomorphic Laplacian of a vector bundle over a hermitian manifold in terms of its complex conjugate and the curvature of the bundle and the torsion of the metric of the manifold. It is named after Salomon Bochner, Kunihiko Kodaira, and Hidegorô Nakano.
A function of the complex variable z defined on an open set in the complex plane is said to be antiholomorphic if its derivative with respect to exists in the neighbourhood of each and every point in that set, where is the complex conjugate.
In Euclidean CFT, one has both a holomorphic and an antiholomorphic copy of the Virasoro algebra. In Lorentzian CFT, one has a left-moving and a right moving copy of the Virasoro algebra (spacetime is a cylinder, with space being a circle, and time a line).
("n,0")-forms are pure spinors, as they are annihilated by antiholomorphic tangent vectors and by holomorphic one-forms. Thus this line bundle can be used as a canonical bundle to define a generalized complex structure. Restricting the annihilator from (Tformula_1
If harmonic Maass forms are interpreted as harmonic sections of the line bundle of modular forms of weight equipped with the Petersson metric over the modular curve, then this differential operator can be viewed as a composition of the Hodge star operator and the antiholomorphic differential.
so that formula_6 is a map which increases the holomorphic part of the type by one (takes forms of type ("p", "q") to forms of type ("p"+1, "q")), and formula_7 is a map which increases the antiholomorphic part of the type by one. These operators are called the Dolbeault operators.