Holomorphic function
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Though the term
Holomorphic functions are also sometimes referred to as regular functions.[2] A holomorphic function whose domain is the whole complex plane is called an entire function. The phrase "holomorphic at a point z0" means not just differentiable at z0, but differentiable everywhere within some neighbourhood of z0 in the complex plane.
Definition
Given a complex-valued function f of a single complex variable, the derivative of f at a point z0 in its domain is defined as the limit[3]
This is the same definition as for the
A function is holomorphic on an open set U if it is complex differentiable at every point of U. A function f is holomorphic at a point z0 if it is holomorphic on some neighbourhood of z0.[5] A function is holomorphic on some non-open set A if it is holomorphic at every point of A.
A function may be complex differentiable at a point but not holomorphic at this point. For example, the function is complex differentiable at 0, but not complex differentiable elsewhere (see the Cauchy–Riemann equations, below). So, it is not holomorphic at 0.
The relationship between real differentiability and complex differentiability is the following: If a complex function f(x + i y) = u(x, y) + i v(x, y) is holomorphic, then u and v have first partial derivatives with respect to x and y, and satisfy the Cauchy–Riemann equations:[6]
or, equivalently, the
which is to say that, roughly, f is functionally independent from the complex conjugate of z.
If continuity is not given, the converse is not necessarily true. A simple converse is that if u and v have continuous first partial derivatives and satisfy the Cauchy–Riemann equations, then f is holomorphic. A more satisfying converse, which is much harder to prove, is the Looman–Menchoff theorem: if f is continuous, u and v have first partial derivatives (but not necessarily continuous), and they satisfy the Cauchy–Riemann equations, then f is holomorphic.[8]
Terminology
The term holomorphic was introduced in 1875 by
Today, the term "holomorphic function" is sometimes preferred to "analytic function". An important result in complex analysis is that every holomorphic function is complex analytic, a fact that does not follow obviously from the definitions. The term "analytic" is however also in wide use.
Properties
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.
If one identifies C with the real
Every holomorphic function can be separated into its real and imaginary parts f(x + i y) = u(x, y) + i v(x, y), and each of these is a harmonic function on R2 (each satisfies Laplace's equation ∇2 u = ∇2 v = 0), with v the harmonic conjugate of u.[12] Conversely, every harmonic function u(x, y) on a simply connected domain Ω ⊂ R2 is the real part of a holomorphic function: If v is the harmonic conjugate of u, unique up to a constant, then f(x + i y) = u(x, y) + i v(x, y) is holomorphic.
Here γ is a
Cauchy's integral formula states that every function holomorphic inside a disk is completely determined by its values on the disk's boundary.[13] Furthermore: Suppose U ⊂ C is a complex domain, f : U → C is a holomorphic function and the closed disk D = {z : |z − z0| ≤ r} is completely contained in U. Let γ be the circle forming the boundary of D. Then for every a in the interior of D:
where the contour integral is taken counter-clockwise.
The derivative f′(a) can be written as a contour integral
for any simple loop positively winding once around a, and
for infinitesimal positive loops γ around a.
In regions where the first derivative is not zero, holomorphic functions are conformal: they preserve angles and the shape (but not size) of small figures.[14]
Every
From an algebraic point of view, the set of holomorphic functions on an open set is a
From a geometric perspective, a function f is holomorphic at z0 if and only if its exterior derivative df in a neighbourhood U of z0 is equal to f′(z) dz for some continuous function f′. It follows from
that df′ is also proportional to dz, implying that the derivative f′ is itself holomorphic and thus that f is infinitely differentiable. Similarly, d(f dz) = f′ dz ∧ dz = 0 implies that any function f that is holomorphic on the simply connected region U is also integrable on U.
(For a path γ from z0 to z lying entirely in U, define in light of the Jordan curve theorem and the generalized Stokes' theorem, Fγ(z) is independent of the particular choice of path γ, and thus F(z) is a well-defined function on U having F(z0) = F0 and dF = f dz.)
Examples
All
As a consequence of the Cauchy–Riemann equations, any real-valued holomorphic function must be constant. Therefore, the absolute value |z |, the argument arg (z), the real part Re (z) and the imaginary part Im (z) are not holomorphic. Another typical example of a continuous function which is not holomorphic is the complex conjugate (The complex conjugate is antiholomorphic.)
Several variables
The definition of a holomorphic function generalizes to several complex variables in a straightforward way. A function in n complex variables is analytic at a point p if there exists a neighbourhood of p in which f is equal to a convergent power series in n complex variables;
More generally, a function of several complex variables that is
Functions of several complex variables are in some basic ways more complicated than functions of a single complex variable. For example, the region of convergence of a power series is not necessarily an open ball; these regions are logarithmically-convex
A complex differential (p, 0)-form α is holomorphic if and only if its antiholomorphic Dolbeault derivative is zero: ∂α = 0.
Extension to functional analysis
The concept of a holomorphic function can be extended to the infinite-dimensional spaces of functional analysis. For instance, the Fréchet or Gateaux derivative can be used to define a notion of a holomorphic function on a Banach space over the field of complex numbers.
See also
- Antiderivative (complex analysis)
- Antiholomorphic function
- Biholomorphy
- Holomorphic separability
- Meromorphic function
- Quadrature domains
- Harmonic maps
- Harmonic morphisms
- Wirtinger derivatives
References
- ^ Analytic functions of one complex variable, Encyclopedia of Mathematics. (European Mathematical Society ft. Springer, 2015)
- ^ "Analytic function", Encyclopedia of Mathematics, EMS Press, 2001 [1994], retrieved February 26, 2021
- ^ Ahlfors, L., Complex Analysis, 3 ed. (McGraw-Hill, 1979).
- ^ Henrici, P., Applied and Computational Complex Analysis (Wiley). [Three volumes: 1974, 1977, 1986.]
- ^ Peter Ebenfelt, Norbert Hungerbühler, Joseph J. Kohn, Ngaiming Mok, Emil J. Straube (2011) Complex Analysis Springer Science & Business Media
- ^ a b Markushevich, A.I.,Theory of Functions of a Complex Variable (Prentice-Hall, 1965). [Three volumes.]
- ^ Zbl 0141.08601.
- JSTOR 2321164.
- ^ The original French terms were holomorphe and méromorphe.
Bouquet, Jean-Claude (1875). "§15 fonctions holomorphes". Théorie des fonctions elliptiques (2nd ed.). Gauthier-Villars. pp. 14–15.of the denominator; it is a holomorphic function in all that part of the plane which does not contain any poles. ¶ When a function is holomorphic in part of the plane, except at certain poles, we say that it is meromorphic in that part of the plane, that is to say it resembles rational fractions.]
Lorsqu'une fonction est continue, monotrope, et a une dérivée, quand la variable se meut dans une certaine partie du plan, nous dirons qu'elle est holomorphe dans cette partie du plan. Nous indiquons par cette dénomination qu'elle est semblable aux fonctions entières qui jouissent de ces propriétés dans toute l'étendue du plan. [...] ¶ Une fraction rationnelle admet comme pôles les racines du dénominateur; c'est une fonction holomorphe dans toute partie du plan qui ne contient aucun de ses pôles. ¶ Lorsqu'une fonction est holomorphe dans une partie du plan, excepté en certains pôles, nous dirons qu'elle est méromorphe dans cette partie du plan, c'est-à-dire semblable aux fractions rationnelles.
[When a function is continuous, monotropic, and has a derivative, when the variable moves in a certain part of the plane, we say that it is holomorphic in that part of the plane. We mean by this name that it resembles entire functions which enjoy these properties in the full extent of the plane. [...] ¶ A rational fraction admits as poles the roots
Harkness, James; Morley, Frank (1893). "5. Integration". A Treatise on the Theory of Functions. Macmillan. p. 161. - Bouquet, Jean-Claude (1859). "§10". Théorie des fonctions doublement périodiques. Mallet-Bachelier. p. 11.
- Zbl 1107.30300.
- ^ Evans, Lawrence C. (1998), Partial Differential Equations, American Mathematical Society.
- ^ Springer Verlag
- MR 0924157
- ^ Gunning and Rossi, Analytic Functions of Several Complex Variables, p. 2.
Further reading
- Blakey, Joseph (1958). University Mathematics (2nd ed.). London: Blackie and Sons. OCLC 2370110.
External links
- "Analytic function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]