Examples of "holomorphic"
In mathematics, almost holomorphic modular forms, also called nearly holomorphic modular forms, are a generalization of modular forms that are polynomials in 1/Im(τ) with coefficients that are holomorphic functions of τ. A quasimodular form is the holomorphic part of an almost holomorphic modular form. An almost holomorphic modular form is determined by its holomorphic part, so the operation of taking the holomorphic part gives an isomorphism between the spaces of almost holomorphic modular forms and quasimodular forms. The archetypal examples of quasimodular forms are the Eisenstein series E(τ) (the holomorphic part of the almost holomorphic modular form E(τ) – 3/πIm(τ)), and derivatives of modular forms.
In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A holomorphic line bundle is a rank one holomorphic vector bundle.
with holomorphic (complex linear) and anti-holomorphic (conjugate linear). For , since is holomorphic, is anti-holomorphic. Direct examination of the explicit expressions for and in equation below shows that they are holomorphic and anti-holomorphic respectively. Closer examination of the expression also allows for identification of and for as and .
of formula_4 acting by holomorphic contractions. Here, a "holomorphic contraction"
are holomorphic maps. The holomorphic structure on the tangent bundle of a complex manifold is guaranteed by the remark that the derivative (in the appropriate sense) of a vector-valued holomorphic function is itself holomorphic.
Because complex differentiation is linear and obeys the product, quotient, and chain rules; the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is not zero.
Such a `local' quadratic differential is holomorphic if and only if formula_4 is holomorphic.
Since "f" is the derivative of the holomorphic function "F", it is holomorphic. The fact that derivatives of holomorphic functions are holomorphic can be proved by using the fact that holomorphic functions are analytic, i.e. can be represented by a convergent power series, and the fact that power series may be differentiated term by term. This completes the proof.
If "f" is "complex differentiable" at "every" point "z" in an open set "U", we say that "f" is holomorphic on "U". We say that "f" is holomorphic at the point "z" if it is holomorphic on some neighborhood of "z". We say that "f" is holomorphic on some non-open set "A" if it is holomorphic in an open set containing "A".
is said to be holomorphic. The vector space of holomorphic quadratic differentials on a Riemann surface
where the formula_15 are holomorphic functions. Equivalently, the ("p",0)-form α is holomorphic if and only if
Holomorphic Fock space is the Hilbert space formula_119 of holomorphic functions "f"("z") on C with finite norm
But "g" is holomorphic, hence the composition φ("g"): "G" ⊂ C → C is holomorphic and therefore by Cauchy's theorem
and is therefore holomorphic wherever the logarithm log("z") is. The function 1/"z" is holomorphic on
The first summand is called the holomorphic part, and the second summand is called the non-holomorphic part of .
In general holomorphic functions along a subvariety "V" of "W" are defined by gluing together holomorphic functions on affine subvarieties.
The weight gives rise to a character (one-dimensional representation) of the Borel subgroup , which is denoted . Holomorphic sections of the holomorphic line bundle over may be described more concretely as holomorphic maps
Let be a holomorphic vector bundle. A "local section" is said to be holomorphic if, in a neighborhood of each point of , it is holomorphic in some (equivalently any) trivialization.
For each "p", a holomorphic "p"-form is a holomorphic section of the bundle Ω. In local coordinates, then, a holomorphic "p"-form can be written in the form
In mathematics, a canonical connection of a holomorphic vector bundle with a Hermitian structure, is the unique metric connection D, such that the part which increases the anti-holomorphic type D`` annihilates holomorphic sections.