In mathematical analysis, an analytic function is a function that is locally represented by a convergent power series. More precisely, a real or complex function is analytic at a point if, in some neighborhood of that point, it is equal to a power series centered there. The analytic function is therefore locally determined by coefficients of the series, or equivalently by derivatives of the function evaluated at the center of the series expansion. In other words, an analytic function is a function that is locally represented by a convergent Taylor series.
Analytic functions occur in both real analysis and complex analysis, in slightly different ways. A real or complex analytic function is necessarily smooth, having derivatives of all orders. But a smooth real function need not be analytic. By contrast, a complex function on an open set is analytic if and only if it is holomorphic, that is, complex differentiable at every point of the set. For this reason, in complex analysis the terms analytic function and holomorphic function are often used interchangeably. The terms complex analytic and real analytic distinguish between these cases. In signal processing, a complex analytic function is sometimes called an analytic signal.
Analyticity is a strong regularity condition. Analytic functions have rigid local behavior: for example, on a connected domain, an analytic function whose zeros have an accumulation point must vanish identically. Standard examples include polynomials, the exponential function, and the trigonometric functions on their domains of analyticity.
Definitions
Formally, a function is real analytic on an open set in the real line if for every one can write
in which the coefficients , , ...are real numbers and this series (the right-hand side of this equation) is convergent to for in a neighborhood of (that is a set containing an open set including ).
Alternatively, a real analytic function is an infinitely differentiable function such that the Taylor series at each point in its domain
converges to for in a neighborhood of pointwise. The set of all real analytic functions on a given set is often denoted by , or just by if the domain is understood.
A function defined on some subset of the real line is said to be real analytic at a point if there is a neighborhood of on which a power series centered at converges to .
The definition of a complex analytic function is obtained by replacing, in the definitions above, "real" with "complex" and "real line" with "complex plane". A function is complex analytic if and only if it is holomorphic i.e. it is complex differentiable. For this reason the terms "holomorphic" and "analytic" are often used interchangeably for such functions.
In complex analysis, a function is called analytic in an open set if it is (complex) differentiable at each point in .
Examples Typical examples of analytic functions are The following elementary functions: All polynomials: if a polynomial has degree , any terms of degree larger than in its Taylor series expansion must immediately vanish to 0, and so this series will be trivially convergent. Furthermore, every polynomial is its own Maclaurin series. The exponential function is analytic. Any Taylor series for this function converges not only for close enough to (as in the definition) but for all values of (real or complex). The trigonometric functions are analytic on any open set of their domain. The natural logarithm is analytic on any open set where its branch is single-valued, such as or the complement in the Riemann sphere of any simple arc connecting to . Power functions are analytic everywhere for non-negative integral powers, away from zero for negative integral powers, and for arbitrary complex powers on any open subset of the complex plane where the logarithm is analytic. Many special functions are analytic on a suitable domain: hypergeometric functions on suitable domains Bessel functions on suitable domains The gamma function away from its poles at zero and the negative integers The Riemann zeta function except for a simple pole at Algebraic functions are analytic away from any poles and branch points they may have. Near a branch point, an algebraic function can be represented by a convergent Puiseux series; equivalently, after a change of variable , it becomes analytic as a function of .
Typical examples of functions that are not analytic are The absolute value function on the real numbers is not analytic at . The corresponding function on the complex numbers is not complex analytic on any nonempty open subset of . Piecewise defined functions (functions given by different formulae in different regions) are typically not analytic where the pieces meet. The complex conjugate function is not complex analytic, although its restriction to the real line is the identity function and therefore real analytic, and it is real analytic as a function from to . Other non-analytic smooth functions, and in particular any smooth function with compact support that is not identically zero, i.e. , cannot be analytic on all of .
Alternative characterizations
The following conditions are equivalent: is real analytic on an open set . There is a complex analytic extension of to an open set that contains . is smooth and for every compact set there exists a constant such that for every and every non-negative integer the following bound holds;
Complex analytic functions are exactly equivalent to holomorphic functions, and are thus much more easily characterized.
For the case of an analytic function with several variables (see below), the real analyticity can be characterized using the Fourier–Bros–Iagolnitzer transform.
In the multivariable case, real analytic functions satisfy a direct generalization of the third characterization. Let be an open set, and let . Then is real analytic on if and only if and for every compact there exists a constant such that for every multi-index the following bound holds
Properties of analytic functions The sums, products, and compositions of analytic functions are analytic. The reciprocal of an analytic function that is nowhere zero is analytic. In one variable, the inverse of a one-to-one analytic function whose derivative is nowhere zero is analytic. In several variables, the corresponding condition is that the derivative be an invertible linear map, or equivalently that the Jacobian determinant be nonzero. Any analytic function is smooth, that is, infinitely differentiable. The converse is not true for real functions; in fact, in a certain sense, the real analytic functions are sparse compared to all real infinitely differentiable functions. For the complex numbers, the converse does hold: any function with a single complex derivative on an open set is analytic on that set (see ). For any open set , the set of all analytic functions is a Fréchet space with respect to the uniform convergence on compact sets. The fact that uniform limits on compact sets of analytic functions are analytic is an easy consequence of Morera's theorem. The set of all bounded analytic functions with the supremum norm is a Banach space. In contrast, the real analytic functions on an open set are not complete under the topology of uniform convergence on compact subsets, since compact-open limits of real analytic functions need not be real analytic. Real analytic functions are dense in the space of continuous functions in the compact-open topology.
A polynomial cannot be zero at too many points unless it is the zero polynomial (more precisely, the number of zeros is at most the degree of the polynomial). A similar but weaker statement holds for analytic functions. If the set of zeros of an analytic function has an accumulation point inside its domain, then is zero everywhere on the connected component containing the accumulation point. In other words, if is a sequence of distinct numbers such that for all and this sequence converges to a point in the domain of , then is identically zero on the connected component of containing . This is known as the identity theorem.
Also, if all the derivatives of an analytic function at a point are zero, the function is constant on the corresponding connected component.
These statements imply that while analytic functions do have more degrees of freedom than polynomials, they are still quite rigid.
Analyticity and differentiability As noted above, any function (real or complex) is infinitely differentiable on a neighborhood where it is equal to a convergent power series. In a neighborhood of a point of analyticity, there is a convergent power series equal to the function in the neighborhood, so the function is infinitely differentiable there. There exist smooth real functions that are not analytic: see Non-analytic smooth function. In fact there are many such functions.
The situation is quite different when one considers complex analytic functions and complex derivatives. Any complex-differentiable function in an open disc centered at (holomorphic function) is analytic there, and conversely any function given by a convergent power series in the complex variable is complex differentiable (and infinitely differentiable) on the disk of convergence. Consequently, in complex analysis, the term analytic function is synonymous with holomorphic function.
Real versus complex analytic functions Real and complex analytic functions have important differences (one could notice that even from their different relationship with differentiability). Analyticity of complex functions is a more restrictive property, as it has more restrictive necessary conditions and complex analytic functions have more structure than their real-line counterparts.
According to Liouville's theorem, any bounded complex analytic function defined on the whole complex plane is constant. The corresponding statement for real analytic functions, with the complex plane replaced by the real line, is clearly false; this is illustrated by
Also, if a complex analytic function is defined in an open ball around a point , its power series expansion at is convergent in the whole open ball (holomorphic functions are analytic). This statement for real analytic functions (with open ball meaning an open interval of the real line rather than an open disk of the complex plane) is not true in general; the function of the example above gives an example for and a ball of radius exceeding , since the power series diverges for .
Any real analytic function on some open set on the real line can be extended to a complex analytic function on some open set of the complex plane. However, not every real analytic function defined on the whole real line can be extended to a complex function defined on the whole complex plane. The function defined in the paragraph above is a counterexample, as it is not defined for . This explains why the Taylor series of diverges for , i.e., the radius of convergence is because the complexified function has a pole at distance from the evaluation point and no further poles within the open disc of radius around the evaluation point.
Taylor series and radius of convergence
If a function is analytic at , then its Taylor series converges to the function in some open neighborhood of . More generally, for any power series
there is a number called the radius of convergence, which can be any non-negative number or , such that the power series converges absolutely for and diverges for . Thus, when a Taylor series converges, it does so in an open interval centered at in the real case, or a disc centered at in the complex case. A Taylor series may converge absolutely or conditionally at some, all, or none of the boundary points of the open interval or disc.