General Identities and Formulas

STILL UNDER CONSTRUCTION

Notations

Traditional Notation

Mathematica StandardForm Notation

Primary Definition

The arbitrary function f(z) in this chapter is defined for all z. It asume to be an analytical function.

General Characteristics

Domain and Analyticity

f(z) is an analytical function of z which is defined in the whole complex z plane.

Symmetries and Periodicities

Evenness:

Oddness:

Periodicity (if f(z) is periodical with period p):

Singularities

No

Series Representations

There are three major techniques for representing an arbitrary function f(z) as an infinite sum of simpler functions. The first is the power series expansion and its two important generalizations, the Laurent series and the Puiseux series. The second is the Dirichlet series, and the third is the Fourier series. These representations provide convenient, very general methods for studying a wide range of functions.

The terms of a power series expansion include power functions in the form (((z-(Subscript[z, 0])))^k); the terms in a Dirichlet series include exponential functions in the form exp(-(Subscript[λ, k])s). The Fourier series usually provide expansions in terms of cos(k x) and sin(k x), although expansions are possible in terms of other orthogonal systems of functions, such as the orthogonal polynomials. With each of these methods, we can express an enormous number of non-elementary functions in terms of only simple elementary functions.

Generalized power series

Expansions at z==0

Taylor series expansion (first investigated by Brook Taylor in 1715).

The Taylor series, first investigated by Brook Taylor in 1715, gives us the following expansion for an arbitary function f(z) at a finite point (Subscript[z, 0]):

The sign ↔ means that the Taylor series should not be taken as purely equal to f(z), since it may not converge everywhere on the complex plane. The series converges absolutely in some disk of radius R centered on (Subscript[z, 0]), where R is called the radius of convergence. On the disk |z-(Subscript[z, 0])|<R, we can state for most functions that

R=(lim_(k->∞))(1/((|(Subscript[c, k])|)^(1/k))).

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

If R=0, this series converges only at the point (Subscript[z, 0]), and the Taylor series offers little analytical benefit. For example, the series (∑_(k=0)^∞)k!(((z-(Subscript[z, 0])))^k) has no disk of convergence around (Subscript[z, 0]), since the k! coefficient makes the terms grow infinitely large no matter how small |z-(Subscript[z, 0])| becomes.

If R=∞, the series converges in the entire plane. In such cases it either represents an entire transcendental function—as, for example, the series (∑_(k=0)^∞)(z-(Subscript[z, 0]))/k! does for the exponential function exp(z-(Subscript[z, 0]))—or else contains only a finite number of terms and therefore represents a polynomial.

If 0<R<∞, the sum of the series defines a regular analytic function having at least one singular point on the circle |z-(Subscript[z, 0])|=R. There may be finitely or infinitely many singular points on the circle, but there must be at least one. The power series (∑_(k=1)^∞)((((z-(Subscript[z, 0])))^k)/k)=-log((Subscript[z, 0])+1-z), for example, has only one singular point, at z=(Subscript[z, 0])+1. Other power series have dense sets of singular points on the circle, such as (∑_(k=0)^∞)(z^(k!)), 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 the InverseEllipticNomeQ,

log ((((q^(-1))(z))/16)z)=(∑_(k=1)^∞)log((((((1+(z^(2k)))/(1+(z^(2k+1))))))^8)).

Inside its region of convergence, we can exchange in the sign ↔ for = in the first formula of this section.

Expansions at z==(Subscript[z, 0])

Laurent Series

The Laurent series, first studied by Pierre Laurent in 1843, gives the following expansion for the function f(z) around a finite point (Subscript[z, 0]):

f(z) ↔ (∑_(k=-∞)^∞)(((Subscript[c, k])(z-(Subscript[z, 0])))^k)    where    (Subscript[c, k])=(1/(2π i)) (∫_(Subscript[C, ρ]))((f(t))/(((t-(Subscript[z, 0])))^(k+1)))dt,   k∈ℤ,

where (Subscript[C, ρ]) is the circle |t-(Subscript[z, 0])|=ρ with radius ρ, r<ρ<R. Here R is the radius of convergence of its regular part (∑_(k=0)^∞)(((Subscript[c, k])(z-(Subscript[z, 0])))^k) and 1/r is the radius of convergence of its principal part (∑_(k=-∞)^(-1))(((Subscript[c, k])(z-(Subscript[z, 0])))^k)=(∑_(k=1)^∞)(((Subscript[c, -k])(z-(Subscript[z, 0])))^(-k)). The coefficient (Subscript[c, -1]) of the (((z-(Subscript[z, 0])))^(-1)) term is called the residue of f(z) at (Subscript[z, 0]).

If r<R, then the Laurent series converges absolutely in the ring  r<\[LeftBracketingBar]z-(Subscript[z, 0])\[RightBracketingBar]<R, where its sum defines an analytic function. If r=0 and the principal part includes only a finite number of terms (that is, if (Subscript[c, k])=0 for k<-n<0 for some n∈(SuperPlus[ℕ])), then f(z) has a pole of order n at the point z=(Subscript[z, 0]).

For example, the function cot(z) has a pole of order 1 (also called a simple pole) at the point z=0, since it has only one term in the principal part of its Laurent series expansion:

cot(z)=(1/z)+(∑_(k=1)^∞)(((((-1))^k)(2^(2k))(Subscript[B, 2k]))/((2k)!))(z^(2k-1))=(1/z)-(z/3)-((z^3)/45)+⋯.

If r=0 and the principal part includes an infinite number of terms, this analytic function has an essential singularity at the point z=(Subscript[z, 0]). For example, the sum (∑_(k=-∞)^0)(z^k)/(-k)!=exp(1/z) has an essential singularity at the point z=0.

Puiseux Series

Puiseux series, first studied by Victor-Alexandre Puiseux in 1850, expand f(z) around a finite point (Subscript[z, 0]) as f(z)↔(∑_(k=-∞)^∞)(Subscript[c, k])(Subscript[φ, k])(z-(Subscript[z, 0])), with (Subscript[φ, k])(w) usually chosen as (Subscript[φ, k])(z)=(w^(k/m)). (Other choices for (Subscript[φ, k])(z) are possible through the use of iterated logarithms of the form (Subscript[φ, k])(w)=(w^(k/m))(log^r)(w), (Subscript[φ, k])(w)= (w^(k/m))(log^r)(w)(log^s)(log(w)), and so on.) Choosing (Subscript[φ, k])(z)=(w^(k/m)) gives us

f(z) ↔ (∑_(k=-∞)^∞)(((Subscript[c, k])(z-(Subscript[z, 0])))^(k/m)).

If this series contains only a finite number of nonzero coefficients (Subscript[c, k]) with negative indices k, the point (Subscript[z, 0]) is an algebraic branch point of order m-1; otherwise it is a transcendental branch point, such as the point z=0 for the function (Sqrt[exp(1/z)]).

The 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 behavior near their singular points; the following examples show that such behavior can be studied using their expansions in Puiseux series.

Example 1. The point  z=1 is an algebraic branch point of order 1 for the function (cos^(-1))(z) = ((Sqrt[1-z])/(Sqrt[z-1]))log(z+(Sqrt[z-1]) (Sqrt[z+1])), which has for its main branch at z=1 the Puiseux representation

(cos^(-1))(z)= (Sqrt[2]) (Sqrt[1-z]) (∑_(k=0)^∞)(((((-1))^k) (Subscript[((1/2)), k]) (((z-1))^k))/((2^k) (2 k+1) k!))    where   |z-1|<1.

A similar representation occurs near the branch point z=-1. Near the branch point z=∞,  (cos^(-1))(z) can be expanded through the following series:

(cos^(-1))(z)=(π/2)-((z )/(2 (Sqrt[-(z^2)])))(log(-4 (z^2))-(1/2) (∑_(k=1)^∞)(( (Subscript[((1/2)), k]) (z^(-2 k)))/(k k!)))

where \[LeftBracketingBar]z\[RightBracketingBar]>1, which shows that (cos^(-1))(z) has a logarithmic-type singularity that can provide infinitely many values depending on the direction of approach.

Example 2. The Puiseux series for (z^z) near the point z=0 includes powers of logarithmic functions:

(z^z)=(∑_(k=0)^∞)(((z^k)(log^k)(z))/(k!)).

Example 3. The Puiseux series for (log^2)(log(z)) near the point z=0 coincides with this function.

Example 4. The Puiseux series for the ProductLog function W(1/z) near the point z=∞ has a more complicated structure involving iterated logarithms:

W(z)=log(z)-log(log(z))+(∑_(n=1)^∞)((((-1))^(n-1))/((log^n)(z)))  (∑_(k=1)^n)(((Subscript[S, n]^((n-k+1))) (log^k)(log(z)))/(k!))
=log(z)-log(log(z))+((log(log(z)))/(log(z)))+(( (log^2)(log(z))-log(log(z)))/(2(log^2)(z)))+⋯

where |z|≫1. The complicated structure is to be expected from inverse functions like ProductLog.

Generalized Fourier Series

Define {(Subscript[ψ, k])(x)}, k∈ℕ, as a system of functions defined on an interval (a,b). If

(∫_a^b)(Subscript[ψ, m])(t)(Subscript[ψ, n])(t)dt={({{0,, m≠n,}, {1,, m=n,}})

then the system {(Subscript[ψ, k])(x)} is an orthogonal system on (a,b). The Fourier series of a function f(x) for this system is the series of the form

f(x)↔(∑_(k=0)^∞)(Subscript[c, k])(Subscript[ψ, k])(x),    where    (Subscript[c, k])=(∫_a^b)f(t)(Subscript[ψ, k])(t)dt,k∈ℕ.

Under some additional conditions (such as piecewise differentiability), this series with Fourier coefficients (Subscript[c, k]) converges to f(x) at the points of continuity of f, and to (1/2)(f(x+0)+f(x-0)) at the points of discontinuity of f, where f(x±0)=(Underscript[lim , ϵ->+0])f(x±ϵ)).

The Fourier series of a function f(x) for this system satisfies Bessel 's inequality:

(∑_(k=0)^∞)(Subscript[c, k]^2)≤(∫_a^b)((f(t))^2)dt.

which can be transformed into Parseval's equality if this orthogonal system is complete orthogonal system:

(∑_(k=0)^∞)(Subscript[c, k]^2)==(∫_a^b)((f(t))^2)dt.

The best-known examples of orthogonal systems are the orthogonal polynomials and the trigonometric system including cos(k x) and sin(k x).

Trigonometric Fourier Series

The series expansion by trigonometric system

first appeared in an 1807 paper of Joseph Fourier, where he used a similar expansion on the interval (0,2π), although Leonhard Euler had discovered the formulas for Fourier coefficients in 1777.

An arbitrary interval (a,b) can be transformed into the interval (-π,π) by the change of variable x->x(b-a)/(2 π)+(a+b)/2. This allows the following Fourier series expansion of absolutely integrable functions on an arbitrary interval (a,b):

f(x)↔((Subscript[a, 0])/2)+(∑_(k=1)^∞)(Subscript[a, k])cos(k (x-((a+b)/2))2(π/(b-a)))+(Subscript[b, k])sin(k (x-((a+b)/2))2(π/(b-a)))

where k∈ℕ and

(Subscript[a, k])=(2/(b-a))(∫_a^b)f(t)cos(k (t-((a+b)/2))2(π/(b-a)))dt, (Subscript[b, k])=(2/(b-a))(∫_a^b)f(t)sin(k (t-((a+b)/2))2(π/(b-a)))dt.

The coefficients (Subscript[a, k])->0,(Subscript[b, k])->0 as k->∞.

In the internal points x,  a<x<b, where the piecewise-differentiable function f(x) is continuous, the sum of the previous Fourier series is equal to f(x) and we can change the symbol ↔ to =. At the points of discontinuity, this sum is equal to (1/2)(f(x+0)+f(x-0)). Outside the interval (a,b), this sum is a periodic function with the period b-a, but the behavior of this sum is not necessarily related to the actual function f(x) outside the interval of expansion.

Example. Let f(x)=x and (a,b)=(-1,1). Then

(Subscript[a, k])=0, (Subscript[b, k])=2 ((sin(k π))/((k^2) (π^2)))-((2 cos(k π))/(k π)),k∈ℕ,f(x)↔(∑_(k=1)^∞)(((2 sin(k π))/((k^2) (π^2)))-((2 cos(k π))/(k π))) sin(k π x)=((i )/π)log((e^(-π i x))).

Inspection shows that the Fourier sum ((i )/π)log((e^(-π i x))) has period 2 and coincides with x on the interval (-1,1). But at the endpoint x=1 the Fourier sum is equal to 0, since we are evaluating (∑_(k=1)^∞)(((2 sin(k π))/((k^2) (π^2)))-((2 cos(k π))/(k π))) sin(k π) with sin(k π)=0. This result coincides with

(1/2)((Underscript[lim , x->1-0])((i )/π)log((e^(-π i x)))+(Underscript[lim , x->1+0])((i )/π)log((e^(-π i x))))=(1/2)(1-1).

Dirichlet Series

Dirichlet series for the function f(z) take the form

f(z)↔(∑_(k=0)^∞)(Subscript[a, k])exp(-(Subscript[λ, k])z).

If all the exponents (Subscript[λ, k])>0, the Dirichlet series converges in some open half-plane Re(z)>c. In this case, for many Dirichlet series the abscissa of absolute convergence c is

c=(lim_(k->∞))((log(|(Subscript[a, k])|))/(Subscript[λ, k])).

In the more general case 0<\[LeftBracketingBar](Subscript[λ, 0])\[RightBracketingBar]≤\[LeftBracketingBar](Subscript[λ, 1])\[RightBracketingBar]≤…, 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.

Examples. If (Subscript[λ, k])=2k and (Subscript[a, k])=1 or (Subscript[a, k])=(((-1))^k) we have, respectively, the Dirichlet series

(∑_(k=0)^∞)(e^(-2k z))=(e^( z)) ((csch(z))/2),
(∑_(k=0)^∞)(((-1))^k)(e^(-2k z))=-(((e^( z)) sech(z))/2)    when    Re(z)>0.

In the case (Subscript[λ, k])=log(k), we have the ordinary Dirichlet series f(s)↔(∑_(k=1)^∞)((Subscript[a, k])/(k^s)), which in its simplest case ((Subscript[a, k])=1, Re(s)>1) is the Riemann zeta function  ζ(z)=(∑_(k=1)^∞)(1/(k^z)).  Other interesting examples of Dirichlet series include the series for elliptic theta functions. For instance, if (Subscript[a, k])=(1/2)(q^(k^2)) and (Subscript[λ, k])=∓2 k i, we have the representations

(∑_(k=1)^∞)(q^(k^2))cos(2 k s)=(1/2)(∑_(k=1)^∞)(q^(k^2))(e^(2 k i s))+(1/2)(∑_(k=1)^∞)(q^(k^2))(e^(-2 k i s))=(((Subscript[ϑ, 3])(s,q)-1 )/2)  when  |q|<1.

Asymptotic Series Expansions

Other Series Representations

No

Integral Representations

On the Real Axis

Fourier Integral Representations

The Fourier integral is the continuous analogue of a Fourier series, defined as

These formulas can be derived from the Fourier series expansion of the function f(x) on interval (-l,l) as l->∞.

The substitution of a(t) and b(t) gives the Fourier integral formulas

The first definition above can be re-written in exponential form leading to Fourier transforms and inverse Fourier transforms:

If the function f(x) is absolutely integrable on the real axis, we have the equality

Analytic Continuations

To give information from corresponding article for poster

Continued Fraction Representations

None

Differential Equations

Ordinary Linear Differential Equations and Wronskians

Transformations

Transformations and Argument Simplifications

Argument involving Basic Arithmetic Operations

Argument involving Related Functions

Products, Sums, and Powers of the Direct Function

Lagrange's identity

Related Transformations (composition and inversion)

Operations

Differentiation and Fractional Differentiation

Faá di Bruno's formula

Indefinite Integration

For the Direct Function

Definite Integration

For the Direct Function

First mean value theorem

Second mean value theorems

(Poincare-Bertrand formula)

Ortogonality

Orthogonal polynomials available in Mathematica are LegendreP (including associated Legendre polynomials),  GegenbauerC, ChebyshevT, ChebyshevU, HermiteH, LaguerreL (including generalized Laguerre polynomials), and JacobiP. The orthogonal polynomials arise from eigenvalue problems for Sturm-Liouville differential operators of the form: ((p(x)Ψ'(z)))'+q(z)Ψ(z)=λ Ψ(z), z∈(a,b).

The classical orthogonal polynomials (Subscript[f, n])(z), n∈ℕ possess a number of interesting properties:

• (Subscript[f, n])(z) is a polynomial of z of degree n.
• (Subscript[f, n])(z) satisfies a linear second–order differential equation with respect to z.
• (Subscript[f, n])(z) satisfies a linear second–order difference equation with respect to n.
• (Subscript[f, n])(z) can be expressed as the nth derivative of a closed form expression.
• (Subscript[f, n])(z) has a simple generating function.

Each family of polynomials described here is a system of orthogonal polynomials; that is, for some interval (a,b), nonnegative function w(x), and k, n∈ ℕ, then

(∫_a^b)(Subscript[f, k])(z)(Subscript[f, n])(z)w(z)dz=={({{0,,  k≠n,}, {(Subscript[c, n]^2),,  k=n.}})

Then (Subscript[f, n])(z) forms a complete function system in (a,b) with respect to the weight function w(z) and orthogonalization constants (Subscript[c, n]):

(Subscript[ψ, n])(z)==((Sqrt[w(z)])/(Subscript[c, n]))(Subscript[f, n])(z),
(∫_a^b)(Subscript[ψ, k])(z)(Subscript[ψ, n])(z)dz==(Subscript[δ, k, ,n]),
(∑_(n=0)^∞)(Subscript[ψ, n])(z)(Subscript[ψ, n])(z')dz==δ(z-z').

This means that an arbitrary smooth enough function f(z) in the points of its continuity on the interval (a,b) can be expanded in the following manner.

f(z)==(∑_(k=0)^∞)(Subscript[c, k])(Subscript[ψ, k])(z)
(Subscript[c, k])==(∫_a^b)f(z)(Subscript[ψ, k])(z)dz,  k∈ℕ.

If the function f(z) is square–integrable on the interval (a,b) the following Parseval equality  takes place

(∫_a^b)f(z)(OverBar[f(z)])\[DifferentialD]z==(∑_(k=0)^∞)(Subscript[c, k])(OverBar[(Subscript[c, k])]).

Below we can see the table of corresponding orthogonal polynomials with their charecteristics

({{(Subscript[f, n])(z), {a,b}, w(z), (Subscript[c, n]), conditions}, {(Subscript[P, n]^((a,b)))(z), {-1,1}, (((1-z))^a) (((1+z))^b) , (Sqrt[(((2^(a+b+1)) Γ(a+n+1) Γ(b+n+1))/(n! (a+b+2 n+1) Γ(a+b+n+1)))]), Re(a)>-1∧Re(b)>-1}, {(Subscript[C, n]^λ)(z), {-1,1}, (((1-(z^2)))^(λ-1/2)), (Sqrt[((π (2^(1-2 λ)) Γ(n+2 λ))/(n! (n+λ) ((Γ(λ))^2)))]), Re(λ)>-(1/2)∧λ≠0}, {(Subscript[C, n]^((0)))(z), {-1,1}, 1/(Sqrt[1-(z^2)]), (Sqrt[2 π])/n, n>0}, {(Subscript[T, n])(z), {-1,1}, 1/(Sqrt[1-(z^2)]), (Sqrt[π/2])((Subscript[δ, n,0])+1), \[Null]}, {(Subscript[U, n])(z), {-1,1}, (Sqrt[1-(z^2)]), (Sqrt[π/2]),  }, {(Subscript[P, n]^m)(z) , {-1,1}, 1, (Sqrt[((2 (m+n)!)/((2 n+1) (n-m)!))]), m∈ℕ}, {(Subscript[P, n])(z), {-1,1}, 1, (Sqrt[2/(2n+1)]),  }, {(Subscript[L, n]^λ)(z), {0,∞}, (z^λ) (E^(-z)), (Sqrt[Γ(n+λ+1)/n!]), Re(λ)>-1}, {(Subscript[L, n])(z), {0,∞}, (E^(-z)), 1,  }, {(Subscript[H, n])(t), {-∞,∞}, (E^(-(z^2))), (Sqrt[(2^n) n! (Sqrt[π])]),  }})

Cauchy Integrals

Suppose L is a closed, smooth contour enclosing the region (SuperPlus[D]) on the complex plane, and that the function φ(z) is analytic over (SuperPlus[D])and continuous over (SuperPlus[D])⋃L. Then we know from the Cauchy integral formula, fundamental to the theory of analytical functions, that

(1/(2π i))(∮_L)((φ(τ))/(τ-z))dτ=={({{φ(z),, z∈(SuperPlus[D]),}, {0,, z∈(SuperMinus[D]),}})

where (SuperMinus[D]) represents the region outside L.

Cauchy-Type Integrals

A more general form of this integral, removing the demand that the curve L be closed, is the Cauchy-type integral,

Φ(z)==(1/(2π i))(∫_L)((φ(τ))/(τ-z))dτ,

where the density function φ(τ) is defined on the piecewise smooth contour L. The function φ(τ) must be continuous along L, but it must also meet a more stringent test known as the Hölder condition. The function φ(t) satisfies the Hölder condition if, for two arbitrary points (Subscript[t, 1]), (Subscript[t, 2]) on the curve, |φ((Subscript[t, 2]))-φ((Subscript[t, 1]))|<A((|(Subscript[t, 2])-(Subscript[t, 1])|)^λ), for some positive constants A and 0<λ≤1.

The Cauchy-type integral is analytic everywhere on the complex plane except on the contour L itself, which is a singular line for this integral. Since the integral contains a factor (called the kernel) in the form of 1/(τ-z), it diverges at τ==z for any z on L.

Singular Integrals

Because of the non-integrable singularity it contains, this improper integral is called a singular integral. It can be evaluated, however, if we remove from the path of integration a small neighborhood around this singularity; the limit as the size of the neighborhood shrinks to zero is the Cauchy principal value of this divergent integral. For example, suppose that L is simply a straight segment on the real axis from a to b; in other words,  L=(a,b). Rather than integrating from a through the point (Subscript[τ, 0])∈L to the point b, we can integrate on the intervals (a,(Subscript[τ, 0])-ϵ) and ((Subscript[τ, 0])+ϵ,b) and then add these results to arrive at a value. By taking the limit of this calculation as ϵ->0, we can state the principle value as:

𝒫(∫_a^b)((φ(τ))/(τ-(Subscript[τ, 0])))dτ==(lim_(ϵ->+0))((∫_a^((Subscript[τ, 0])-ϵ))((φ(τ))/(τ-(Subscript[τ, 0])))dτ+(∫_((Subscript[τ, 0])+ϵ)^b)((φ(τ))/(τ-(Subscript[τ, 0])))dτ)/;                                    a< (Subscript[τ, 0]) <b .

Here 𝒫∫ represents the Cauchy principal value; other symbols used in the mathematical literature include ∫', vp ∫, and the ordinary integration sign. The result can be generalized to singular curvilinear integrals by considering the curve L with a circular neighborhood l, centered on (Subscript[τ, 0]) and of radius ϱ, removed; then the value of the Cauchy principal value is

𝒫(∫_L)((φ(τ))/(τ-(Subscript[τ, 0])))dτ==(lim_(ϱ->0))(∫_(L-l))((φ(τ))/(τ-(Subscript[τ, 0])))dτ/;                                                                                                  (Subscript[τ, 0]) ∈L.

Sokhotskii formulas

When L is a closed contour and (Subscript[τ, 0])∈L, we can think of Φ(z) as a piecewise analytic function where

Φ(z)==({{(SuperPlus[Φ])(z),, z∈(SuperPlus[D]),}, {(SuperMinus[Φ])(z),, z∈(SuperMinus[D]),}})

with a jump discontinuity as z->(Subscript[τ, 0]) ∈L. So we can write

({{(SuperPlus[Φ])((Subscript[τ, 0]))==(lim_(z->(Subscript[τ, 0])))Φ(z)/;z∈(SuperPlus[D]),}, {(SuperMinus[Φ])((Subscript[τ, 0]))==(lim_(z->(Subscript[τ, 0])))Φ(z)/;z∈(SuperMinus[D]).}})

If L is an open contour with endpoints a and b, we can add an additional arbitrary curve segment connecting b to a (with the sense that (SuperPlus[Φ])(τ) corresponds to the limiting value from the left) and assign φ(τ)=0 along this new segment. This extension allows us to apply the above definitions for (SuperPlus[Φ]) and (SuperMinus[Φ]) to open contours.

Then, according to the Sokhotskii formulas,

(SuperPlus[Φ])((Subscript[τ, 0]))==((φ((Subscript[τ, 0])))/2)+(1/(2π  i))𝒫(∫_L)((φ(τ))/(τ-(Subscript[τ, 0])))dτ,
(SuperMinus[Φ])((Subscript[τ, 0]))==-((φ((Subscript[τ, 0])))/2)+(1/(2π  i))𝒫(∫_L)((φ(τ))/(τ-(Subscript[τ, 0])))dτ.

By adding and subtracting these two formulas, we get

(SuperPlus[Φ])((Subscript[τ, 0]))+(SuperMinus[Φ])((Subscript[τ, 0]))==(1/( π  I))𝒫(∫_L)((φ(τ))/(τ-(Subscript[τ, 0])))\[DifferentialD]τ,
(SuperPlus[Φ])((Subscript[τ, 0]))-(SuperMinus[Φ])((Subscript[τ, 0]))==φ((Subscript[τ, 0])).

In particular, if φ(z) is analytical over (SuperPlus[D])⋃L, then (SuperPlus[Φ])((Subscript[τ, 0]))==φ((Subscript[τ, 0])) and (SuperMinus[Φ])((Subscript[τ, 0]))==0. If the contour L is a finite or infinite segment of the real axis, L=(a,b), these formulas hold for all a<(Subscript[τ, 0])<b, and so (SuperPlus[Φ])((Subscript[τ, 0]))==(lim_(ϵ->+0))Φ((Subscript[τ, 0])+i  ϵ), (SuperMinus[Φ])((Subscript[τ, 0]))==(lim_(ϵ->+0))Φ((Subscript[τ, 0])-i  ϵ). Thus Φ(z) is an analytic function with a jump discontinuity at L; the size of the jump is determined by the Sokhotskii formulas.

The Sokhotskii formulas play a basic role in solving boundary value problems, problems in function theory, and singular integral equations. They were first derived by Y.V. Sokhotskii in 1873; because they were later given a more rigorous treatment by J. Plemelj in 1908, they are often refered to as the Plemelj formulas.

For example

Φ(z)==(1/(2 π  I))(∫_0^∞)((E^(-τ))/(τ-z))\[DifferentialD]τ==((E^(-z))/(2 π  I)){({{π  I-Ei(z),, Im(z)>0,}, {-π  I-Ei(z),, Im(z)<0,}, {-Ei(z),, Im(z)==0.}})

For arbitrary x>0 the Sokhotskii formulas give us the following:

(SuperPlus[Φ])(x)==(((E^(-x)) (π  I-Ei(x)))/(2 π  I))==((E^(-z))/(2 ))+((-(E^(-x))Ei(x))/(2 π  I)),
(SuperMinus[Φ])(x)==(((E^(-x)) (-π  I-Ei(x)))/(2 π  I))==-((E^(-z))/(2 ))+((-(E^(-x))Ei(x))/(2 π  I)),
(SuperPlus[Φ])(x)+(SuperMinus[Φ])(x)==(1/(π  I))𝒫(∫_0^∞)((E^(-τ))/(τ-x))\[DifferentialD]τ==-(((E^(-x))Ei(x) )/(π  I)),
(SuperPlus[Φ])(x)-(SuperMinus[Φ])(x)==(E^(-x)).

It is important to note that for z<0 the function Φ(z) is analytic and its limit values taken from either side of the real axis should agree with each other. This gives us:

(lim_(ϵ->+0))(π  I-Ei(x+I  ϵ))==(lim_(ϵ->+0))(-π  I-Ei(x-I  ϵ))==-Ei(x).

This leads to the following behavior for Ei(x):

(lim_(ϵ->+0))Ei(x+I  ϵ)==Ei(x)+π  I,
(lim_(ϵ->+0))Ei(x-I  ϵ)==Ei(x)-π  I.

Another example

Φ(z)==(1/(2π  I))(∫_0^∞)((τ^(α-1))/(τ-z))\[DifferentialD]τ==(1/(2 π  I)){({{π (((-z))^(α-1))csc(α  π),, 0<Re(α)<1∧Arg(z)≠0,}, {-π  (z^(α-1)) cot(α  π),, 0<Re(α)<1∧z>0.}})

For arbitrary x>0, the Sokhotskii formulas give us the following results when we take into account the fact that, as ϵ->+0, (((-z))^(α-1))is continuous for -z=-x+i  ϵ but has a jump of size (e^(-i  α  π)) compared to when approached from the other side, -z=-x-i  ϵ.

(SuperPlus[Φ])(x)==(lim_(ϵ->+0)) (( (((-x-I  ϵ))^(α-1)) csc(α  π))/(2  I ))==-(((x^(α-1)) (E^(-I  α  π)) csc(α  π))/(2  I )) ==((x^(α-1))/2)+((I  (x^(α-1)) cot(α  π))/(2 )),
(SuperMinus[Φ])(x)==(lim_(ϵ->+0)) (((((-x+I  ϵ))^(α-1)) csc(α  π))/(2  I ))==-(((x^(α-1))(E^(I  α  π))csc(α  π))/(2  I ))==-((x^(α-1))/2)+((I  (x^(α-1)) cot(α  π))/(2 )),
(SuperPlus[Φ])(x)+(SuperMinus[Φ])(x)==-(((x^(α-1)) (E^(-I  α  π)) csc(α  π))/(2  I ))-(((x^(α-1))(E^(I  α  π))csc(α  π))/(2  I ))==I  (x^(α-1)) cot(α  π)==(1/(π  I))(∫_0^∞)((τ^(α-1))/(τ-x))\[DifferentialD]τ,
(SuperPlus[Φ])(x)-(SuperMinus[Φ])(x)==-(((x^(α-1)) (E^(-I  α  π)) csc(α  π))/(2  I ))+(((x^(α-1))(E^(I  α   π))csc(α  π))/(2 I ))==(x^(α-1)).

Finite Summation

Dirichlet formula for Fourier series

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Parseval's lemma which reflects completness trigonometric system

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Last property can be valid under special restrictions on z.

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