Holomorphic function

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Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane \mathbb{C} with values in \mathbb{C} that are complex-differentiable at every point. This is a much stronger condition than real differentiability and implies that the function is infinitely often differentiable and can be described by its Taylor series.

The term analytic function is often used interchangeably with holomorphic function, although the term analytic is also used in a broader sense of any function (real, complex, or of more general type) that is equal to its Taylor series in a neighborhood of each point in its domain. The fact that the class of analytic functions coincides with the class of holomorphic functions is a major theorem in complex analysis.

Holomorphic functions are sometimes called regular functions.[1] A function that is holomorphic on the whole complex plane is called an entire function. The phrase "holomorphic at a point z" means not just differentiable at z, but differentiable everywhere within some open disk centered at z in the complex plane.

If the inverse function is also holomorphic, such function is called biholomorphism. The domain of biholomorphism should be specified, because in the whole complex plane, only linear functions are biholomorphic.


[edit] Definition

If U is an open subset of \mathbb{C} and f : U\mathbb{C} is a complex function on U, we say that f is complex differentiable at a point z0 of U if the limit

f'(z_0) = \lim_{z \rightarrow z_0} {f(z) - f(z_0) \over z - z_0 }


The limit here is taken over all sequences of complex numbers approaching z0, and for all such sequences the difference quotient has to approach the same number f '(z0). Intuitively, if f is complex differentiable at z0 and we approach the point z0 from the direction r, then the images will approach the point f(z0) from the direction f\,'(z_0) r, where the last product is the multiplication of complex numbers. This concept of differentiability shares several properties with real differentiability: it is linear and obeys the product, quotient and chain rules.

If f is complex differentiable at every point z0 in U, we say that f is holomorphic on U. We say that f is holomorphic at the point z0 if it is holomorphic on some neighborhood of z0. We say that f is holomorphic on some non-open set A if it is holomorphic in an open set containing A.

The relationship between real differentiability and complex differentiability is the following. If a complex function f(x + iy) = u(x, y) + iv(x, y) is holomorphic, then u and v have first partial derivatives with respect to x and y, and satisfy the Cauchy-Riemann equations:

\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \qquad \mbox{and} \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}

If no continuity is 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, and they satisfy the Cauchy-Riemann equations, then f is holomorphic.

[edit] Terminology

The word "holomorphic" was introduced by two of Cauchy's students, Briot (1817–1882) and Bouquet (1819–1895), and derives from the Greek őλoς (holos) meaning "entire", and μoρφń (morphe) meaning "form" or "appearance".[2]

Today, many mathematicians prefer the term "holomorphic function" to "analytic function", as the latter is a more general concept. This is also because an important result in complex analysis states that every holomorphic function is complex analytic, a fact that does not follow directly from the definitions. The term "analytic" is however also in wide use.

[edit] 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 R2, then the holomorphic functions coincide with those functions of two real variables with continuous first derivatives which satisfy the Cauchy-Riemann equations (see above).

Every holomorphic function can be separated into its real and imaginary parts, and each of these is a solution of Laplace's equation on R2. In other words, if we express a holomorphic function f(z) as u(xy) + i v(xy) both u and v are harmonic functions.

In regions where the first derivative is not zero, holomorphic functions are conformal in the sense that they preserve angles and the shape (but not size) of small figures.

Cauchy's integral formula states that every function holomorphic inside a disk is completely determined by its values on the disk's boundary.

Every holomorphic function is analytic. That is, a holomorphic function f has derivatives of every order at each point a in its domain, and it coincides with its own Taylor series at a in a neighborhood of a. In fact, f coincides with its Taylor series at a in any disk centered at that point and lying within the domain of the function.

From an algebraic point of view, the set of holomorphic functions on an open set is a commutative ring and a complex vector space. In fact, it is a locally convex topological vector space, with the seminorms being the suprema on compact subsets.

[edit] Examples

All polynomial functions in z with complex coefficients are holomorphic on C, and so are sine, cosine and the exponential function. (The trigonometric functions are in fact closely related to and can be defined via the exponential function using Euler's formula). The principal branch of the complex logarithm function is holomorphic on the set C \ {zR : z ≤ 0}. The square root function can be defined as

\sqrt{z} = e^{\frac{1}{2}\log z}

and is therefore holomorphic wherever the logarithm log(z) is. The function 1/z is holomorphic on {z : z ≠ 0}.

Typical examples of continuous functions which are not holomorphic are complex conjugation and taking the real part.

As a consequence of Cauchy-Riemann equations, a real-valued holomorphic function must be constant. Therefore, the absolute value of z and the argument of z are not holomorphic.

[edit] References

  1. Springer Online Reference Books, Wolfram MathWorld
  2. Markushevich, A.I.; Silverman, Richard A. (ed.) [1977] (2005). Theory of functions of a Complex Variable, 2nd ed.. New York: American Mathematical Society. ISBN 0-8218-3780-X. 
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