Holomorphic function in the context of Hardy space

In mathematics, a Hilbert space is a real or complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space, to infinite dimensions. The inner product, which is the analog of the dot product from vector calculus, allows lengths and angles to be defined. Furthermore, completeness means that there are enough limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space.

Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions.

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value ${\displaystyle \infty }$ for infinity. With the Riemann model, the point ${\displaystyle \infty }$ is near to very large numbers, just as the point ${\displaystyle 0}$ is near to very small numbers.

The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as ${\displaystyle 1/0=\infty }$ well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere.

In mathematics, a modular form is a holomorphic function on the complex upper half-plane, ${\displaystyle {\mathcal {H}}}$, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modular forms has origins in complex analysis, with important connections with number theory. Modular forms also appear in other areas, such as algebraic topology, sphere packing, and string theory.

Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group ${\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )\subset \mathrm {SL} _{2}(\mathbb {R} )}$. Every modular form is attached to a Galois representation.

The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space ${\displaystyle \mathbb {C} ^{n}}$, that is, n-tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables (and analytic space), which the Mathematics Subject Classification has as a top-level heading.

As in complex analysis of functions of one variable, which is the case n = 1, the functions studied are holomorphic or complex analytic so that, locally, they are power series in the variables z_i. Equivalently, they are locally uniform limits of polynomials; or locally square-integrable solutions to the n-dimensional Cauchy–Riemann equations. For one complex variable, every domain(${\displaystyle D\subset \mathbb {C} }$), is the domain of holomorphy of some function, in other words every domain has a function for which it is the domain of holomorphy. For several complex variables, this is not the case; there exist domains (${\displaystyle D\subset \mathbb {C} ^{n},\ n\geq 2}$) that are not the domain of holomorphy of any function, and so is not always the domain of holomorphy, so the domain of holomorphy is one of the themes in this field. Patching the local data of meromorphic functions, i.e. the problem of creating a global meromorphic function from zeros and poles, is called the Cousin problem. Also, the interesting phenomena that occur in several complex variables are fundamentally important to the study of compact complex manifolds and complex projective varieties (${\displaystyle \mathbb {CP} ^{n}}$) and has a different flavour to complex analytic geometry in ${\displaystyle \mathbb {C} ^{n}}$ or on Stein manifolds, these are much similar to study of algebraic varieties that is study of the algebraic geometry than complex analytic geometry.

In complex analysis, a complex-valued function ${\displaystyle f}$ of a complex variable ${\displaystyle z}$:

• is said to be holomorphic at a point ${\displaystyle a}$ if it is differentiable at every point within some open disk centered at ${\displaystyle a}$, and
• is said to be analytic at ${\displaystyle a}$ if in some open disk centered at ${\displaystyle a}$ it can be expanded as a convergent power series $${\displaystyle f(z)=\sum _{n=0}^{\infty }c_{n}(z-a)^{n}}$$ (this implies that the radius of convergence is positive).

One of the most important theorems of complex analysis is that holomorphic functions are analytic and vice versa. Among the corollaries of this theorem are

Every meromorphic function on ${\displaystyle D}$ can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on ${\displaystyle D}$: any pole must coincide with a zero of the denominator.

In mathematics, specifically in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. It can be understood as the subset of phase space covered by the trajectory of the dynamical system under a particular set of initial conditions, as the system evolves. As a phase space trajectory is uniquely determined for any given set of phase space coordinates, it is not possible for different orbits to intersect in phase space, therefore the set of all orbits of a dynamical system is a partition of the phase space. Understanding the properties of orbits by using topological methods is one of the objectives of the modern theory of dynamical systems.

For discrete-time dynamical systems, the orbits are sequences; for real dynamical systems, the orbits are curves; and for holomorphic dynamical systems, the orbits are Riemann surfaces.

In mathematics, a critical point is the argument of a function where the function derivative is zero (or undefined, as specified below).The value of the function at a critical point is a critical value.

More specifically, when dealing with functions of a real variable, a critical point is a point in the domain of the function where the function derivative is equal to zero (also known as a stationary point) or where the function is not differentiable. Similarly, when dealing with complex variables, a critical point is a point in the function's domain where its derivative is equal to zero (or the function is not holomorphic). Likewise, for a function of several real variables, a critical point is a value in its domain where the gradient norm is equal to zero (or undefined).

In mathematics, the gamma function (represented by ${\displaystyle \Gamma }$, capital Greek letter gamma) is the most common extension of the factorial function to complex numbers. First studied by Daniel Bernoulli, the gamma function ${\displaystyle \Gamma (z)}$ is defined for all complex numbers ${\displaystyle z}$ except non-positive integers, and ${\displaystyle \Gamma (n)=(n-1)!}$ for every positive integer ${\displaystyle n}$. The gamma function can be defined via a convergent improper integral for complex numbers with positive real part:

$${\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\,dt,\ \qquad \Re (z)>0.}$$The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic except at zero and the negative integers, where it has simple poles.

In mathematics, the error function (also called the Gauss error function), often denoted by erf, is a function ${\displaystyle \mathrm {erf} :\mathbb {C} \to \mathbb {C} }$ defined as:$${\displaystyle \operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\int _{0}^{z}e^{-t^{2}}\,dt.}$$

The integral here is a complex contour integral which is path-independent because ${\displaystyle \exp(-t^{2})}$ is holomorphic on the whole complex plane ${\displaystyle \mathbb {C} }$. In many applications, the function argument is a real number, in which case the function value is also real.