General Identities General Mathematical Identities for Analytic Functions

Series representations

General remarks

There are three main possibilities to represent an arbitrary function as an infinite sum of simple functions. The first is the power series expansion and its two important generalizations, the Laurent series and the Puiseux series. The second is the series and Dirichlet series (general and periodic), and the third is the Fourier series (exponential, trigonometric, and generalized Fourier series by the orthogonal systems). These representations provide very general convenient methods for studying a wide range of functions.

The terms of a power series expansion or its generalizations include power functions in the form or ; the terms in the series include expressions like ; the terms in a Dirichlet series include exponential functions in the form . The Fourier trigonometric series usually provide expansions in terms of and (for trigonometric series) and in terms of (for exponential series). Generalized Fourier series provide expansions in terms of other orthogonal systems of functions, such as the classical orthogonal polynomials. With each of these methods, you can express in closed form an enormous number of non-elementary functions in terms of only simple elementary functions.

Generalized power series

Expansions at z==z0

For the function itself

The Taylor series, first investigated by B. Taylor in 1715, gives the above Taylor expansion for an arbitrary function at a finite point .

The sign ↔ means that the Taylor series should not be taken as purely equal to , since it may not converge everywhere on the complex plane. The series converges absolutely in some disk of radius centered on , where is called the radius of convergence. On the disk , you can exchange the sign ↔ for ⩵. You can state for most functions that The Taylor series coefficients can also be evaluated by the Cauchy integral formula .

There are then three cases for : either is infinite, is zero, or is some finite positive number greater than 0.

If , this series converges only at the point , and the Taylor series offers little analytical benefit. For example, the series has no disk of convergence around , since the coefficient makes the terms grow infinitely large no matter how small becomes.

If , the series converges in the entire plane. In such cases it either represents an entire transcendental function—as, for example, the series does for the exponential function —or it contains only a finite number of terms and therefore represents a polynomial.

If , the sum of the series defines a regular analytic function having at least one singular point on the circle . There may be finitely or infinitely many singular points on the circle, but there must be at least one. The power series , for example, has only one singular point, at . Other power series have dense sets of singular points on the circle, such as the series , which has many singular points on the unit circle, the edge of its natural region of analyticity. The same holds true, with the same region of analyticity, for the series representation of the logarithm of InverseEllipticNomeQ: .

Inside the region of convergence of the series, you can exchange the sign ↔ for ⩵ in the preceding Taylor expansion for an arbitrary function at a finite point :

Infinite Taylor series expansion can be approximated by the truncated version, the finite Taylor polynomial expansions. But in these cases, instead of the equality sign ⩵, you will use the sign ∝ and an ellipsis … or an explicit Landau O term, for example:

The Landau O term in the preceding relation means that the next expression is bounded near point :

For compositions with elementary functions

Including power functions

Including logarithmic and power functions

Expansions at z==0

For the function itself

Laurent series

The Laurent series, first studied by P. Laurent in 1843, gives the Laurent expansion for the function around a finite point , where is the circle with radius . Here is the radius of convergence of its regular part and is the radius of convergence of its principal part . The coefficient of the power in the Laurent expansion of function is called the residue of at the point :

If , then the Laurent series converges absolutely in the ring , where its sum defines an analytic function. If and the principal part includes only a finite number of terms (that is, if for for some ), then has a pole of order at the point .

For example, the function has a pole of order 1 (also called a simple pole) at the point , since it has only one term in the principal part of its Laurent series expansion:

If and the principal part includes an infinite number of terms, this analytic function has an essential singularity at the point . For example, the sum has an essential singularity at the point .

Puiseux series

The Puiseux series, first studied by V.-A. Puiseux in 1850, expand around a finite point as , with usually chosen as . (Other choices for are possible through the use of iterated logarithms of the form , , and so on.) Choosing gives the Puiseux series for algebraic bivariate functions (because such functions should not include logarithms like ).

If this series contains only a finite number of nonzero coefficients with negative indices , the point is an algebraic branch point of order ; otherwise it is a transcendental branch point, such as the point for the function .

Puiseux series are widely used for describing solutions of differential equations near singular points and, in particular, for representations of elementary and special functions near their singular points. The named elementary functions can have rather complicated behaviors near their singular points.

The point is an algebraic branch point of order 1 for the function . This function has the Puiseux representation for its fundamental branch near point .

A similar representation occurs near the branch point .

This expansion near the branch point shows that has a logarithmic-type singularity that can provide infinitely many values depending on the direction of approach of the variable to ∞.

This Puiseux series for near the point includes powers of logarithmic functions.

The Puiseux series for near the point coincides with itself.

This Puiseux series for the ProductLog function near the point has a more complicated structure involving iterated logarithms. The complicated structure is to be expected from inverse functions like ProductLog.

q-series

This q‐series is widely used in applications. In the case , it coincides with exponential Fourier series.

In very particular cases, q‐series can coincide with some trigonometric functions. For example, if , and all other are zero, then .

In more complicated cases, q‐series can define different special functions, for example elliptic theta functions:

Dirichlet series

General Dirichlet series

If all coefficients , this Dirichlet series for the function converges in some open half-plane . In this case, for many Dirichlet series the abscissa of absolute convergence can be found by the formula .

In the more general case , the Dirichlet series absolutely converges in some open convex domain. In both cases, the sum of the Dirichlet series is an analytic function in the domain of convergence.

For example, if and or , you have, respectively, the Dirichlet series:

In the case , you have the ordinary Dirichlet series which in its simplest case () is the Riemann zeta function . Other interesting examples of the Dirichlet series include the series for elliptic theta functions. For instance, if and , you have the representations:

Generalized Fourier series

Under some additional conditions (such as piecewise differentiability), this Fourier series of an arbitrary function by the orthogonal system with Fourier coefficients converges to on an interval at the points of continuity of , and to at the points of discontinuity of , where ).

The inequality takes place for all orthogonal systems ; it is called Bessel's inequality. If it can be transformed into Parseval's equality:

The corresponding system is called the complete system.

The best-known examples of orthogonal systems are the classical orthogonal polynomials and the trigonometric system including and .

Generalized Fourier series through classical orthogonal polynomials

For example, any sufficiently smooth function can be expanded in the Hermite orthogonal system or in other orthogonal systems with corresponding weight factors as a generalized Fourier series, with its sum converging to almost everywhere in corresponding intervals of variable . The following represents this type of Fourier expansion through all classical orthogonal systems of polynomials.

Exponential Fourier series

This exponential Fourier series expansion can be transformed into a trigonometric Fourier series by using the Euler formula .

Following is an example of the exponential Fourier series for the simple function .

Outside the interval , the sum forms a periodic function and the following expansion holds for all real :

Trigonometric Fourier series

This series expansion into a trigonometric system first appeared in an 1807 paper of J. Fourier, who used a similar expansion on the interval . However, L. Euler had discovered similar formulas for Fourier coefficients in 1777.

An arbitrary interval can be transformed into the interval by changing the variable . This allows you to write the following Fourier series expansion of the arbitrary function on an arbitrary interval :

In the preceding formulas the coefficients should vanish at infinity as .

In the internal points where the piecewise differentiable function is continuous, the preceding sum of the Fourier series is equal to and you can change the symbol ↔ to ⩵.

At the points of discontinuity, this Fourier sum is equal to .

Outside the interval , the Fourier sum is a periodic function with period , but the behavior of this sum is not necessarily related to the actual behavior of the function outside the interval of expansion .

For example, let and . Then , and:

Inspection shows that the Fourier sum has period 2 and coincides with on the interval . But at the endpoint the Fourier sum is equal to 0, since you are evaluating with . This result coincides with .

Asymptotic series expansions

Expansions at z==ComplexInfinity

This asymptotic series expansion of the function at includes the main term and other terms of the form . The corresponding formal asymptotic series is, by definition, formed in such a way that the following inequality holds for all and sufficiently large : . This asymptotic series can be a divergent or convergent series. If this series converges, it coincides with the Taylor power series expansion at infinity.

For example, the confluent hypergeometric function has the following asymptotic series expansion through the divergent series :

The relation takes place because the following inequality holds for each and sufficiently large :

The function has a rather interesting asymptotic expansion at through the following divergent series:

The next example includes a convergent series in an asymptotic expansion. So, you can use the sign ⩵ instead of ∝:

In the particular case , the first generic formula for asymptotic expansion can be rewritten in the following form:

Expansions at z==z0

This asymptotic series expansion of the function at includes the main term and other terms . The corresponding formal asymptotic series by definition is formed in such a way that the following inequality holds for all and sufficiently small : . This asymptotic series can be a divergent or convergent series. If this series converges, it coincides with the Taylor power series expansion at the point .

For example, the functions and have the following asymptotic expansions near the points and through convergent series. So, you can use the sign ⩵ instead of ∝ here:

In the particular case , the first generic formula for asymptotic expansion can be rewritten in the following form:

Residue representations

For many analytic functions , you can establish so-called residue representations through infinite sums of residues of another analytic function in the points . The residue of the analytic function in the pole of order can be calculated by the formula:

If the function can be represented as a quotient , where the functions and are analytic at the point and is the simple root of the equation , then the following formula holds:

Very often, residue representations appear from the theory of the Meijer G function. The majority of functions of the hypergeometric type can be equivalently defined as corresponding to infinite sums of residues from products of ratios of gamma functions on power function .

For example, the following residue representation formula for a logarithm function takes place: