Chapter Name

15.1-1. Functional analysis regards suitable classes of functions as “points” in topological spaces (Chap. 12) and, in particular, as multidimensional vectors admitting the definition of inner products (Sec. 15.2-1) and expansions in terms of orthogonal functions (base vectors, Sec. 15.3-4). The resulting elegant and powerful geometrical analogy relates a large class of operations, including linear integral transformations and differentiation, to the theory of linear transformations introduced in Chap. 14. The solution of linear ordinary and partial differential equations and of linear integral equations is thus found to be a more or less simple generalization of the solution of linear simultaneous equations; in particular, the solution may involve an eigenvalue problem.

Sections 15.3-1 to 15.3-10 review linear integral equations; Secs. 15.4-1 to 15.4-12 introduce linear boundary-value problems and eigenvalue problems involving differential equations. The remainder of the chapter deals with various methods of solving linear boundary-value problems, viz.

1. Eigenfunction expansions (Sec. 15.4-12); this method may be extended to include the various integral-transform methods (Sec. 10.5-1)

2. Green's functions (Secs. 15.5-1, 15.5-3, 15.6-6, and 15.6-9)

3. Reduction to integral equations (Sec. 15.3-2)

4. Variation methods (Sec. 15.4-7; see also Sees. 11.7-1 to 11.7-3)

In particular, Secs. 15.6-1 to 15.6-10 deal with boundary-value problems involving Laplace's and Poisson's differential equations (potential theory) and the space form of the wave equation.

Although many practical problems will yield only to numerical solution (Sec. 20.9-4), the general and intuitively suggestive viewpoint of functional analysis offers far-reaching insight into the behavior of vibrating systems, atomic phenomena, etc.

15.1-2. Notation (see also Sec. 15.4-1). Throughout Secs. 15.2-1 to 15.5-4, Φ(x), f(x), F(x), . . . symbolize either functions of a single independent variable x or, for brevity, functions of a set of n independent variables x¹, x², . . . , xⁿ(see also Secs. 6.2-1 and 16.1-2). In the one-dimensional case, dx is a simple differential; in the multidimensional case, dx = dx¹ dx² . . . dxⁿ. An integral

stands either for a one-dimensional definite integralover a bounded or unbounded interval V = (a, b), or for an n-dimensional integral

over a region V in n-dimensional space. As a rule it is possible to introduce volume elements so that each integral (2) becomes a volume integral (Secs. 6.2-3, 15.4-1b, and 16.10-10).

15.2. FUNCTIONS AS VECTORS. EXPANSIONS IN TERMS OF ORTHOGONAL FUNCTIONS

15.2-1. Quadratically Integrable Functions as Vectors, Inner Product and Normalization.* (a) A real or complex function f(x) defined on the measurable set E of “points” (x)or (x¹, x²,. . . , x³) is quadratically integrable on E if and only if exists in the sense of Lebesgue (Sec. 4.6-15). The class L² [more accurately L²(V)] of all real or complex functions quadratically integrable on a given interval or region V constitute, respectively, an infinite-dimensional real or complex unitary vector space (Sec. 14.2-6) if one regards the functions f(x), h(x), . . . as vectors and defines

The vector sum of f(x) and h(x) as f(x) + h(x)

The product of f(x) by a scalar α as αf(x)

The inner product of f(x) and h(x) as

* See Sec. 15.1-2 for notation.

where γ(x) is a given real nonnegative function (weighting function) quadratically integrable on V. A change of weighting function corresponds to a change in the independent variable; in many applications, one has γ(x) ≡ 1, or ≡(ξ) dξ is a volume element (Sec. 15.4-1b).

Linear independence of a set of functions (vectors) in L² is defined in the manner of Sec. 1.9-3 (see also Sec. 14.2-3). A set of quadratically integrable functions f₁(x), ƒ₂(x), . . . , f_m(x) are linearly independent if and only if Gram's determinant det [(ƒ_i,ƒ_k)] differs from zero (see also Sec. 14.2-6a).

(b) As in Sec. 14.2-7, the norm of a function (vector) f(x) in L² is the quantity

A (necessarily quadratically integrable) function f(x) is normalizable if and only if ||ƒ|| exists and is different from zero. Multiplication of a normalizable function f(x) by 1/||ƒ|| yields a function f(x)||ƒ|| of unit norm [normalization of ƒ(x)].

(c) The inner product defined by Eq. (1) has all the properties listed in Sec. 14.2-6. In particular, if f(x), h(x), and the real nonnegative weighting function γ{x) are quadratically integrable on V

and

15.2-2. Metric and Convergence in L². Convergence in Mean (see also Secs. 12.5-2 to 12.5-5, 14.2-7, and 18.6-3). (a) As a unitary vector space, L² admits the distance function (metric, Sec. 12.5-2)

The root-mean-square difference (5) between f(x) and h(x) equals zero (metric equality) if and only if f(x) = h(x) for almost all x in V (Sec. 4.6-14b).

(b) Convergence in Mean. The metric (5) leads to the following definition of metric convergence in L². For a given interval or region V, a sequence of quadratically integrable functions s₀(x), s₁(x), s₂(x), . . .

converges in mean (with index 2) to the limit s as n → ∞] if and only if

In this case, the sequence defines its limit-in-mean uniquely almost everywhere in V. Convergence in mean does not necessarily imply actual convergence of the sequence s₀(x), s₁(x), s₂(x), . . . at every point, nor does convergence of a sequence at all points of V imply convergence in mean. Equation (6) does imply convergence of a subsequence of s_n(x) for almost every x in V.

In particular, an infinite series a₀(x) + a₁(x) + a₂(x) + ... of quadratically integrable functions converges in mean to the limit s(x) if and only if

only if . In this case,one write

(c) Completeness of L²: the Generalized Riesz-Fischer Theorem. The space L² associated with a given interval or region V is complete (Sec. 12.5-4a); i.e., every sequence of quadratically integrable functions s₀(x), s₁(x), s₂(x), . . . such that(Cauchy sequence)

converges in mean to a quadratically integrable function s(x) and defines s(x) uniquely for almost all x in V (generalized Riesz-Fischer Theorem).

NOTE: The completeness property expressed by the generalized Riesz-Fischer theorem establishes L² as a Hilbert space (Sec. 14.2-7c), permitting the introduction of orthonormal bases with all the properties noted in Secs. 14.7-4 and 15.2-4. This is an important reason for the use of Lebesgue integration and convergence in mean.

(d) One defines as α →> a if and only if

15.2-3. Orthogonal Functions and Orthonormal Sets of Functions (see also Sec. 14.7-3). (a) Two quadratically integrable functions f(x), h(x) are mutually orthogonal [orthogonal with respect to the real non-negative weighting function γ(x)]* on V if and only if

A set of functions u₁(x), u₂(x), ... is an orthonormal (normal orthogonal, normalized orthogonal) set if and only if

* Some authors call f(x) and h(x) mutually orthogonal only if also γ(x) ≡ 1 in V. so that

Every set of normalizable mutually orthogonal functions (and, in particular, every orihonormal set) is linearly independent.

(b) Bessel's Inequality. Given a finite or infinite orihonormal set u₁(x), u₂(x), . . . and any function f(x) quadratically integrable over V,

The equal sign applies if and only if f(x) belongs to the linear manifold spanned by u₁(x), u₂(x), . . . (see also Secs. 14.2-2, 14.7-3, and 15.2-4).

15.2-4. Complete Orthonormal Sets of Functions (Orthonormal Bases, see also Sec. 14.7-4). An orthonormal set of functions u₁(x), u₂(x), ... in L²(V) is a complete orthonormal set (orthonormal basis) if and only if it satisfies the following conditions:

Each of these four conditions implies the three others.

Given an interval or region V with a complete orthonormal set of functions u₁(x), u₂(x), . . . and any set of complex numbers f₁ f₂, . . . such that converges, there exists a quadratically integrable function f(x) such that f₁u₁(x) + f₂u₂(x) + … converges in mean to f(x) (Riesz-Fischer Theorem, see also Sec. 15.2-2c); the ƒ_k define f(x) uniquely almost everywhere in V, and in particular wherever ƒ(x) is continuous in V (Uniqueness Theorem, see also Sec. 4.11-5).

15.2-5. Orthogonalization and Normalization of a Set of Functions (see also Sec. 14.7-4b). Given any countable (finite or infinite) set of linearly independent (Sec. 1.9-3) functions φ₁(x), φ₂(x), . . . normalizable on V, there exists an orthonormal set u₁(x), u₂(x), . . . spanning the same manifold of functions. Such an orthonormal set may be constructed with the aid of the following recursion formulas (Gram-Schmidt Orthogonalization Process):

See also Sec. 21.7-1 for examples.

15.2-6. Approximations and Series Expansions in Terms of Ortho-normal Functions (see also Secs. 4.11-2c, 4.11-4b, 15.4-12, 20.6-2, 20.6-3, 20.9-9, and 21.8-12). Given a quadratically integrable function f(x) and an orthonormal set u₁(x), u₂(x), . . . , an approximation to f(x) of the form

yields the least mean-square error

Note that the choice of the coefficients a_k is independent of n. This property, together with the relative simplicity of the formulas listed in Sec. 15.2-4a, establishes the great importance of series expansions

in terms of suitable normalized orthogonal functions u₁(x), u₂(x), ....

15.2-7. Linear Operations on Functions. Sections 8.2-1, 8.6-1 to 8.6-4, 15.3-1, 15.4-1, and 20.4-2 introduce various linear operations (Sec. 14.3-1) associating a function

with a given function Φ(ξ) so that

Φ(ξ) and φ(x) may or may not be defined on the same domain. As in Sec. 14.1-3, there exist two distinct interpretations of the functional transformation (12):

1. Equation (12) describes an operation on the object function Φ(ξ) (“alibi” point of view).

1. Φ(ξ) and φ(x) represent the same abstract vector (in the same sense as two matrices, Sec. 14.6-1b), and Eq. (12) describes a change of representation (“alias” point of view, used especially in quantum mechanics; see also Sec. 8.1-1).

15.3. LINEAR INTEGRAL TRANSFORMATIONS AND LINEAR INTEGRAL EQUATIONS

15.3-1. Linear Integral Transformations.* (a) Sections 15.3-1 to 15.3-10 deal with linear integral transformations

relating a pair of functions ƒ(ξ) and F(x). The function K(x, ξ) is called the kernel of the linear integral transformation. All integrals will be assumed to exist in the sense of Lebesgue (Sec. 4.6-15).

The domains of the object function ƒ(ξ) and the result function f(x) in Eq. (1) are not necessarily identical (note, for example, the Laplace transformation_} Sec. 8.2-1). Throughout Secs. 15.3-lb to 15.3-10 it is, however, assumed that x and ξ range over the same interval or region V. The “symbolic” integral transformation

represents the identity transformation (see also Secs. 15.5-1 and 21.9-2).

Linear integral transformations (1) may be interpreted from either the “alibi” or the “alias” point of view (Sec. 15.2-7); in the former case each kernel K(x, ξ) represents a linear operator in the same sense as a matrix (Sec. 14.5-2; see also Sec. 15.3-lc).

(b) For a given kernel K(x, ξ), K(x, ξ) ≡ K(ξ, x) is called the transposed kernel, and † ;(x, ξ) ≡ K*(ξ, x) is the adjoint (hermitian conjugate) kernel (see also Sec. 14.4-3). A given kernel K(x, ξ) is

15.3-3. Homogeneous Fredholm Integral Equations of the Second Kind: Eigenfunctions and Eigenvalues (see also Secs. 14.8-3 and 15.4-5). (a) A function φ = φ (x) which is not identically zero in V and satisfies the homogeneous Fredholm integral equation of the second kind

* Refer again to Sec. 15.1-2 for notation. See also Sec. 15.4-1b.

Symmetric if and only if K(ξ, x) = K(x, ξ)

Hermitian if and only if K*(ξ, x) = K(x, ξ)

Normalizable if and only if exists and is different from zero

Continuous in the mean over V if and only if

(see also Sec. 12.5-lc)

Separable (degenerate) if and only if K(x, ξ) can be expressed as a as a finite sum

A normalizable kernel represents a bounded operator (Sec. 14.4-1), so that f(x) is normalizable if ƒ(ξ) is normalizable (Sec. 15.2-1b). Separable kernels represent operators of finite rank (Sec. 14.3-2). If K(x, ξ) is hermitian, normalizable, and continuous in the mean over V, then F(x) is continuous in V whenever f(ξ) is quadratically integrable over V.

(c) Matrix Representation. Product of Two Integral Transformations. Given a normalizable kernel K(x, ξ) and an orthonormal basis (Sec. 15.2-4) u₁(x), u₂(x), ... in the space of functionsf (z), let

Then Eq. (1) is equivalent to the matrix equation {F_i} = [k_ik]{f_k}. Note that the matrix product [m_ik][k_ik] corresponds to the kernel ∫_v M (x, n)K(n, ξ) dn representing the product of two successive integral transformations (1) whose kernels K(x, ξ), M(x, ξ) correspond to [k_ik], [m_ik].

If K(x, ξ) is a separable kernel, then it is possible to choose the uk(x) so that the matrix [k_ik] is finite.

15.3-2. Linear Integral Equations: Survey. An integral equation is a functional equation (Sec. 9.1-2) involving integral transforms of the unknown function Φ(x) [if the functional equation also involves a derivative of Φ(x) one speaks of an integro-differential equation]. A given integral equation is homogeneous if and only if every multiple αξ(x) of any solution Φ(x) is a solution.

Sections 15.3-2 to 15.3-10 deal with linear integral equations of the general form

where K(x, ξ), β(x), and f(x) are given functions; the integration domain V may be given (Fredholm-type integral equations) or variable (e.g., Volterra-type integral equations, Sec. 15.3-10). Three important types of problems arise:

1. A linear integral equation of the first kind [β(x) ≡ 0, λ = 1, Sec. 15.3-9] requires one to find an unknown function Φ (x) with the given integral transform F(x). The corresponding operator equation KΦ = F is analogous to a matrix equation [k_ik]{Φ_k} = {F_i}.

2. A homogeneous linear integral equation of the second kind [F(x) ≡ 0, β(x) ≡ 1, λ unknown; Secs. 15.3-3 to 15.3-6] represents an eigenvalue problem. The corresponding operator equation λKΦ= Φ is analogous to a matrix equation λ[k_ik]{Φ_k} = {Φ_i}.

3. A nonhomogeneous linear integral equaion of the second kind [β(x) = 1, λ given; Sec. 15.3-7] may be written as ϕ - λKϕ = F and represents a problem of the type discussed in Sec. 14.8-10.

If β(x) is a real positive function throughout V, one can reduce the general linear intrgral equation (3) to a linear integral equation of the second kind with the aid of the transformation

15.3-5. Homogencous Fredholm Integral Equation of the Second Kind: Eigenfunctions and Eigenvalues (see also Secs. 14.8-3 and 15.4-5). (a) A function = (x) withis not identically zero in V and satisfies the homogeneous Fredholm integral equationof the secon kind

for a suitably determined value of the parameter λ is called an eigen-function (characteristic function) of the linear integral equation (5) or of the kernel K(x, ξ). The corresponding value of λ is an eigenvalue of the integral equation.

If φ(x) and φ(x) are eigenfunctions corresponding to the same eigenvalue λ, the same is true for every linear combination α₁ φ₁;(x) + α₂φ₂(x). The number m of linearly independent eigenfunctions corresponding to a given eigenvalue λ is its degree of degeneracy; if K(x, ξ) is a normaliz-able kernel, each m is finite. Eigenfunctions associated with different eigenvalues of a linear integral equation (5) are linearly independent. The total number of linearly independent eigenfunctions is finite if and only if K(x, ξ) is separable.

If the linear operator K described by the linear integral transformation (5) has a unique inverse L = K^-1, then the φ(x) and λ are eigenfunctions and eigenvalues of the nonsingular linear operator L (see also Sec. 15.4-5), and all the eigenvalues λ are different from zero. In many applications L is a differential operator, and K(x, ξ) is a Green's function (Sec. 15.5-1).

(b) Eigenfunctions and Eigenvalues for Hermitian Kernels (see also Secs. 14.8-4 and 15.4-6). If K(x, ξ) is a hermitian kernel, then

1. All eigenvalues of the integral equation (5) are real.

2. Eigenfunctions corresponding to different eigenvalues are mutually orthogonal (with weighting function 1, Sec. 15.2-3). if the kernel K(x, ξ) is normalizable as well as hermitian, then

1. All eigenvalues are different from zero.

2. There exists at least one eigenvalue different from zero; the eigenvalues constitute a discrete set comprising at most a finite number of eigenvalues in any finite interval, and every eigenvalue has a finite degree of degeneracy.

15.3-4. Expansion Theorems, (a) Expansion Theorems for Hermitian Kernels. Given the definitions of inner products (Sec. 15.2-1) and convergence in mean (Sec. 15.2-2) with γ(x) ≡ 1, every normalizable hermitian kernel K(x, ξ) admits an orthonormal set of eigenfunctions φ(x), φ(x), . . . such that, for every given function F(x) representable as the integral transform (1) of a function f(ξ),

The series (6) converges absolutely and uniformly to F(x) for all x in V (1) if f(ξ) is piecewise continuous in V, or (2) if f(ξ) is quadratically integrable and K(x, ξ) is continuous in the mean. A hermitian kernel K(x, ξ) is complete if and only if every quadratically integrable function f(x) can be represented in the form (1) or (6), so that the eigenfunctions φ_k(x) constitute a complete orthonormal set (Sec. 15.2-4); this is true, for example, if K(x, ξ) is definite (Sec. 15.3-6).

Every normalizable hermitian kernel K(x, ξ) can be expanded in the form

The series converges uniformly to K(x, ξ) for all x, ξ in V (1) if K(x, ξ) is continuous in the mean, or (2) if V is a bounded interval or region such that K(x, ξ) is continuous for all x, ξin V, and either the positive or the negative eigenvalues of Eq. (5) are finite in number (Mercer's Theorem).

(b) Auxiliary Kernels and Expansion Theorems for Nonhermitian Kernels. For every normalizable kernel K(x, ξ), the auxiliary hermitian kernels

∫_vK†(x, n)K(n, ξ) dn and ∫_v K(x, n)K†(n, ξ) dn yield identical real eigenvalues u_k² and the respective orthonormal sets of eigenfunctions v_k(x) and w_k(x). Note

where u_k 0.

Given any normalizable kernel K(x, ξ), every function F{x) expressible as an integral transform (1) with kernel K or K† can be expanded in the respective form

Each series converges absolutely and uniformly to F(x) for all x in V (1) if f(ξ) is piecewise continuous in V, or (2) if f(ξ) is quadratically integrable, and K(x, ξ) is continuous in the mean; the weighting function is 1 (Sec. 15.2-1).

For every normalizable kernel K(x, ξ)

15.3-5. Iterated Kernels. The iterated kernels K_p(x, ξ) defined by

represent the powers K^p of the linear operator K of Eq. (1). The linear integral equation

has the same eigenfunctions φ(x) as Eq. (5), with corresponding eigenvalues λ^p (see also Sec. 14.8-3d). Conversely, every solution φ(x) of Eq. (12) solves Eq. (5).

If K(x, ξ) is hermitian and normalizable, then

and this series converges absolutely and uniformly in V.

15.3-6. Hermitian Integral Forms. The Eigenvalue Problem as a Variation Problem (see also Secs. 11.1-1, 13.5-2, 13.5-3, 14.8-8a, and 15.4-7). (a) Given a normalizable hermitian kernel K(x, ξ), the (necessarily real) inner product

is called a hermitian integral form or, in particular, a real symmetric quadratic integral form if K(x, ξ) is real and symmetric.* The hermitian integral form (14), and also the hermitian kernel K{x, ξ), is positive definite, negative definite, non-negative, or nonpositive if and only if the expression (14) is, respectively, positive, negative, nonnegative, or nonpositive for every function Φ(x) not identically zero in V and such that the integral exists. The integral form (14) is positive definite or negative definite if and only if all eigenvalues of K(x, ξ) are, respectively, positive or negative.

b) The problem of finding the eigenfunctions φ(x) and the eigenvalues λfor a normalizable hermitian kernel may be expressed as a stationary-value problem in the manner of Sec. 14.8-8a, e.g.,

Find a quadratically integrable function Φ(x) such that the hermitian integral form (14) has a stationary value subject to the constraint

Φ = φ(x) yields the stationary value λ_k.

All the other theorems of Sec. 14.8-8a apply; it is only necessary to remember that the operator K represented by the given kernel K(x, ξ) has the eigenfunctions φ_k(x) and the eigenvalues l/φ_k. It is, then, often possible to solve an integral equation of the form (5) exactly or approximately by the methods of the calculus of variations.

15.3-7. Nonhomogeneous Fredholm Equations of the Second Kind (see also Sec. 14.8-10). (a) Existence and Uniqueness of Solutions. Fredholm's linear integral equation of the second kind

has the following “alternative property”:

1. If the given parameter λ is not an eigenvalue of K(x, ξ), then Eq. (16) has at most one unique solution Φ (x).

If λ equals an eigenvalue ₁;of Eq. (5), then the “adjoint” homogeneous integral equation

has a solution x(x) not identically zero in V, and the given integral equation (16) has a solution Φ(x) only if F(x) is orthogonal (with

* Note that, in terms of the matrix elements defined in Sec. 15.3-lc,

weighting function 1) to every solution x(x) of Eq. (17). Note that Eqs. (17) and (5) are identical for hermitian kernels.

Solutions subject to the above conditions actually exist, in particular, whenever K(x, ξ) is piecewise continuous and normalizable and f(x) is continuous and quadratically integrable over V. If a solution φ(x) exists in case 2, then Eq. (16) has infinitely many solutions, since every sum of a particular solution and a linear combination of eigenfunctions Φ(x) corresponding to Xi is a solution; in particular, there exists a unique solution Φ(x) orthogonal to all these eigenfunctions.

(b) Reduction to an Integral Equation with Hermitian Kernel. If K(x, ξ) is normalizable, every solution Φ(x) of Eq. (16) is a solution of the integral equation

where H(x,ξ) is the hermitian kernel denned by

It is thus sufficient to study solution methods for hermitian kernels.

NOTE: H(x,ξ)and K(x,ξ) have identical eigenfunctions, with corresponding eigenvalues |A*| and X*.

(c) The Resolvent Kernel. A solution Φ(x) of the linear integral equation (16) is conveniently written in the form

The function Γ(x,ξ; λ) is called the resolvent kernel (sometimes known as the reciprocal kernel) for the integral equation (16); Eqs. (16) and (19) represent mutually inverse linear transformations.

Whenever the resolvent kernel Γ(x,ξ; λ) exists, it satisfies the integral equations

for arbitrary λ, λ′ which are not eigenvalues of K(x, ξ).

15.3-8. Solution of the Linear Integral Equation (16) (see also Sec. 20.8-5 for numerical methods), (a) Solution by Successive Approximations. Neumann Series. Starting with a trial solution

one may attempt to compute successive approximations

to the desired solution Φ(x) of Eq. (16). The functions (21) may be regarded as partial sums of the infinite series

If K(x, ξ) is normalizable, and F(x) is quadratically integrable over V,then there exists a real numbersuch that the Neumann series (22) converges in mean to a solution (19) for |λ|< r_c.

If, in addition, and are uniformly

bounded in V, then the power series (22) and the corresponding power series for the resolvent kernel

actually converge uniformly to the indicated limits for x,ξin V and |λ|< r_c.The function (23) is then an analytic function of λ for |λ|< r_c and can be continued analytically (Sec. 7.8-1) to yield a resolvent kernel for other suitable values of λ. The series (23), as well as the series (22), is known as a Neumann series.

If the normalizable kernel K(x,ξ) is hermitian, then the radius of convergence (or mean convergence) r_c is given by r_c = |λ₁, where λ₁ is the eigenvalue of K(x,ξ) with the smallest absolute value.

(b) The Schmidt-Hilbert Formula for the Resolvent Kernel (see also Sec. 14.8-10). For every normalizable and hermitian kernel K(x, ξ) the solution of the linear integral equation (16) is given by Eq. (19) with the resolvent kernel

where the Ψ_k(x) are orthonormal eigenfunctions* of K(x,ξ). The series converges uniformly for x,ξ in V and λ≠ λ_k.

* If λ_i is an ra-fold degenerate eigenvalue (Sec. 15.3-3a), one writes

so that the series (24) contains m terms involving the orthonormal eigenfunctions Ψ_i(x), Ψ_i+1(x), . . ,Ψ_i+m-1(x) corresponding to λ_i.

Γ(x, ξ; λ) is a meromorphic function of λ in the finite part of the λ plane; the residues of Γ(x,ξ; λ) at its poles λ = λ_k are simply related to the eigenfunctions Ψ_k(x) (Secs. 7.6-8 and 7.7-1).

SCHMIDT-HILBERT SOLUTION IF λ EQUALS AN EIGENVALUE λ_i. If the given parameter λ equals an eigenvalue λ_i of the given kernel K(x, ξ), omit the term or terms involving λ_i in the sum (24) (which is now no longer a resolvent kernel) and add an arbitrary eigenfunction (x) associated with λ to the right side of Eq. (19). The resulting function φ(x) will solve the given integral equation (16), subject to the conditions of case 2 in Sec. 15.3-7a.

(c) Fredholm's Formulas for the Resolvent Kernel. If K(x, ξ) is normalizable, the resolvent kernel Γ(x, ξ; λ) can be expressed as the ratio of two integral functions of λ (see also Sec. 7.6-7), viz.,

Both power series converge for all finite λ the power series for D(x, ξ; λ) converges uniformly in V. The poles of Γ(x, ξ; λ) coincide with the zeros of D(λ) .

Note the analogy between the functions D(x, ξ; λ), D(λ) and the determinants used

to find the solution Xk of the analogous finite-dimensional problem

(i = 1, 2, . . . , n) by Cramer's rule (1.9-4).

(d) Singular Kernels. If the given kernel K(x, ξ) becomes unbounded in a neighborhood of x = ξ while the iterated kernel K₂(x, ξ) remains bounded, find the solution φ (x) of Eq. (16) by solving the integral equation

Equation (27) is obtained by substitution of

on the left of Eq. (16). This procedure may be repeated if K₃(x ξ), K₄(x, ξ),... is the first iterated kernel which remains bounded.

15.3-9. Solution of Fredholm's Linear Integral Equation of the First Kind (see also Secs. 14.8-10b, 15.4-12, and 15.5-1). If the linear integral equation

with a given normalizable kernel K(x, ξ) has a solution φ(x), the expansion theorem of Sec. 15.3-4b yields

where the ν_k(x), w_k(x), and μ_k are defined as in Sec. 15.3-4b, and φ(x) is an arbitrary function orthogonal to all v_k(x).

If the Vk(x) constitute a complete orthonormal set in L₂(V) (Sec. 15.2-4), then φ(x) = 0 for almost all x in V, and the solution (29) is unique almost everywhere in V. If the wk(x) as well as the Vk(x) form a complete orthonormal set in L₂(V), then the integral transformation (28) is nonsingular and has the unique inverse

K_-1(x, ξ) is called the reciprocal kernel associated with K(x, ξ) and represents the linear operator K^_1.

For hermitian kernels K(x, ξ) one has v_k(x) ≡ wk(x), μ_k = |λ_k|.

15.3-10. Volterra-type Integral Equations, (a) Let x and ξ be one-dimensional real variables. Then

1. Volterra's integral equation of the second kind

reduces to a Fredholm-type integral equation (16) over the interval V s (0, ∞) if one introduces the new kernel

2. Volterra's integral equation of the first kind

reduces by differentiation to the form (31) with

(b) The following example illustrates a method of dealing with a class of Volterra-type integral equations having unbounded kernels. To solve

multiply both sides by (y — x)^{α - 1} and integrate with respect to x from a to y; the resulting integral equation has a bounded kernel. It follows that

In the special case a = 1/2, Eq. (35) is known as AbeVs integral equation.

15.4. LINEAR BOUNDARY-VALUE PROBLEMS AND EIGENVALUE PROBLEMS INVOLVING DIFFERENTIAL EQUATIONS

15.4-1. Linear Boundary-value Problems: Problem Statement and Notation (see also Secs. 9.3-1, 10.3-4, 15.1-2, and 15.2-7). (a) A linear homogeneous function of a function φ(x) and its derivatives will be written as the product Lφ(x) of φ(x) and a linear differential operator L, such as d/dx or — ∇² — q(x^l, x², x³). One desires to find unknown functions φ(x) which satisfy a linear differential equation

throughout a given open interval or region V of points (x), subject to a set of N linear boundary conditions

to be satisfied on the boundary S of V; each B_iφ(x) is to be a linear homogeneous function of φ(x) and its derivatives.

(b) Notation. Volume Integrals and Inner Products (see also Secs. 4.6-12, 15.1-2, 15.2-1, and 16.10-10). In the one-dimensional case, x is a real variable, Eq. (la) is an ordinary differential equation, and V is a bounded or unbounded open interval whose end points x = a, x = b constitute the boundary S.

In the n-dimensional case, x stands for a “point” (x₁ x₂, . . . , x_n) in the n-dimensional space, and Eq. (la) is a partial differential equation. One assumes the possibility of introducing a volume element

in the manner of Secs. 6.2-3b and 16.10-10;is to be real and positive throughout V. Then is an n-dimensional volume

integral over V, and one can define the inner product of two functions u(x), υ(x) on V (Sec. 15.2-1) as

Note that depends on the coordinate system.

In particular, for n = 3, V is a bounded or unbounded open region of space with boundary surface S; S is to be a regular surface (Sec. 3.1-14). For n = 2, V is usually a plane region with (regular) boundary curve S.

15.4-2. Complementary Homogeneous Differential Equation and Boundary Conditions for a Linear Boundary-value Problem. Superposition Theorems (see also Secs. 9.3-1, 10.4-2, 14.3-1, and 15.2-7). With each linear boundary-value problem (1), one can associate a unique homogeneous complementary or reduced differential equation

and a unique ordered set of homogeneous “complementary boundary conditions”

The following important theorems may be applied in turn to relate solutions φ(x) of a given linear boundary-value problem (1) to functions satisfying simpler relations of the form (3).

1.The solution of every linear boundary-value problem (1) can be reduced to that of a linear boundary-value problem involving the same operator L and the homogeneous boundary conditions (3b) (see also Sec. 15.5-4).*

2.The most general function satisfying a linear differential equation (la) can be written as the sum of a particular solution of Eq. (la) and the most general solution of the complementary equation (3a).

* Write φ(x) = φ(x) + v(x), where v(x) is a conveniently chosen function satisfying the given boundary conditions (16). Then φ(x) is the solution of the linear boundary-value problem

Lv(x) is determined by the choice of v(x) but may contain 5-function terms; it is often possible to choose v(x) so that Lv(x) = 0 (see also Sec. 10.4-2).

3.A homogeneous linear differential equation is satisfied by every linear combination of solutions.

4.Let φ₁(x) and φ₂(x) satisfy the respective linear differential equations Lφ₁(x) = f₁(x) and Lφ₂(x) = f₂(x) with identical homogeneous linear boundary conditions B_iφ(x) = 0. Then αφ₁(x) + βφ₂(x) satisfies the differential equation Lφ(x) = αf₁(x) + βf₂ (x) and the given boundary conditions.

5.Let φ₁(x) and φ₂(x) satisfy the homogeneous linear differential equation Lφ(x) = 0 subject to the respective linear boundary conditions Biφ₁ = b₁_i(x) and Biφ₂(x) = b_2i(x). Then αφ₁{x) + βφ₂(x) satisfies the given differential equation and the boundary condition

15.4-3. Hermitian-conjugate and Adjoint Boundary-value Problems. Hermitian Operators (see also Secs. 14.4-3, 14.4-4, and 15.3-1b). (a) Given a homogeneous linear boundary-value problem (3) and the definition (2) of inner products, one defines the hermitian-conjugate boundary-value problem

by the condition

where u = u(x), v = v(x) is any pair of suitably differentiable functions such that u(x) satisfies the given boundary conditions (36), and v(x) satisfies the hermitian-conjugate boundary conditions (56). u and v (or v*) may then be said to represent vectors in adjoint vector spaces (Sec. 14.4-9).

Two linear second-order differential operators L and L † will be called hermitian-conjugate operators if and only if the function v*Lu — w(L†ν)* has the form of an

n-dimensional divergence (Table 16.10-1), where each

P^k can be a function of u_t v, u*_f v* and their first-order derivatives.* It is then pos-

sible to express the volume integral as an integral over the

boundary S; formulas of this type are known as generalized Green's formulas (see also Sec. 15.4-3c). Suitable sets of hermitian-conjugate boundary conditions B_iu = 0, B_i†ν = 0 {x in S) are now defined by the fact that the boundary integral vanishes, so that Eq. (6) holds.

In case of real functions and operators, hermitian-conjugate boundary-value problems, operators, and boundary conditions are usually known as mutually adjoint.

* The k's are superscripts, not exponents, as in Chaps. 6 and 16.

(b) Hermitian Operators. A differential operator L is hermitian (self-adjoint) if and only if

for every pair of suitably differentiable functions u = u(x), v = v(x) which satisfy identical homogeneous boundary conditions defining a linear manifold of functions. Hermitian-conjugate boundary-value problems with hermitian operators have identical boundary conditions and are thus identical (self-adjoint boundary-value problems).

(c) Special Cases. Real Sturm-Liouville Operators and Generalization of Green's Theorem (see also Secs. 5.6-1 and 15.5-4). In the one-dimensional case, Eqs. (3a) and (5a) are ordinary linear differential equations subject to given linear boundary conditions at the end points of an interval (a, b) = V. If inner products are defined by

(Sec. 15.2-la), one has, for real second-order differential operators,

P(x) is sometimes called the conjunct of u(x) and v(x) with respect to the operator L.

The condition a₁(x) ≡ a^′₀(x) makes L a self-adjoint Sturm-Liouville operator

(see also Sec. 15.4-8), and partial integration yields the generalized Green's formula

In the three-dimensional case, define inner products by Eq. (2), and define the operator V² in the manner of Tables 6.4-1 or 16.10-1. Then for any differentiable real function q = q(x^l, x², x³), the real differential operator

is self-adjoint and satisfies the generalized Green's formula

(see also Table 5.6-1). An analogous formula may be written for the two-dimensional case.

15.4-4. The Fredholm Alternative Theorem (see also Secs. 14.8-10, 15.3-7a, and 15.4-12). The linear boundary-value problem defined by the differential equation

with homogeneous boundary conditions

cannot have a unique solution φ(x) if the hermitian-conjugate (adjoint) homogeneous boundary-value problem (5) has a solution χ(x) other than _χ(x) ≡ 0. In this casej the given problem (13) can be solved only if f(x) is orthogonal to every χ(x), i.e., if Eq. (5) implies

If this condition is satisfied, then the given problem (13) has either no solution or an infinite number of solutions.

NOTE: In many applications, L is a hermitian operator, and the hermitian-conjugate problem (5) is identical with the given problem (13).

15.4-5. Eigenvalue Problems Involving Linear Differential Equations (see also Secs. 10.4-2c, 14.8-3, and 15.1-1). (a) For a given set of linear homogeneous boundary conditions, an eigenfunction (proper function, characteristic function) of the linear differential operator L is a solution (x), not identically zero in V, of the differential equation

where λ is a suitably determined constant called the eigenvalue (propervalue, characteristic value) of L associated with the eigenfunction (x).

(b) More general eigenvalue problems require one to find eigenfunc-tions (x) ≡ 0 and eigenvalues λ satisfying a linear differential equation

* In the one-dimensional case, dV is simply identical with dx.

subject to given linear homogeneous boundary conditions; B{x) is to be real and positive throughout V.

(c) If (x) is an eigenfunction associated with the eigenvalue X, the same is true for every function α(x)≠ 0. If ₁(x), ₂(x), . . . , _s(x) are eigenfunctions associated with λ, the same is true for every function

This theorem also applies to uniformly convergent infinite series of eigenfunctions.

Eigenfunctions associated with different eigenvalues are linearly independent (Secs. 1.9-3 and 9.3-2). An eigenvalue λ associated with exactly m > 1 linearly independent eigenfunctions is said to be ra-fold degenerate; m is the degree of degeneracy.

(d) The Spectrum of a Linear Eigenvalue Problem. Continuous Spectra and Improper Eigenfunctions (see also Secs. 14.8-3d and 15.4-12). For a given set of homogeneous linear boundary conditions and B(x) > 0, the spectrum of a linear eigenvalue problem (16) is the set of all complex numbers λ such that the differential equation

with a given normalizable “forcing function ” f(x) does not have a unique normalizable solution φ(x) subject to the given boundary conditions. The spectrum may comprise continuous and residual spectra (Sec. 14.8-3d) as well as a discrete spectrum defined by Eq. (16) with normalizable eigenfunctions (x). In the special case B{x) = 1, one speaks of the spectrum of the differential operator L.

If L is a hermitian operator and B{x) > 0, both the discrete and the continuous spectrum of the eigenvalue problem (16) are included in the approximate spectrum (Sec. 14.8-3d). One can often obtain the approximate spectrum by approximating the given eigenvalue problem with a sequence of eigenvalue problems yielding purely discrete spectra. In the course of such a limit process, the discrete eigenfunctions are replaced by a set of functions labeled by the continuously variable spectral parameter λ; such functions are known as improper eigenfunctions and will satisfy Eq. (16).

EXAMPLE: The ordinary differential equation with V ≡ (0, ∞)and the boundary conditions

yields the continuous spectrum 0 ≤ λ < ∞. This spectrum is approximated by the discrete spectrun of the eigenvalue problem

as a → ∞. The eigenfunctions of the latter problem approximate the improper eigenfunctions as a→ ∞ (transition from Fourier series to Fourier integral).

Similarly, the quantum-mechanical wave functions and energy levels of a particle in free space are approximated by those for particles confined to increasingly larger “boxes.”

15.4-6. Eigenvalues and Eigenfunctions of Hermitian Eigenvalue Problems. Complete Orthonormal Sets of Eigenfunctions (see also Secs. 14.8-4, 14.-8-7, 15.2-4, 15.3-3, and 15.4-3b). (a) If the operator L in Eq. (15) is hermitian, then

1.All spectral values λ are real.

2.Normalized eigenfunctions _i, _k corresponding to different eigenvalues are mutually orthogonal in the sense that

If L is a hermitian operator having a purely discrete spectrum, the following expansion theorem holds:

3.There exists an orthonormal set of eigenfunctions ₁(x), ₂(x), . . .permitting a series expansion

of every quadratically integrable function p(x) satisfying the boundary conditions of the eigenvalue problem and such that Lφ(x) exists for almost all x in V.

(b) The same theorems apply to the “generalized“ eigenvalue problem (16), where L is hermitian and B(x) > 0, provided that one redefines orthogonality and normalization in terms of the new inner product*

Thus Eq. (18a) is replaced by the more general expansion

and orthonormality of the eigenfunctions k(x) is denned by

These relations apply to the simpler eigenvalue problem (15) as a special case.

* In the one-dimensional case, dV is, again, simply identical with dx.

(c) For a hermitian eigenvalue problem with a (necessarily real) continuous spectrum admitting improper eigenfunctions (Sec. 15.4-5d), there exists a set of improper eigenfunctions (x, λ) such that

15.4-7. Hermitian Eigenvalue Problems as Variation Problems (see also Secs. 11.7-1 to 11.7-3, 14.8-8, 15.3-6b, 15.4-8b, 15.4-9b, and 15.4-10). (a) The eigenvalue problem (16) for a hermitian differential operator L with discrete eigenvalues λ1, λ2, . . . is equivalent to each of the following variation problems:*

1. Find functions (x) ≠ 0 in V which satisfy the given boundary conditions and reduce the variation (Secs. 11.5-1 and 11.5-2) of

to zero.

2. Find functions (x) which satisfy the given boundary conditions and reduce the variation of

to zero subject to the constraint

In each case, = k(x) yields the stationary value λ_k.

It is, then, possible to utilize the direct methods of the calculus of variations, notably the Rayleigh-Ritz method (Sec. 11.7-2) for the solution of eigenvalue problems involving ordinary or partial differential equations.

(b) Given a hermitian operator L with a discrete spectrum which includes at most a finite number of negative eigenvalues (Secs. 15.4-8 and 15.4-9), let the eigenvalues be arranged in increasing order, with an m-fold degenerate eigenvalue repeated m times, or λ₁ ≤ λ₂ ≤ . . . . The smallest eigenvalue λ₁ is the minimum value of Rayleigh's quotient (20) for an arbitrary “admissible” function (x), i.e., for an arbitrary normaliz-able {x) which satisfies the given boundary conditions and such that Rayleigh's quotient exists.

Similarly, the r^th eigenvalue λ^r in the above sequence will not exceed Ray-leigh's quotient for any “admissible” function φ(x) such that

* In the one-dimensional case, dV — dx.

for all eigenfunctions _k associated with λ₁, λ₂, . . . , λ_{r_i} (Courant's Minimax Principle).

15.4-8. One-dimensional Eigenvalue Problems of the Sturm-Liouville Type, (a) Consider a one-dimensional real variable x, and let V be the bounded interval (a, b). Then the most general real her-mitian second-order differential operator L has the form (9). The real differential equation

defines a self-adjoint eigenvalue problem if one adds either of the linear homogeneous boundary conditions

It will be assumed that p(x), q(x), and B{x) are differentiate in [a, b], that p(x) as well as B(x) is positive in [a, b], and that no eigenfunction corresponding to λ = 0 satisfies the given boundary conditions. These assumptions ensure that the eigenvalues λ are discrete and positive; they are nondegenerate for the boundary conditions (246). If the eigenvalues are arranged in increasing order λ₁ ≤ λ₂ ≤ . . . , then λ_n is asymptotically proportional to n² as n → ∞ (Sec. 4.4-3).

Various simplifying transformations and methods for solving homogeneous differential equations of the type (24) are treated in Secs. 9.3-4 to 9.3-10; see also Secs. 21.7-1 to 21.8-12. Eigenvalue problems of the Sturm-Liouville type are met, in particular, upon separation of variables in boundary-value problems involving linear partial differential equations (Secs. 10.4-3 to 10.4-9) and are of great importance in quantum mechanics. Note that every differential equation of the form

can be reduced to the form (24a) through multiplication by exp

(see also Sec. 9.3-8a). Hence the examples of Secs. 10.4-3 to 10.4-9 may be treated as Sturm-Liouville problems.

(b) The generalized Green's formula (10) applies; if one writes q(x) ≡ q^′(x) – q₂(x),

The integrals (26) have physical significance in many applications (see also Sec. 15.4-7).

(c) Generalizations and Related Problems. Related more general problems involve unbounded intervals V = (0, ∞) or V = (—∞, ∞) with boundary conditions which specify the asymptotic behavior of (x) at infinity or simply demand normaliz-ability. Again, one may admit singularities of p(x), q(x), and/or B(x) for x = a or x — b (Sec. 21.8-10). If the spectrum is purely discrete, λ_n is still asymptotically proportional to n² even if one admits singularities of p(x), q(x), or B(x) at the end points of the bounded interval (a, b).

Nonhomogeneous boundary-value problems involving the Sturm-Liouville operator L may be solved by the methods of Secs. 9.3-3, 9.3-4, 15.4-12, and 15.5-1.

15.4-9. Sturm-Liouville Problems Involving Second-order Partial Differential Equations (see also Sec. 15.4-3c). (a) Let x ≡ (x¹, x², . . . , xⁿ), and define inner products by Eq. (2) with dV ≡ dx¹ dx² . . . dxⁿ. Then the real partial differential equation

all differentiable on V and S, defines a self-adjoint eigenvalue problem for given homogeneous boundary conditions of the form If the given region V is bounded, the resulting eigenvalue spectrum is discrete and includes at most a finite number of negative eigenvalues. If the eigenvalues are arranged in a sequence λ₁ ≤ λ₂ ≤. . . , then X_n → ∞ as n → ∞.

(b) In the three-dimensional case, the same theorem applies to the eigenvalues of the differential equation

with boundary conditions of the form and B > 0 and q both differ- entiable on V and S. The generalized Green's formula (12) applies, and

The integral (28) has physical significance in many applications (see also Sec. 15.4-7 and Table 5.6-1).

15.4-10. Comparison Theorems (see also Secs. 5.6-1b, 14.8-9c, and 15.4-7). The following comparison theorems apply to ordinary or partial differential equations of the Sturm-Liouville types denned in Secs. 15.4-8 and 15.4-9. Given a differential equation (24a) or (27) and an interval or region V with boundary conditions (246) or (24c) (β/α ≥ 0),

1.An increase in p, increase in q, and/or decrease in B will surely not increase any eigenvalue λk; similarly, a decrease in p, decrease in q, and/or increase in B will not decrease any eigenvalue λc.

2.A reduction in the size of the given interval or region V will never decrease any eigenvalue λk generated by the boundary conditions

or decrease any eigenvalue λk generated by the boundary conditions

3.Each eigenvalue λk generated by the boundary conditions

with β α ≥ 0 is a nondecreasing function of β α in particular, the Neumann condition (29b) will not yield larger eigenvalues than the Dirichlet condition (29a).

A modification of the eigenvalue problem imposing constraints (acces sory conditions) on will not decrease any eigenvalue λk.

The comparison theorems are of special interest in vibration theory (effects of mass, stiffness, and geometry on natural frequencies).

EXAMPLES (see also Sec. 10.4-9): ″(x) = -λ(x) {vibrating string) and V²λ(x, y, z) = – λ (x_} y, z) (vibrating membrane).

15.4-11. Perturbation Methods for the Solution of Discrete Eigenvalue Problems. Given the eigenvalues k and the orthonormal eigenfunctions k of a her-mitian eigenvalue problem defined by

one desires to approximate the eigenvalues λk and the eigenfunctions k fo of the “perturbed“ hermitian problem

with unchanged boundary conditions; ∊L′ is to be a small perturbation term (∊<< 1). (a) Nondegenerate Case. Corresponding to each nondegenerate eigenvalue λ_i of the unperturbed problem, one has

(b) Degenerate Case. For each m-fold degenerate unperturbed eigenvalue A, with the m eigenfunctions ₁, ₂, . . . , _m, one may have up to m distinct eigenvalues of the perturbed operator. The corresponding values of are approxi-mated by the roots of the m^th-degree secular equation

and may or may not coincide; i.e., a perturbation may remove degenracies.

The eigenfunction or eigenfunctions corresponding to each value of λ = λ,- + λ

are approximated ty where the a* are obtained from

Note that no eigenfunctions other than the m eigenfunctions of the degenerate eigenvalue λ, affect the approximations to λ and given for the degenerate case; the approximation given for is a “zero-order“ approximation not proportional to e. See Ref. 15.18 for higher-order effects.

15.4-12. Solution of Boundary-value Problems by Eigenfunction Expansions (see also Secs. 10.4-2c, 14.8-10, 15.4-4, and 15.5-2; refer to Sec. 10.4-9 for examples), (a) A very important class of physical boundary-value problems (e.g., elastic vibrations, electromagnetic theory) involves a real linear ordinary or partial differential equation

subject to given homogeneous boundary conditions, where L is a her-mitian operator (Sec. 15.4-3b), and B(x) > 0 in V. If f(x) s 0 in V (no applied forces, currents, etc.), then Eq. (36) reduces to the complementary homogeneous differential equation

which can be satisfied only by proper or improper eigenfunctions (x)with spectral values λ. In the nonhomogeneous case (36) (e.g., forced oscillations), λ will be a given parameter.

Consider first that the eigenvalue problem (37) has a purely discrete spectrum of (not necessarily distinct) eigenvalues λ1, λ2, . . . associated with corresponding orthonormal eigenfunctions 1(x), 2(x), . . . (Sec. 15.4-6b). Assuming that the “forcing function“ can be expanded in the form

for almost all x in V, the desired solution of Eq. (36) is given by the“normal-mode expansion”

The series (39) defines the solution φ(x) uniquely at every point of continuity in V, provided that the given parameter λ does not equal an eigenvalue λ (resonance!). In the latter case, a solution exists only if f(x) is orthogonal to all eigenfunctions associated with λk, so that f_k = 0. One has then an infinite number of solutions given by the series (39) plus any linear combination of eigenfunctions associated with λ_k.

(b) If the eigenvalue problem (37) has a purely continuous spectrumD_A with improper eigenfunctions (x, λ) having the orthonormality property (196) (e.g., Sturm-Liouville operators with singularities in V,or V unbounded), one may attempt to expand the solution φ(x) as a “generalized Fourier integral “ over the (necessarily real) spectrum DA.One has then, for almost all x in V,

provided that D_A does not contain λ (see also Sec. 10.5-1, integral-transform methods).

If the eigenvalue problem (37) has both a discrete and a continuous spectrum, the solution will contain terms of the form (39) as well as an integral (40); both types of terms can be combined into a Stieltjes integral over the spectrum.

(c) The solution methods of Secs. 15.4-12a and b may apply even if the given operator L is not hermitian, provided that valid orthonormal- eigenfunction expansions can be shown to exist.

15.5. GREEN'S FUNCTIONS. RELATION OF BOUNDARY-VALUE PROBLEMS AND EIGENVALUE PROBLEMS TO INTEGRAL EQUATIONS

15.5-1. Green's Functions for Boundary-value Problems with Homogeneous Boundary Conditions (see also Secs. 9.3-3, 9.4-3, and 15.4-4; for examples, see Table 9.3-1, Sec. 15.6-6, and Sec. 15.6-9). (a) A linear boundary-value problem

expresses the given function f(x) as the result of a linear operation on an unknown function φ(x) subject to the given boundary conditions. If it is possible to write the corresponding inverse operation as a linear integral transformation (Sec. 15.3-1)

for every suitable f(x), the kernel G(x, ξ) is called the Green's function for the given boundary-value problem (1). G(x, ξ) must satisfy the given homogeneous boundary conditions (1b) together with

where δ(x, ξ) is the delta function for the specific coordinate system used (Sec. 21.9-7).

Equation (2) describes the solution φ(x) of the given boundary-value problem (1) as a superposition of elementary solutions G(x, ξ)f(ξ) having singularities at x = ξ. These elementary solutions can be interpreted as effects of impulse forces, point charges, etc., f(ξ)δ(x, ξ) at x = ξ (Secs. 9.4-3, 15.5-4, 15.6-6, and 15.6-9).

Green's functions can often be found directly by formal integration of the “ symbolic differential equation“ (36) subject to the given boundary conditions by the methods of Secs. 9.4-5, 10.5-1, or 15.4-12. See Table 9.3-3, Sec. 15.6-6, and Sec. 15.6-9 for examples of Green's functions.

(b) Modified Green's Functions (see also Sec. 15.4-4). A given boundary-value problem (1) will not admit a Green's function satisfying Eq. (3) if the hermitian-conjugate homogeneous problem

possesses a solution χ(x) other than χ(x) ≡ 0 (Sec. 15.4-4). In this case, it may still be possible to find a modified Green's function G(x, ξ) permitting the expansion (2) for suitable functions f(x) orthogonal to every χ{x). G(x, ξ) must satisfy the condition

* In the one-dimensional case, dV(ξ) ≡ dξ.

together with the given boundary conditions (1b), where the χ_ok(x) are any complete orthonormal set spanning all solutions of the problem (4). Any resulting solution (2) cannot possibly be unique (Sec. 15.4-4), but there exists at most one modified Green's function which satisfies the added orthogonality conditions

If L is a hermitian operator, then the χ_ok are simply its eigenfunctions for λ = 0.

(c) Green's functions for hermitian-conjugate boundary-value problems (Sec. 15.4-3a) are hermitian-conjugate kernels (Sec. 15.3-1b). For every Green's function G(x, ξ) associated with a hermitian operator L (Sec. 15.4-3b), G*, (ξ x) ≡ G(x, (ξ).

15.5-2. Relation of Boundary-value Problems and Eigenvalue Problems to Integral Equations. Green's Resolvent, (a) If there exists a Green's function G(x ξ) for the boundary-value problem (1), Eq. (2) implies that the more general boundary-value problem

Note that the integral equation takes the given boundary conditions into account. In the one-dimensional case, dV(ξ) ≡ dξ, and |g(ξ)| ≡ 1.

The integral-equation representation brings the theory of Secs. 15.3-1 to 15.3-9 and the numerical methods of Sec. 20.9-10 to bear on linear boundary-value problems and eigenvalue problems. In particular, one may employ the Neumann series (15.3-23) and analytic continuation to introduce a resolvent kernel (Sec. 15.3-7c) Γ(x, ξ; λ), so that the solution appears in the form

Γ(x, ξ; λ) is called Green's resolvent; it is the kernel of a linear integral transformation representing the resolvent operator (L – λ)^-1 (Sec. 14.8-3d). Γ(x, ξ; λ) can often be obtained by the methods of Sec. 15.3-8; but note carefully that K{xξ) is not necessarily a normalizable kernel. The set of singularities λ of Γ(x ξ; λ) is precisely the spectrum of the operator L (Sec. 15.4-5d), which may include a continuous spectrum. Specifically, the poles of Γ(x, ξ; λ) correspond to discrete eigenvalues of L, while branch points indicate the presence of a continuous spectrum (Ref. 15.2).

(b) Problems Involving Hermitian Operators. Eigenfunction Expansion of Green's Function (see also Secs. 15.3-3, 15.3-4, 15.4-6, and 15.4-12). In the important special cases where L is a hermitian operator and B(x) is real and positive in V, one may introduce a new unknown function

to replace Eq. (8) by a linear integral equation with hermitian kernel:

The hermitian eigenvalue problems

yield identical spectra; corresponding eigenfunctions(x) amd (x) are related by . If the eigenvalue spectrum is purely discrete, there exist complete sets of eigenfunctions i(x) and i(x) such that

In this case, K(x, ξ) is a normalizable kernel, and

which does not depend explicitly on B(x) or g(x). In the case of Sturm-Liouville problems with purely discrete spectra (Secs. 15.4-8a and 15.4-9), the series (14) converges absolutely and uniformly in V (see also Mercer's theorem, Sec. 15.3-4).

(c) If the given boundary-value problem (7) is expressed in terms of differential invariants (Sec. 16.10-7), then G(x, ξ), K(x, ξ), and K(x, ξ) are point functions invari ant with respect to changes in the coordinate system used.

15.5-3. Application of the Green's-function Method to an Initial-value Problem: The Generalized Diffusion Equation (see also Secs. 10.4-7s, 10.5-3, and 10.5-4). It is desired to find the solution Φ = Φ(x, t) of the initial-value problem

with given constant coefficients a², b and homogeneous boundary conditions Φ = 0 or ∂Φ/∂n = 0 on the boundary of a given n-dimensional region V of points (x), where n = 1, 2, or 3. Then

where the Green's function G(x, t; ξ, r) satisfies the given boundary conditions and

Given the eigenfunctions _k(x) and eigenvalues λ_k of the space form of the wave equation

subject to the given boundary conditions (Secs. 10.4-4 and 10.4-5b), one has

If V encompasses the entire space, then

where |r – p| is the distance between the points (x) = (r) and (ξ) = (p). The resulting solution (16) is known as the Poisson-integral solution of the diffusion problem.

15.5-4. The Green's-function Method for Nonhomogeneous Boundary Conditions (see also Sec. 15.4-2). (a) The solution Φ(x) of a three-dimensional linear boundary-value problem of the form

can often be written as a surface integral

for every given function b(x) integrable over S. G_s(x, ξ) must satisfy the given differential equation for x in V, ξ in S, and

Gs(x, ξ) is variously defined as a Green's function of the second kind or simply as a Green's function (see also Sec. 15.6-6). Analogous relations hold in the two-dimensional case (see also Sec. 15.6-9).

Linear boundary-value problems involving nonhomogeneous differential equations as well as nonhomogeneous boundary conditions can often be solved by superposition of a volume integral (2) and a surface integral (22).

(b) As shown in Sec. 15.4-2, every boundary-value problem (21) can be rewritten as a boundary-value problem of the type (1). Hence, Gs(x, ξ) can be related to the ordinary Green's function G(x, ξ) defined in the manner of Sec. 15.5-1 for the problem with “complementary” homogeneous boundary conditions. In particular, consider two-dimensional or three-dimensional boundary-value problems involving real self-adjoint differential equations of the form

where q = q(x) is a real differentiate function. Given a Green's function G(x, ξ) satisfying

for suitably given homogeneous boundary conditions (Sec. 15.5-1), Green's formula (15.4-12) yields

where ∂/∂v denotes normal differentiation with respect to ξ. Hence for boundary conditions of the form

the solution (22) requires

where G(x, ξ) satisfies Eq. (25) in V and vanishes on S.

For boundary conditions of the form

the solution (22) of the differential equation (24) requires one to use the “Neumann function”

where G(x, ξ) satisfies Eq. (25) in V, and ∂G(x, ξ)∂n = 0 for x in S.

Sections 15.6-6 and 15.6-9 show the application of these relations to the solution of true boundary-value problems for elliptic differential equations. Section 10.3-6 illus- trates a similar method for the solution of initial-value problems for hyperbolic differential equations (see also Sec. 10.3-5).

15.6. POTENTIAL THEORY

15.6-1. Introduction. Laplace's and Poisson's Differential Equations (see also Secs. 5.7-3, 10.4-3, and 10.4-5). Many important applications involve solutions Φ(r) of the linear partial differential equations

where Φ(r) and Q(r) are functions of position in a three-dimensional Euclidean space of points (r) ≡ (x,y,z) ≡ (x¹, x², x³), or in a two-dimensional Euclidean space points (r) ≡ (x, y) ≡ (x^l, x²). Φ(r) is most frequently interpreted as the potential of an irrotational vector field

due to a distribution of charge or mass such that V . F(r) = 4πQ(r) (Secs. 5.7-2, 5.7-3, and 15.6-5). The study of such potentials and, in particular, of solutions of Laplace's differential equation (1) is known as potential theory.

15.6-2. Three-dimensional Potential Theory: The Classical Boundary-value Problems, (a) The Dirichlet Problem. A bounded region V admitting a solution of the boundary-value problem

for any given continuous single-valued function b(r) is called a Dirichlet region; whenever a solution exists, it is necessarily unique. If V is an unbounded region, one must specify the asymptotic behavior of the solution at infinity, say Φ(r) = 0(l/r) as r →∞; the latter condition ensures uniqueness whenever a solution exists. Section 15.6-6d further discusses the existence of solutions .

The solution Φ(r) of the Dirichlet problem (3) yields a stationary value of the Dirichlet integral where Φ(r) is assumed to be twice continuously dif-ferentiable in V and S and to satisfy the given boundary conditions (see also Sec. 15.4-7a). Dirichlet problems are of particular importance in electrostatics (Ref. 15.6).

(b) The Neumann Problem. For the second classical boundary-value problem

where b(r) is a given continuous single-valued function, the existence of solutions requires(see also Gauss's theorem, Table 5.6-1). If V is an unbounded region, one requires Φ(r) = 0(l/r) and ∂Φ/∂n = 0(l/r²) as r→ ∞. The solution of a Neumann problem for a bounded region V is unique except for an additive constant.

Neumann problems occur, in particular, in connection with the flow of incompressible fluids (Ref. 15.6).

15.6-3. Kelvin's Inversion Theorem. IfΦ(r) is a solution of Laplace's differential equation in a region V inside the sphere |r – a| = R, then

is a solution in a corresponding region V outside the sphere, and conversely (Kelvin's Inversion Theorem). In terms of spherical coordinates r, ϑ, φ one may say that, ifΦ(r, ϑ, φ) is a solution for r < R, then is a solution for r, andconversely.

Hence, the so-called exterior boundary-value problem

for a bounded region V can be transformed into a corresponding boundary-value problem for the interior of an “inverted” region V through a transformation

15.6-4. Properties of Harmonic Functions, (a) Mean-value and Maximum-modulus Theorems. Solutions Φ(r) of Laplace's differential equation (1) are called harmonic functions. Every function Φ(r) harmonic in the open region V is analytic (Sec. 4.10-5b) and has harmonic derivatives of every order throughout V. Each value Φ(r1) equals the arithmetic mean (Sec. 4.6-3) of Φ(r) over the surface (and hence over the volume) of any sphere centered at r = ri and contained in V (Mean-value Theorem). Conversely, a continuous function Φ(r) is harmonic in every open region V where the above mean-value property holds.

A function Φ(r) harmonic throughout the bounded region V and its boundary S cannot have a maximum or minimum in the interior of V (Maximum-modulus Theorem, see also Sec. 7.3-5). Conversely, a continuous function without maxima and minima in the interior of a sphere is harmonic throughout the sphere. If Φ(r) is harmonic in V, continuous in V and S, and equal to zero on S, then Φ(r) ≡ 0 in V. If Φ(r) is harmonic in V, continuously differentiate in V and S, and ∂Φ/∂n = 0 on S, then Φ(r) is constant in F. The inversion theorem of Sec. 15.6-3 yields analogous theorems for the unbounded region V outside S.

(b) Harnack's Convergence Theorems. The following convergence theorems are of interest in connection with series approximations for harmonic functions. If the sequence s₀(r), s₁(r), S₂(r), . . . of functions all harmonic in V and continuous on the boundary SofV converges uniformly on S, then the sequence converges uniformly in V to afunction s(r) which is harmonic in V and such that on S. The sequencefor any partial derivative of s_n(r) converges uniformly to the corresponding partial derivative of s(r) in every closed subregion of V (Harnack's First Convergence Theorem).

Given a sequence of functions s₀(r), S₁(r), S₂(r), . . . harmonic in V and such that So(r) > S_o(r) ≥ ₁ ≥ s₂(r) > ≥. . . for all r in V, convergence at any point of V implies convergence throughout V and uniform convergence in every closed subregion of V; the limit is a harmonic function in V (Harnack's Second Convergence Theorem).

15.6-5. Solutions of Laplace's and Poisson's Equations as Potentials. (a) Potentials of Point Charges, Dipoles, and Multipoles. The following particular solutions of Laplace's equation have especially simple physical interpretations:

More generally, the potential of a multipole of order j at the origin( = 0) is

where the so-called multipole-moment components Q_ikl^(j) are constants determined by the j vectors pi, p2, . . . , P_j defining the multipole. In terms of spherical coordinates r, ϑ, φ

where Y_j(φ, ζ) is a spherical surface harmonic of degree j (Sec. 21.8-12). Multipoles of order 2 and 3 are respectively known as quadrupoles and octupoles.

(b) Potentials of Charge Distributions. Jump Relations. Other particular solutions of Laplace's equation are obtained by linear superposition or integration of simple and/or dipole potentials. Of particular interest are volume potentials of charge and dipole distributions,

and surface-distribution potentials for charges and dipoles (potentials of single- and double-layer distributions)

Potentials due to one-dimensional charge distributions (line integrals) are of interest mainly in two-dimensional potential theory (Sec. 15.6-7).

If the charge density Q(r) is bounded and integrable in V, the simple volume potential (12) and all its derivatives exist and are uniformly continuous for all r; the derivatives may be obtained by differentiation under the integral sign. The potential satisfies Poisson's equation (2) if Q(r) is bounded and continuously differentiable.

If the surface-density functions σ(r) and p(r) are twice differentiable on S,

1. The single-layer potential (14) is continuous at every regular point r₀ of the surface S. The same is true for the directional derivative ∂Φ/∂t in any given direction tangent to S at r₀; but the directional derivative ∂Φ/∂n along the normal of S at r₀ satisfies the jump relation

where the subscripts + and – indicate the respective unilateral limits as r → r₀ on the positive-normal side of S and on the negative-normal side of S.

2. The double-layer potential (15) and its tangential derivatives satisfy the jump relations

at every regular point r₀ of S, while the normal derivative ∂Φ/∂n is continuous.

NOTE: In the special case p(r) = p = constant, the double-layer potential (15) equals p times the solid angle subtended by S at the point (r); this angle is taken to be positive if (r) is on the side of the positive normals to S. For a closed surface S such a potential equals –4πpif (r) is inside S, and 0 if (r) is outside S.

(c) Multipole Expansion and Gauss's Theorem. Consider the potential Φ(r) due to any combination of charge distributions confined to a bounded sphere |r| ≤R; let QT be the finite total charge. Φ(r) will be a linear combination of potentials of the types (12) to (15); for |r| > R, one can expand Φ(r) as a Taylor series (5.5-4) of terms (10), or as a series of spherical harmonics (11). For sufficiently large r, the potential is thus successively approximated by the potential of a point charge QT at the origin, by a point-charge potential plus a dipole potential, etc. (multipole expan sion). For every regular surface enclosing the entire charge distribution, Gauss's theorem (Table 5.6-1) takes the special form

(d) General Solutions of Laplace's and Poisson's Equations as Potentials. Let V be a singly connected, bounded or unbounded three- dimensional region with regular boundary surface S, let Φ(r) be twice continuously differentiable in V and continuously differentiate on S, and let Φ(r) = 0(l/r) as r → ∞. Then Green's theorem (Table 5.6-1 or Sec. 15.4-3c) permits one to represent Φ(r) in the form

where the operator ∇_p implies differentiations with respect to the components of p; note that Equation (20) expresses every solution Φ(r) of Poisson's differential equation (2) as a potential due to three distributions:

1. A single-layer surface distribution (14) of density

2. A double-layer surface distribution (15) of density

3. A volume distribution (12) of density

The last potential vanishes if Φ(r) satisfies Laplace's differential equation (1) in V.

NOTE: The expression (20a) vanishes if r is outside V and S, and equals Φ(r)/2 for r in S.

15.6-6. Solution of Three-dimensional Boundary-value Problems by Green's Functions. The Greene-function methods of Secs. 15.5-1 and 15.5-4 express the solution Φ(r) of the Dirichlet problem (3) and the Neumann problem (4) for suitable regions V as surface integrals

Section 15.5-4b relates each “surface” Green's function Gs(r, 9) simply to the ordinary Green's function G(r, 9) which yields the solution

of Poisson's differential equation (2) subject to “complementary“ homogeneous Dirichlet or Neumann conditions. Superposition of Eqs. (21) and (22) yields solutions of Poisson's equation with given boundary values b(9) of Φ or ∂Φ/∂n (Sec. 15.4-2). Note G(g, r) ≡ G(r, 9) (Sec. 15.5-1). The Green's functions are easily found in the following special cases; note that the positive surface normals point outward.

(a) Green's Function for the Entire Space. If V is the entire three-dimensional space, Eq. (20) yields the unique solution (22) of Poisson's equation subject to the “boundary conditions” Φ(r) = 0(1/r) as r → ∞. The required Green's function is

(b) Infinite Plane Boundary with Dirichlet Conditions. Image Charges. If V is the half-space z > 0, the Green's functions for Dirichlet conditions are

where the point (9) = (ξ, n, – ζ) is the reflected image of (9) ≡ (ξ, n, ζ) in the boundary plane. The solution (22) can be interpreted as a volume potential due to the given charges and induced image charges; equation (21) expresses the effect of nonhomogeneous Dirichlet conditions as the potential of a double layer on the boundary.

(c) Sphere with Dirichlet Conditions. Poisson's Integral Formulas. If V is the interior of a sphere |r| = R, the Green's functions for Dirichlet conditions are

where γ is the angle between r and or

if the spherical coordinates of the points (r) and (p) are respectively denoted by r, ∂, φ and φ, ∂′, φ′.The second term in Eq. (26) may be regarded as the effect of an induced image charge located with the aid of Kelvin's inversion theorem (Sec. 15.6-3). The solution (21) of the Dirichlet problem

becomes

which may again be interpreted as a double-layer potential. The expression (27) can be expanded in spherical surface harmonics with the aid of Eq. (21.8-68) (multipole expansion, Sec. 15.6-5c; see also Sec. 10.4-9).

Equations (26) to (29) yield solutions in the interior of the sphere (r <R). If V is the exterior of the sphere (r > R), G(r, ) is still given by Eq. (26), but now, so that the signs on the right sides of Eqs. (27) and (29) must be reversed.

(d) An Existence Theorem. A Green's function G(r, g) for Poisson's equation with Dirichlet conditions—and hence a solution of the Dirichlet problem (3) with reasonable boundary values—exists for every region V bounded by a finite number of regular surface elements such that every boundary point can be the vertex of a tetrahedron outside V.

15.6-7. Two-dimensional Potential Theory. Logarithmic Potentials. (a) In many three-dimensional potential problems Φ(x, y, z) is independent of the z coordinate, and V is a right-cylindrical region whose boundary surface S intersects the xy plane in a boundary curve C. V is then represented by a plane region D bounded by C (e.g., potentials and flows about infinite cylinders, plane boundaries). Such problems form the subject matter of two-dimensional potential theory. The divergence theorem and Green's theorem take the form (4.6-33); one may use rectangular cartesian coordinates x, y, or plane polar coordinates r, φ to write Laplace's and Poisson's equations as

with Φ = Φ(r), Q ≡ = Q(r), r ≡ (x,y) ≡ (r,φ)Theorems entirely analogous to those of Sec. 15.6-4 apply to two-dimensional harmonic functions; it is only necessary to substitute circles for spheres in each mean-value theorem. Kelvin's inversion formula (5) holds for every circle |r – a| = R thus, if Φ(r, φ is a solution for r < R, the same is true for ,and conversely.

(b) The two-dimensional Laplace equation (30) has the elementary particular solution

which describes the potential of a uniformly charged straight line in the z direction. The function (32) takes the role of the function (8) in the two-dimensional theory, so that the potentials (12), (14), and (15) of Sec. 15.6-5b are replaced by corresponding logarithmic potentials

Equation (20) is replaced by

15.6-8. Two-dimensional Potential Theory: Conjugate Harmonic Functions (see also Secs. 7.3-2 and 7.3-3). (a) The xy plane may be regarded as a complex-number plane with points z = x + iy. A pair of (necessarily harmonic) functions Φ(x, y), φ(x, y) are harmonic conjugate functions in a region D of the plane if and only if

is an analytic function of z = x + iy in D. Harmonic conjugate functions are related by the Cauchy-Riemann equations

and define each other uniquely throughout D, except for arbitary additive constants. Given a function Φ(x, y) harmonic in D, one obtains (x, y) as the line integral

where x₀ and y_o are arbitrary constants, and the path of integration is located in D. If Ψ(x, y) is given, one has

The curves Φ(x,y) = constant and Ψ(x,y) = constant are mutually orthogonal families. These curves have important physical interpretations (equipotential lines and gradient lines in electrostatics, lines of constant velocity potential and streamlines for incompressible flow). Ξ is often called the complex potential.

(b) Every transformation

which is analytic and such that dz/dz ≠ 0 in V (conformal mapping, Sec. 7.9-1) transforms the harmonic conjugates Φ(x, y) Ψ(x, y) into a new pair of harmonic conjugates with mutually orthogonal contour lines. This theorem permits one to simplify boundaries and contour lines by conformal mapping (see also Sec. 15.6-9).

such that Ψ(x, y) and its derivatives are continuous on C as well as in D. Then the complex conjugate (39b) of Ψ(x, y) is the solution of the Dirichlet problem

The solution Φ(x, y) of the Dirichlet problem (42) similarly yields the solution (39a) of the Neumann problem (41), provided that

so that Gauss's theorem is satisfied (see also Table 5.6-1 and Sec. 15.6-5c).

15.6-9. Solution of Two-dimensional Boundary-value Problems. Green's Functions and Conformal Mapping (see also Secs. 15.5-1, 15.5-4, and 15.6-6). The Green's-function methods of Secs. 15.5-1 and 15.5-3 express the solution Φ(r) of Poisson's differential equation (31) with homogeneous linear boundary conditions in the form

and the solution Φ(r) of Laplace's differential equation (30) with given boundary values b(r) of Φ or ∂Φ/∂n as

Solutions of Poisson's equation subject to nonhomogeneous linear boundary conditions may be obtained by superposition of suitable integrals (45) and (46). Note G(r, ) = G(, r). The Green's functions are easily found in the following special cases.

(a) Green's Function for the Entire Plane. If D is the entire plane, Eq. (36) yields the unique solution (45) of Poisson's equation subject to the “boundary conditions“ Φ(r) → 0 as r → ∞. The required

Green's function is

(b) Half Plane with Dirichlet Conditions. If D is the half plane x > 0, the Green's functions for Dirichlet conditions are

where the point is the reflected image of in the boundary line.

(c) Circle with Dirichlet Conditions. Poisson's Integral Formulas. If D is the interior of a circle r = R, the Green's functions for Dirichlet conditions are

where p, φ^′ are the polar coordinates of the point The solution (45) of the Dirichlet problem

becomes

which may be expanded as a Fourier series in p.

Equations (50) to (53) yield solutions in the interior of the circle (r < R). If D is the exterior of the circle (r > R), G(r, ) is still given by Eq. (50), but now so that the signs on the right sides of Eqs. (51) and (53) must be reversed.

(d) Existence of Green's Functions and Conformal Mapping.

Section 15.6-9c and Riemann's mapping theorem (Sec. 7.10-1) imply the existence of a Green's function (and hence of a solution of the Dirichlet problem) for any region D which, together with its boundary C, can be mapped conformally onto the unit circle. More specifically, let w = w(z) be the analytic function mapping the points z = x + iy of D and C onto the unit circle so that the point z = ζ = ξ + in of D is transformed into the origin, or

Then the Dirichlet problem for the region D admits the Green's function

15.6-10. Extension of the Theory to More General Differential Equations. Retarded and Advanced Potentials (see also Sec. 10.4-4).

The theory of Secs. 15.6-5 to 15.6-7 and 15.6-9 attempts to construct solutions of Laplace's and Poisson's differential equations by superposition of simple and dipole potentials. The theory is readily generalized to deal with the more general differential equations

which include the space form of the wave equation k real, Sec. 10.4-4) and the space form of the Klein-Gordon equation used in nuclear physics (k =ik). The differential equation (57) is of the type discussed in Sec. 15.5-4b.

(a) Three-dimensional Case. The three-dimensional equation (56) admits the elementary particular solution

For real k the positive sign corresponds to outgoing waves, and the negative sign to incoming waves; for imaginary k = ik, only negative exponents –|K| are of general interest. Substitution of the appropriate expression (58) for in Eqs. (8) to (15), (23), and (24) yields solutions of the differential equation (56) or (57) instead of the corresponding solutions of Laplace's or Poisson's equation.

The resulting solutions of the time-dependent wave equation (Sec. 10.4-8) are special cases of retarded potentials [positive sign in Eq. (58)] and advanced potentials [negative sign in Eq. (58)].

In particular, if Φ(r) is a twice continuously differentiate solution of the homogeneous differential equation (56), Eq. (20) is replaced by Helmholtz's theorem

(b) Two-dimensional Case. In the case of two-dimensional differential equations of the form(56) or (57), the elementary particular solution (32) is replaced by

where the H₀^(r)(z) are Hankel functions (Sec. 21.8-1).

Functions of a complex variable Chap. 7

The Laplace transformation Chap. 8

Ordinary differential equations Chap. 9

differential equations Chap. 10

Calculus of variations Chap. 11

Linear vector spaces, linear operators Chap. 14

Numerical solutions Chap. 20

Special transcendental functions Chap. 21

15.7-2. References and Bibliography (see also Secs. 4.12-2, 10.6-2, 12.9-2, and 14.11-2).

15.1. Akhiezer, N. I.: Theory of Approximation, Ungar, New York, 1956.

15.2. and I. M. Glazman: Theory of Linear Operators in Hilbert Space, Ungar, New York, 1963.

15.3 Banach, S.: Theorie des Operations Lineaires, Chelsea, New York, 1933.

15.4 Berberian, S. K.: Introduction to Hilbert Space, Oxford, Fair Lawn, N.J., 1963.

15.5 Bochner, S.: Lectures on Fourier Integrals, Princeton, Princeton, N.J., 1959.

15.6 Courant, R., and D. Hilbert: Methods of Mathematical Physics, rev. ed., Wiley, New York, 1953/66.

15.7 Dieudonne, J. A.: Foundations of Modern Analysis, Academic, New York, 1960.

15.8 Dunford, N., and J. T. Schwartz: Linear Operators, Interscience, New York, 1964.

15.9 Edwards, R. E.: Functional Analysis, Holt, New York, 1965.

15.10 Feshbach, H., and P. M. Morse: Methods of Theoretical Physics (2 vols.), McGraw-Hill, New York, 1953.

15.11 Friedman, B.: Principles and Techniques of Applied Mathematics, Wiley, New York, 1956.

15.12. : Generalized Functions and Partial Differential Equations, Prentice- Hall, Englewood Cliffs, N.J., 1956.

15.13 Halmos, P. R.: Introduction to Hilbert Space and the Theory of Spectral Multiplicity, Chelsea, New York, 1957.

15.14 Kellogg, O. D.: Foundations of Potential Theory, Ungar, New York, 1943.

15.15 Kolmogorov, A., and S. V. Fomin: Elements of the Theory of Functions and Functional Analysis (2 vols.), Graylock, New York, 1957/61.

15.16 Lanczos, C: Linear Differential Operators, Van Nostrand, Princeton, N.J., 1961.

15.17 Liusternik, L. A., and V. J. Sobolev: Elements of Functional Analysis, Ungar, New Yo.k, 1961.

15.18 Lorch, E. R.: Spectral Theory, Oxford, Fair Lawn, N.J., 1962.

15.19 Madelung, E.: Die mathematischen Hilfsmittel des Physikers, 7th ed., Springer, Berlin, 1964.

15.20 Mikhlin, S. G.: Integral Equations, Pergamon, New York, 1957.

15.21 Riesz, I., and B. Nagy: Functional Analysis, Ungar, New York, 1955.

15.22 Vulikh, B. Z.: Functional Analysis for Scientists and Technologists, Pergamon, New York, 1963.

See also the articles by F. Schlogl and J. Sneddon in vol. I of the Handbuch der Physik, Springer, Berlin, 1956, and the references for Chap. 10).