Analytical continuation of two-dimensional wave fields

1. Introduction

In this paper we study two-dimensional diffraction problems for the Helmholtz equation belonging to a special class: namely those that can be reformulated as problems of propagation on branched surfaces with finitely many sheets. At least two classical canonical diffraction problems belong to this class: the Sommerfeld problem of diffraction by a half-line with ideal boundary conditions, and the problem of diffraction by an ideal segment. There are also some other important problems belonging to this class, they are listed in appendix A. The branched surface for such a problem is referred to as a Sommerfeld surface and denoted by S. This surface has several sheets over the real Cartesian plane (x₁, x₂), and these sheets are connected at several branch points.

The solution of the corresponding problem is denoted by u(x₁, x₂) and is assumed to be known on S. We consider the possibility to continue the solution u into the complex domain of the coordinates $(x_1,x_2)\in {\mathbb{C}}^2$. Namely, we are looking for a function u_c(x₁, x₂) which is analytical almost everywhere, obeys the complex Helmholtz equation, and is equal to u(x₁, x₂) for real (x₁, x₂).

It is shown in the paper that such a continuation can be constructed using Green’s third identity. The integration contours used involve rather complicated loops drawn on S, referred to as double-eight or Pochhammer contours. The integrand comprises the function u on S, a complexified kernel and their first derivatives.

The analytical continuation u_c is a branched (i.e. multi-valued) function in ${\mathbb{C}}^2$. Its branch set T is the union of the complex characteristics of the Helmholtz equation passing through the branch points of S. At each point of ${\mathbb{C}}^2$ not belonging to the branch set T there exists an infinite number of branches of u_c, i.e. infinitely many possible values of u_c in the neighbourhood of this point. However, we prove here that these branches have a finite basis, such that any branch can be expressed as a linear combination of a finite number of basis functions with integer coefficients.

This property of the analytical continuation is an important property of the initial (real) diffraction problem. Namely, it indicates that one can build the so-called coordinate equations, which are ‘multidimensional ordinary differential equations’ [1–3]. Thus, a PDE becomes effectively solved as a finite set of ODEs.

To emphasize the non-triviality of the statements proven in this paper and the importance of studying the analytical continuation of the field, we can say that we have not yet succeeded in generalizing the results to the case of three-dimensional diffraction problems. It is possible to build a three-dimensional analogue of Sommerfeld surfaces (for example, it can be done for the ideal quarter-plane diffraction problem), but at the moment it does not seem possible to show that the analytical continuation possesses a finite basis. Thus, a generalization of the coordinate equations seems not to be possible for three-dimensional problems. In the future, with the quarter-plane problem in mind, we wish to extend these results to wave propagation problems involving the Laplace–Beltrami operator defined on the surface of a sphere with a cut (see e.g. [4–6] for more detail on the links between Laplace–Beltrami and the quarter-plane problem).

The ideas behind this work were inspired in part by our recent investigations of applications of multi-variable complex analysis to diffraction problems [7–11], in part by our work on coordinate equations [1–5] and in part by the work of Sternin and his co-authors [12–14], in which the analytical continuation of wave fields is considered.

In [12–14], the practical problems of finding minimal configurations of sources producing certain fields, or continuation of fields through complicated boundaries, are solved.

Different techniques can be used to study the analytical continuation of a wave field. We are using Green’s theorem as in [12,15]. Alternatives include the use of Radon transforms [12,14], series representations or Schwarz’s reflection principle.

The rest of the paper is organized as follows. In §2 we define the concept of a Sommerfeld surface and introduce the notion of diffraction problems on such an object. In §3, we specify what we mean by the analytical continuation of a wave field u to ${\mathbb{C}}^2$, and discuss the notion of branching of functions of two complex variables. In §4, we give an integral representation that enables us to analytically continue u from a point in S to a point in ${\mathbb{C}}^2$. The obtained function u_c is multi-valued in ${\mathbb{C}}^2$, and, in §5, we study in detail its branching structure and specify all its possible branches by means of Green’s integrals involving double-eight contours. In particular, we show that there exists a finite basis of functions such that any branch of the analytical continuation can be expressed as a linear combination of these basis functions with integer coefficients. In §6, we apply the general theory developed thus far to the specific problem of diffraction by an ideal strip, showing that, in this case, the number of basis functions can be reduced to four; we describe explicitly and constructively, via some matrix algebra, all possible branches of the analytical continuation. Finally, in §7, still for the strip problem, we show that our results imply the existence of the so-called coordinate equations, effectively reducing the diffraction problem to a set of two multi-dimensional ODEs.

2. A diffraction problem on a real Sommerfeld surface

Let us start by defining more precisely the concept of a Sommerfeld surface. Take M copies of the plane ${\mathbb{R}}^2$ (called sheets of the surface), each equipped with the Cartesian coordinates (x₁, x₂). Let there exist N affixes of branch points $P_1,\dots ,P_N\in {\mathbb{R}}^2$ with coordinates $(X_1^{(j)},X_2^{(j)})$, j = 1, …, N. Consider a set of non-intersecting cuts on each sheet, connecting the points P_j with each other or with infinity. Finally, let the sides of the cuts be connected (‘glued’) to each other according to an arbitrary scheme. The connection of the sides should obey the following rules: (a) only points having the same coordinates (x₁, x₂) can be glued to each other; (b) one can glue a single ‘left’ side of a cut to a single ‘right’ side of this cut on another sheet; and (c) a side of a cut should be glued to a side of another cut as a whole.

Upon allowing spurious cuts, i.e. cuts glued to themselves, it is possible for the set of cuts to be the same on all sheets. The result of assembling the sheets is a Sommerfeld surface denoted by S. We assume everywhere that N and M are finite integers. Two examples of Sommerfeld surfaces are shown in figure 1.

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The concept of Sommerfeld surfaces is naturally very close to that of Riemann surfaces of analytic functions of a complex variable. For example, the Sommerfeld surfaces shown in figure 1 can be treated as the Riemann surfaces of the functions $\sqrt{x_1+i{x}_2}$ and $\sqrt{{(x_1+i{x}_2)}^2-a^2}$, respectively. However, we prefer to refer to them as Sommerfeld surfaces since the coordinates x₁ and x₂ are real here, and since we would like to avoid confusion with the complex context that will be developed below.

Sommerfeld surfaces emerge naturally when the reflection method is applied to a two-dimensional diffraction problem with straight ideal boundaries. For example, the surfaces shown in figure 1 help one to solve the classical Sommerfeld problem of diffraction by a Dirichlet half-line [16] and the problem of diffraction by a Dirichlet segment [1]. A connection between the diffraction problems and the Sommerfeld surfaces is discussed in more details in appendix A, where the class of scatterers leading to finite-sheeted Sommerfeld surfaces is described.

There exists a natural projection ψ of a Sommerfeld surface S to ${\mathbb{R}}^2$. For any small neighbourhood $U\subset {\mathbb{R}}^2$ not including any of the branch points P_j the pre-image ψ⁻¹ (U) is a set of M samples of U, as illustrated in figure 2.

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Let u be a continuous single-valued function on some Sommerfeld surface S (thus, u is generally an M-valued function on ${\mathbb{R}}^2$). For any neighbourhood $U\subset {\mathbb{R}}^2\mathrm{\setminus }\{P_1,\dots ,P_N\}$, let u obey the (real) Helmholtz equation

on each sample of ψ⁻¹ (U). The wavenumber parameter $\mathcal{k}$ is chosen to have a positive real part and a vanishingly small positive imaginary part mimicking damping of waves.

Let u also obey the Meixner condition at the branch points P_j. The Meixner condition characterizes the local finiteness of the energy-type integral

near a branch point. In particular, it guarantees the absence of sources at the branch points of S.

A function u obeying the equation (2.1) in the sense explained above and the Meixner condition will be referred to as a function obeying the Helmholtz equation on S.

Let us now formulate a diffraction problem on a Sommerfeld surface S: find a function u obeying the Helmholtz equation on S such that, on the physical sheet, u can be represented as a sum

is a known plane wave incident field. Here φⁱⁿ is the angle of incidence, and an appropriate radiation condition is imposed. Neither uⁱⁿ nor u^sc are continuous, but their sum is. The field u is then constructed on the other sheets of S by means of the reflection principles as explained in appendix A. The resulting field u satisfies the Helmholtz equation on S and is regular everywhere apart from a finite number of branch points.

In the rest of the paper, we assume that the existence and uniqueness theorem is proven for the chosen S and $\mathcal{k}$ and that the field u is fully known on S.

3. Analytical continuation of the field and its branching

Our aim is to build an analytical continuation u_c(x₁, x₂) of the solution u(x₁, x₂) of a certain diffraction problem on a Sommerfeld surface S. The continuation has the following sense.

Let x₁ and x₂ be complex variables, i.e. $(x_1,x_2)\in {\mathbb{C}}^2$. Naturally, ${\mathbb{C}}^2$ is a space of real dimension 4. Let $T\subset {\mathbb{C}}^2$ be a singularity set (built below), which is a union of several manifolds of real dimension 2. Let T be such that the intersection of T with the real plane ${\mathbb{R}}^2\subset {\mathbb{C}}^2$ is the set of branch points P₁, …, P_N.

The continuation u_c should obey three conditions.

—	The continuation uc(x1, x2) should be a multi-valued analytical function on ${\mathbb{C}}^2\mathrm{\setminus }T$. Each branch of uc in any small domain $U\subset {\mathbb{C}}^2$ not intersecting with T should obey the Cauchy–Riemann conditions $${\overline{\mathrm{\partial }}}_{x_{\ell }}{u}_{\text{c}}=0,\text{where}\hspace{0.17em}{\overline{\mathrm{\partial }}}_{x_{\ell }}\equiv \frac{1}{2}(\frac{\mathrm{\partial }}{\mathrm{\partial }\text{Re}[x_{\ell }]}+i\frac{\mathrm{\partial }}{\mathrm{\partial }\text{Im}[x_{\ell }]}),\ell \in \{1,2\}.$$ 3.1
—	The continuation uc(x1, x2) should obey the complex Helmholtz equation in ${\mathbb{C}}^2\mathrm{\setminus }T$: $$({\mathrm{\partial }}_{x_1}^2+{\mathrm{\partial }}_{x_2}^2+{\mathcal{k}}^2)u_{\text{c}}=0,\text{where}\hspace{0.17em}{\mathrm{\partial }}_{x_{\ell }}\equiv \frac{1}{2}(\frac{\mathrm{\partial }}{\mathrm{\partial }\text{Re}[x_{\ell }]}-i\frac{\mathrm{\partial }}{\mathrm{\partial }\text{Im}[x_{\ell }]}),\ell \in \{1,2\}.$$ 3.2 Note that the notation ${\mathrm{\partial }}_{x_{\ell }}$ is used both in the real (2.1) and in the complex (3.2) context. However one can see that if the Cauchy–Riemann conditions are valid, the complex derivative gives just the same result as the real one.1
—	When considering the restriction of uc(x1, x2) onto ${\mathbb{R}}^2$, over each point of ${\mathbb{R}}^2\mathrm{\setminus }\{P_1,\dots P_N\}$, there should exist M branches of uc equal to the values of u on S.

Let us describe, without a proof, the structure of the singularity set T and the branching structure of u_c. Later on, we shall build u_c explicitly, and one will be able to check the correctness of these statements.

The branch points P_j are singularities of the field u on the real plane. According to the general theory of partial differential equations, the singularities propagate along the characteristics of the PDE (the Helmholtz equation here). Thus, it is natural to expect that T is the union of the characteristics passing through the points $P_j=(X_1^{(j)},X_2^{(j)})\in {\mathbb{R}}^2$. Since the Helmholtz equation is elliptic, these characteristics are complex. They are given for j ∈ 1, …, N by

and One can see that $L_{1,2}^{(j)}$ are complex lines having real dimension 2. We will hence refer to them as 2-lines. Their intersection with ${\mathbb{R}}^2$ is the set of the points P_j, i.e. $L_1^{(j)}\cap L_2^{(j)}=P_j$. It is interesting to note that topologically, the 2-lines $L_{1,2}^{(j)}$ are similar to the complex rays considered in [17], in the sense that they are both two-dimensional manifolds embedded in ${\mathbb{C}}^2$, though the main difference between the two is that complex rays are complexified characteristics of the eikonal equation, while the branch 2-lines here are the characteristics of the Helmholtz equation.

We demonstrate below that the 2-lines $L_{1,2}^{(j)}$ are, generally, branch 2-lines of u_c. The branching of functions of several complex variables is not a well-known matter, thus, we should explain what it means. Consider for example a small neighbourhood $U\subset {\mathbb{C}}^2$ of a point on $L_1^{(1)}$, which is not a crossing point of two such lines. Note that the complex variable

is a coordinate transversal to $L_1^{(1)}$. The 2-line $L_1^{(1)}$ corresponds to z₁ = 0. The complex variable is then a coordinate tangential to $L_1^{(1)}$.

Let A be some point in U. Consider a path (oriented contour) σ in U starting and ending at A, and having no intersections with $L_1^{(1)}$. Such a contour, called a bypass of $L_1^{(1)}$, can be projected onto the variable z₁. Denote this projection by σ′. One can continue u_c along σ and obtain the branch u_c(A ;σ). Branches can be indexed by an integer p, which is the winding number of σ′ about zero. If for some σ′ having winding number p

for any such continuation (here we consider the smallest possible p having this property), then the branch line $L_1^{(1)}$ has order of branching equal to p. If there is no such p, the branching is said to be logarithmic.

Thus, generally speaking, the branching of a function of several complex variables is similar to that of a single variable, and it is convenient to study this branching using a transversal complex coordinate. To provide the existence of such a transversal variable, the branch set should be a set (a complex manifold) of complex codimension 1.

For j, k ∈ {1, …N}, the 2-lines $L_1^{(j)}$ and $L_2^{(k)}$ intersect at a single point. For example, if j = k, this point is P_j, while for j ≠ k, this intersection point does not belong to ${\mathbb{R}}^2$. The branching of u_c near each crossing point has a property that does not occur in the one-dimensional complex case: the bypasses about $L_1^{(j)}$ and $L_2^{(k)}$ commute.

Let us prove this in the case j = k by considering a small neighbourhood $U\subset {\mathbb{C}}^2$ of the point $P_j=L_1^{(j)}\cap L_2^{(j)}$. The case j ≠ k is similar. Introduce the local coordinates

which are transversal variables to $L_1^{(j)}$ and $L_2^{(j)}$, respectively. Take a point $A\in U\mathrm{\setminus }(L_1^{(j)}\cup L_2^{(j)})$ and a path σ in $U\mathrm{\setminus }(L_1^{(j)}\cup L_2^{(j)})$ starting and ending at A. Consider the projections σ₁ and σ₂ of σ onto the complex planes of z₁ and z₂, respectively.

Assume that the path σ is parametrized by a real parameter τ ∈ [0, 1], i.e.

or, in the new coordinates, The path σ can be deformed homotopically into a path ${\sigma }^{\ast }$ defined by for some small parameter ϵ. The projection of ${\sigma }^{\ast }$ onto z₁ (resp. z₂) is a small circle ${\sigma }_1^{\ast }$ (resp. ${\sigma }_2^{\ast }$) of radius ϵ turning (possibly many times) around the origin. Therefore, ${\sigma }^{\ast }$ lies on a torus (product of two circles), for which ${\sigma }_1^{\ast }$ and ${\sigma }_2^{\ast }$ are strictly longitudinal and latitudinal paths, respectively. Thus these loops commute (this comes from the fact that the fundamental group of the torus is Abelian). Therefore, the path σ can be homotopically deformed into the concatenations σ = σ₁σ₂ = σ₂σ₁, where the path σ₁ occurs for fixed z₂, and σ₂ occurs for fixed z₁.

4. Integral representation of the analytical continuation

Here we present the technique for analytical continuation of the wave field using Green’s third identity as in e.g. [12,15].

Let $U\subset (S\mathrm{\setminus }{\cup }_j{P}_j)$ be a small neighbourhood of a point A₀ ∈ S such that $\psi (A_0)=(x_1,x_2)\in {\mathbb{R}}^2$, where ψ is the natural projection of S to ${\mathbb{R}}^2$. In what follows, in an abuse of notation, we may sometimes identify A₀ and ψ(A₀) when no ambiguity can arise. Let the contour γ be the boundary of U oriented anticlockwise with unit external normal vector n. Write Green’s third identity for A₀ ∈ U:

where r′ = (x₁′, x₂′) is a position vector along γ, r = (x₁, x₂) points to A₀, ∂/∂n′ corresponds to the normal derivative associated to the unit external normal vector, and where $H_0^{(1)}$ is the zeroth-order Hankel function of the first kind. Note that the point A₀ is the only singularity of the integrand of (4.1) in the real (x₁′, x₂′) plane.

The orientation of γ plays no role in (4.1), however we can use the orientation of contours to set the direction of the normal vector. Namely, let the normal vector point to the right from an oriented contour.

The formula (4.1) can be used to continue u(x₁, x₂) to u_c(x₁, x₂) in a small domain of ${\mathbb{C}}^2$. Namely let $A\equiv \mathbf{\text{r}}=(x_1,x_2)\in {\mathbb{C}}^2$ be complex, while (x₁′, x₂′) remains real and belongs to γ. By small domain we mean that we choose A = (x₁, x₂) to be close enough to A₀ so that if we consider a straight path between A₀ and A, then, for any ${\mathbf{r}}^{\prime }\in \gamma$, this path does not intersect any of the complex singularities of the Green’s function G(r, r′) seen as a function of (x₁, x₂).

Hence, if (x₁, x₂) is close to A₀, the Green’s function G(r, r′) remains regular for each r′ ∈ γ. Moreover, being considered as a function of r, the Green’s function G(r, r′) obeys the Cauchy–Riemann conditions (3.1) and the complex Helmholtz equation (3.2) provided r is a regular point for certain fixed r′. Thus, for a small complex neighbourhood of A₀ the formula (4.1) provides a function obeying all restrictions (listed in §3) imposed on u_c(A).

We should note that the continuation u_c of u in a small complex neighbourhood of A₀ is unique and is provided by letting r become a complex vector in (4.1). The proof is given in appendix B and its structure is as follows. We start by deriving a complexified Green’s formula obeyed by u_c (or by any analytical solution of (3.2)). Then, by Stokes’ theorem, the integration contour for this formula for A belonging to some small complex neighbourhood of A₀ can be taken to coincide with γ. In this case, the complexified Green’s formula for u_c coincides with (4.1).

The procedure of analytical continuation of u using (4.1) fails when G(r, r′) becomes singular for some r′. Let us develop a simple graphical tool to explore the singularities of G(r, r′) for complex r ≡ A. The function (4.2) is singular when

i.e. when A ≡ (x₁, x₂) and A′ ≡ (x₁′, x₂′) both belong to some characteristic of (3.2). Let us fix the complex point A and find the real points A′ ≡ (x₁′, x₂′) such that (4.3) is valid. Obviously, A′ can have two values and These two points coincide when A is a real point. We will call A₁ and A₂ the first and the second real points associated with A and will sometimes use the notation A₁(A) and A₂(A) to emphasize the link between A_1,2 and A. Both points A₁ and A₂ belong to ${\mathbb{R}}^2$, but by the slight abuse of notation mentioned above we can also consider them to be on S.

The Green’s function G(r, r′) is singular at some point of the integral contour γ in (4.1) if A₁(A) ∈ γ or A₂(A) ∈ γ, where r points to A.

Consider the analytical continuation along a simple path σ as a continuous process. Let σ be parametrized by a real parameter τ ∈ [0, 1], i.e. let each point on the contour σ be denoted by A(τ). Let A(0) = A₀ ∈ S be the starting real point, and let A(1) be the ending complex point. For each point A(τ) find the associated real points A_1,2 (τ) ≡ A_1,2 (A(τ)). The position of these points depends continuously on τ.

We are now well-equipped to formulate the first theorem of analytical continuation.

Theorem 4.1.

Let $A_0\in S\mathrm{\setminus }{\cup }_j{P}_j$ and let $A\in {\mathbb{C}}^2$ be within a small neighbourhood of A₀. Let σ be a simple path from A₀ to A parametrized by τ ∈ [0, 1] as above. Let Γ(τ) ⊂ S be a continuous set of closed smooth oriented contours (i.e. a homotopical deformation of a contour) such that

—	Γ(0) = γ;
—	for each τ ∈ [0, 1] $A_1(\tau )\notin \psi (\mathit{\Gamma }(\tau ))$, $A_2(\tau )\notin \psi (\mathit{\Gamma }(\tau ))$.

Then the formula

$$u_{\text{c}}(A(\tau )+\delta \mathbf{\text{r}})={\int }_{\mathit{\Gamma }(\tau )}[\frac{\mathrm{\partial }G}{\mathrm{\partial }{n}^{\prime }}(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime })u({\mathbf{\text{r}}}^{\prime })-\frac{\mathrm{\partial }u}{\mathrm{\partial }{n}^{\prime }}({\mathbf{\text{r}}}^{\prime })G(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime })]d{l}^{\prime },$$ 4.6

defines an analytical continuation u_c of u in a narrow neighbourhood of σ. δr is an arbitrary small-enough complex radius vector, while the radius-vector r points to A(τ) + δr. Here, by arbitrary small, we mean that $\delta \mathbf{r}$ should be small enough so that the two associated real points $A_{1,2}(A(\tau )+\delta \mathbf{r}\mathbf{)}$ remain within the interior of Γ(τ).

Proof.

We present a sketch of the proof on the ‘physical level of rigour’. Let the contour Γ(τ) be changing incrementally, i.e. consider the contours Γ(τ_n), where 0 = τ₀ < τ₁ < τ₂… < τ_K = 1 is a dense grid on the segment [0, 1]. Each fixed contour Γ(τ_n) provides an analytic function u_c in a small neighbourhood of the point A(τ_n). The grid is dense enough to ensure that such neighbourhoods are overlapping. Moreover, for any point belonging to an intersection of neighbourhoods of A(τ_n) and A(τ_n+1) one can deform the contour Γ(τ_n) into Γ(τ_n+1) homotopically without changing the value of the integral, and hence without changing the value of u_c. ▪

Note that, formally, the expression (4.6) defines the field u_c ambiguously. The values of u and ∂_n′ u on S are found in a unique way, but the values of G and ∂_n′ G should be clarified. Namely, according to (4.2), the value depends on the branch of the square root and of the Hankel function (having a logarithmic branch point at zero).

For τ = 0, let G be defined on γ = Γ(0) in the ‘usual’ way: the square root is real positive, and the values of $H_0^{(1)}(\cdot )$ are belonging to the main branch of this function. Then, as τ changes continuously from τ = 0 to 1, define the values of G and ∂_n′ G by continuity. Since the (moving) contour Γ(τ) does not hit the (moving) singular points A₁ (τ), A₂ (τ), the branch of G is defined consistently.

The last theorem in this section extends the local result of theorem 4.1 to a global result.

Theorem 4.2.

Let B be a point of ${\mathbb{C}}^2\mathrm{\setminus }(T\cup {\mathbb{R}}^2)$, where T is the union of all the 2-lines $L_{1,2}^{(j)}$. Let A₀ be a point belonging to $S\mathrm{\setminus }({\cup }_j{P}_j)$ and let σ be a smooth path connecting A₀ with B, such that

$$(\sigma \mathrm{\setminus }{A}_0)\cap (T\cup {\mathbb{R}}^2)=\mathrm{\varnothing }.$$

Then there exists a family of contours Γ(τ) associated with σ and obeying the conditions of theorem 4.1.

We omit the proof of this theorem. It is almost obvious, but not easy to formalize. One should consider the process of changing τ from 0 to 1, and ‘pushing’ the already built contour Γ(τ), which is considered to be movable, by the moving points A₁ (τ) and A₂ (τ) (or, to be more precise, by small discs centred at A₁ (τ) and A₂ (τ)).

Some issues may occur with such deformation. For example, the contour may become pinched between A_1,2(τ) and P_j for some j, or between A₁(τ) and A₂(τ). The condition

guarantees that the points A₁(τ), A₂ (τ) do not pass through P_j, and thus the contour Γ(τ) cannot be pinched between A_1,2 and P_j. The condition guarantees that the contour Γ(τ) cannot be pinched between A₁ (τ) and A₂ (τ).

Theorems 4.1 and 4.2 demonstrate that u_c can be expressed almost everywhere as an integral (4.6) containing u(x₁, x₂) defined on the real surface S, and some known kernel G. The continuation u_c has a more complicated structure than u on S. This is explained by the fact that the set of closed contours on S has a more complicated structure than S itself. This will be investigated in the next section.

5. Analysis of the analytical continuation branching

We will now study the branching of u_c as follows. Let us consider a starting point $A_0\in (S\mathrm{\setminus }{\cup }_j{P}_j$), an ending point B located near a set on which we study the branching (i.e. near ${\mathbb{R}}^2$ or $L_{1,2}^{(j)}$), and a simple path σ connecting A₀ to B satisfying

i.e. σ is such that the condition of theorem 4.2 is valid. According to the theorems 4.1 and 4.2, one can continue u_c from A₀ to B along σ. Let the result of this continuation be denoted by u_c(B). Let σ′ be a local path starting and ending at B and encircling a corresponding fragment of the branch set. For example, one can introduce a local transversal variable near the branch set and build a contour σ′ in the plane of this variable encircling zero for a single time in the positive direction. One can consider a continuation of u_c(B) along σ′ and obtain a branch of u_c that is denoted by u_c(B ;σ′). This is a continuation of u_c from A₀ along the concatenation σσ′ of the contours σ and σ′. Let the parameter τ parametrize the contour σσ′; τ = 0 correspond to A₀; τ = 1/2 correspond to the end of σ; and τ = 1 correspond to the end of σσ′. Consistently with this parametrization, the contour of integration used to obtain u_c(B) and u_c(B ;σ′) will be denoted by Γ(1/2) and Γ(1), respectively.

If u_c(B ;σ′) ≡ u_c(B), then the contour σ′ yields no branching. If u_c(B ;(σ′)^p) ≡ u_c(B), for some integer p > 1, then the branching has order p (the smallest strictly positive integer with such property). Here u_c(B ;(σ′)^p) is the result of continuation along the concatenation σσ′…σ′, where σ′ is taken p times. The two following theorems establish the type of branching of u_c.

Theorem 5.1.

The points of the real plane ${\mathbb{R}}^2$ other than P_j do not belong to the branch set of u_c.

Proof.

Note that ${\mathbb{R}}^2$ is not an analytical set, so the concept of a transversal variable is not fully applicable to it (and of course this is the reason of non-branching at it). Anyway, the scheme sketched above can be applied to this case. Let B be close to ${\mathbb{R}}^2$ but not close to any of the P_j. Consider a local path σ′ starting and ending at B and encircling ${\mathbb{R}}^2$. Consider the motion of A₁(A) and A₂(A) as the point A travels along σ′. To be consistent with the parametrization, we will refer to these points as A_1,2(τ) for τ ∈ [1/2, 1]. These points are close to each other and close to B since B is close to ${\mathbb{R}}^2$. Depending on the particular contour σ′, it may happen that during this motion A₂ encircles A₁ a single time, several times or not at all. Note that A₁(1/2) = A₁ (1) = A₁(B) and A₂(1/2) = A₂ (1) = A₂(B).

One can show that if A₂ does not encircle A₁ then the contour Γ(1) is homotopic to Γ(1/2), and thus u_c(B ;σ′) = u_c(B). This can be seen by considering a small disc around A₂, and thinking of Γ(τ) as an elastic band that remains glued to that disc once in contact and deforms accordingly, without intersecting A₁, as A₂ moves. Since A₂ does not encircle A₁, then neither does the resulting contour. Hence, once A₂ is back to its original position, the contour can easily be deformed back to Γ(1/2).

Let A₂ encircle A₁ a single time. The case of several times follows from this case in a clear way. To consider this case, we follow the principles of computation of ramified integrals that can be found for example in [12,14,18]. Namely, it is known that the integral changes locally during a local bypass. The fragments of the contour Γ(1/2) that are located far from the points A₁(1/2) and A₂(1/2) are not affected by the bypass σ′. The fragments that are close to these points but do not pass between these points are not affected either. The only parts that are affected are the parts of Γ(1/2) passing between the points A₁(1/2) and A₂ (1/2).

In figure 3 we demonstrate the deformation of a fragment of Γ(1/2) (figure 3a into a corresponding fragment of Γ(1) (figure 3b). One can see from figure 3c that the fragment of Γ(1) can be written as the sum of the initial fragment of Γ(1/2) and an additional contour δΓ. The contour δΓ is a ‘double-eight’ contour having the following property: it bypasses each of the points A₁(1/2) and A₂(1/2) zero times totally (one time in the positive and one time in the negative direction). We say that this double-eight contour is based on the points A₁(1/2) and A₂(1/2). Such a contour is also known as a Pochhammer contour.

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Since there may be several parts of Γ(1/2) passing between A₁(1/2) and A₂(1/2), the contour Γ(1) can be written as a sum of Γ(1/2) and several double-eight contours based on the points A₁(1/2) and A₂(1/2). These contours, possibly, drawn on different sheets of S.

Let us show that for a double-eight contour δΓ based on A₁(1/2) and A₂(1/2), we have

$${\int }_{\delta \mathit{\Gamma }}[\frac{\mathrm{\partial }G}{\mathrm{\partial }{n}^{\prime }}(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime })u({\mathbf{\text{r}}}^{\prime })-\frac{\mathrm{\partial }u}{\mathrm{\partial }{n}^{\prime }}({\mathbf{\text{r}}}^{\prime })G(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime })]\text{d}{l}^{\prime }=0.$$ 5.1

This will yield that u_c(B;σ′) = u_c(B) and that there is no branching on ${\mathbb{R}}^2$. In order to do so, connect the points A₁(1/2) and A₂(1/2) with an oriented contour δΓ′ (see figure 3d, δΓ′ is shown by a black line). Squeeze the contour δΓ in such a way that it goes four times along δΓ′.

We now claim that it is not necessary to account for the integral in the close vicinity of A₁(1/2) or A₂(1/2). Indeed, using (4.2), (4.4) and (4.5), one can show that the kernel G has two logarithmic branch points at A₁(1/2) and A₂(1/2), and hence, in principle, the contour δΓ needs to be considered on the Sommerfeld surface of G. This surface has infinitely many sheets, and can be constructed by considering a straight cut between A₁(1/2) and A₂(1/2) and suitable ‘gluing’. What is important is that locally, around A₁(1/2) and A₂(1/2), G behaves like a complex logarithm. Hence, locally, the difference in G between two adjacent sheets is constant. Therefore, when considering the part of δΓ in the vicinity of A₁(1/2) and A₂(1/2), two loops going in opposite directions, the overall contribution of the integral tends to zero as these loops ‘shrink’ to A_1,2(1/2). This is due to the fact that, u being single valued on δΓ, the logarithmic singularities of G cancel out. Thus, a consideration of the integral near A₁(1/2) and A₂(1/2) is not necessary, and it is possible to reduce δΓ to four copies of δΓ′.

As explained in detail in appendix C, according to the formulae (4.4)–(4.5) linking the complex variables (x₁, x₂) with the real coordinates of the points A₁, A₂, and to formulae (4.2), a bypass about A₁ in the positive direction (in the real plane) leads to the following change of the argument of $H_0^{(1)}(z)$: z → e^iπ z. Similarly, a bypass about A₂ in the positive direction changes the argument of the Hankel function as z → e^−iπ z.

As a result, one can rewrite the integral of (5.1) as

$${\int }_{\delta \mathit{\Gamma }}[\frac{\mathrm{\partial }G}{\mathrm{\partial }{n}^{\prime }}(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime })u({\mathbf{\text{r}}}^{\prime })-\frac{\mathrm{\partial }u}{\mathrm{\partial }{n}^{\prime }}({\mathbf{\text{r}}}^{\prime })G(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime })]\text{d}{l}^{\prime }={\int }_{\delta {\mathit{\Gamma }}^{\prime }}[\frac{\mathrm{\partial }{G}^{\prime }}{\mathrm{\partial }{n}^{\prime }}(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime })u({\mathbf{\text{r}}}^{\prime })-\frac{\mathrm{\partial }u}{\mathrm{\partial }{n}^{\prime }}({\mathbf{\text{r}}}^{\prime })G^{\prime }(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime })]\text{d}{l}^{\prime },$$ 5.2

where

$$G^{\prime }(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime })=-\frac{i}{4}(-H_0^{(1)}(e^{-i\pi }\mathcal{k}\hspace{0.17em}r(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime }))+2H_0^{(1)}(\mathcal{k}\hspace{0.17em}r(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime }))-H_0^{(1)}(e^{i\pi }\mathcal{k}\hspace{0.17em}r(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime }))).$$ 5.3

The notation $H_0^{(1)}(e^{i\pi }z)$ means the value of $H_0^{(1)}(\cdot )$ obtained as the result of continuous rotation of the argument z about the origin for the angle π in the positive direction.

Apply the formula (see e.g. [19], sec. 1.33, eqns (202)–(203)) well-known from the theory of Bessel functions:

$$-H_0^{(1)}(e^{-i\pi }z)+2H_0^{(1)}(z)-H_0^{(1)}(e^{i\pi }z)=0,$$ 5.4

to conclude that the expression on the right-hand side of (5.2) is equal to zero and conclude the proof. Here and below, the formula (5.4) seems to play a fundamental role in the process of analytical continuation. ▪

Note also that every branch of u_c is continuous at any point $A_0\in {\mathbb{R}}^2\mathrm{\setminus }{\cup }_j{P}_j$, thus these points are regular points of any branch of u_c. This can be seen using the complexified Green’s theorem of appendix B. Indeed, using the same idea of variable translation, it can be shown that for a point $A\in {\mathbb{C}}^2\mathrm{\setminus }(T\cup {\mathbb{R}}^2)$ in a close complex neighbourhood of A₀ and a given branch u_c(A), we can write $u_c(A)={\int }_{\tilde{\gamma }}(u_c{\mathrm{\nabla }}_{C}G-G{\mathrm{\nabla }}_C{u}_c)$ for some small contour $\tilde{\gamma }$ surrounding A in ${\mathbb{C}}^2$. Since the differential form $u_c{\mathrm{\nabla }}_{C}G-G{\mathrm{\nabla }}_C{u}_c$ is closed (see appendix B), Stokes’ theorem allows us to deform $\tilde{\gamma }$ to a contour γ in ${\mathbb{R}}^2$ surrounding A₀ (see figure 4) without changing the value of the integral. A is chosen close enough to A₀ so that we can let A → A₀ along a simple small straight path without any singularities of the integrand hitting the contour γ, showing, in doing so, the continuity of u_c at A₀.

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Theorem 5.2.

Let $P_j\in {\mathbb{R}}^2$ be a branch point of order p (the definition of the order of the branch point is clarified below). Then both $L_1^{(j)}$ and $L_2^{(j)}$ are branch 2-lines of u_c of order p.

Proof.

First, let us define what it exactly means for $P_j\in {\mathbb{R}}^2$ to be a branch point of S of order p. The situation is simple if ψ⁻¹ (P_j) is a single point. Then p is the order of branching of S at ψ⁻¹(P_j). If ψ⁻¹ (P_j) is a set of several points, then p is the least common multiple of orders of all points ψ⁻¹ (P_j). A bypass encircling P_j p times in the positive direction returns each point located closely to P_j to itself.

Consider a point $B\in {\mathbb{C}}^2\mathrm{\setminus }T$ that is close to $L_1^{(j)}$. Let the loop σ′ bypass $L_1^{(j)}$ once in the positive direction (i.e. its projection onto the local transversal coordinate

$$z_1=x_1+i{x}_2-(X_1^{(j)}+i{X}_2^{(j)})$$

bypasses zero once in the positive direction). Let σ′ be also such that it does not bypass any other 2-line of T. As before, parametrize σ′ by τ ∈ [1/2, 1] and denote by Γ(τ) the associated integration contours used for the analytical continuation along σ′ . Then A₁(τ) bypasses P_j once in the positive direction, and A₂(τ) does not bypass any branch point.

The fragments of Γ(1/2) that are far from A₁ (1/2) or not passing between A₁ (1/2) and P_j are not affected by σ′. For each fragment of Γ(1/2) passing between A₁ (1/2) and P_j, the double-eight contour δΓ is added to obtain a fragment of Γ(1). In this case, the double-eight contour is based on the points A₁ (1/2) and P_j. The graphical proof is similar to that in figure 3.

Let us now show that the order of branching at $L_1^{(j)}$ is equal to p. Consider a fragment of Γ(1/2) passing between A₁ and P_j. A single bypass along σ′ changes (locally)

$$\mathit{\Gamma }(1/2)\stackrel{{\sigma }^{\prime }}{⟶}\mathit{\Gamma }(1/2)+\delta \mathit{\Gamma },$$

where δΓ is illustrated in figure 5. Upon performing a second bypass σ′, we hence get

$$\mathit{\Gamma }(1/2)+\delta \mathit{\Gamma }\stackrel{{\sigma }^{\prime }}{⟶}\mathit{\Gamma }(1/2)+\delta \mathit{\Gamma }+\delta {\mathit{\Gamma }}^{(1)},$$

where δΓ⁽¹⁾ is the double-eight contour based on the points A₁ and P_j, and obtained from δΓ by rotating A₁ about P_j once in the positive direction.

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The projection ψ(δΓ⁽¹⁾) coincides with ψ(δΓ), however, the new contour passes along other sheets of S, and the branch of $H_0^{(1)}$ is different on it. Finally, after p rotations one gets the integration contour Γ(1/2) + δΓ + δΓ⁽¹⁾ + · · · + δΓ^(p−1). In order to show that the branching of $L_1^{(j)}$ is of order p, we need to show that

$${\int }_{\delta \mathit{\Gamma }+\delta {\mathit{\Gamma }}^{(1)}+\cdots +\delta {\mathit{\Gamma }}^{(p-1)}}[\frac{\mathrm{\partial }G}{\mathrm{\partial }{n}^{\prime }}(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime })u({\mathbf{\text{r}}}^{\prime })-\frac{\mathrm{\partial }u}{\mathrm{\partial }{n}^{\prime }}({\mathbf{\text{r}}}^{\prime })G(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime })]\text{d}{l}^{\prime }=0,$$ 5.5

implying that u_c(B ;(σ′)^p) = u_c(B), and that the branching has order p. Algebraically, in terms of homology classes, this can be written as

$$\delta \mathit{\Gamma }+\delta {\mathit{\Gamma }}^{(1)}+\cdots +\delta {\mathit{\Gamma }}^{(p-1)}=0.$$ 5.6

We will show this by decomposing δΓ and all its subsequent ‘copies’ into three main parts: (i) the two circles around P_j; (ii) the two circles around A₁; and (iii) the four straight lines between A₁ and P_j, and showing that the contribution of each of these three parts to the integral (5.5) is indeed zero.

(i)	By Meixner conditions, u and ∂n′u are integrable in the close vicinity of Pj, where G(r, · ) is well behaved. Hence we can ‘shrink’ the circles to Pj, and the integrals over the two circles become zero and do not contribute to (5.5).
(ii)	For the circles around A1, the argument is slightly more subtle. Let (s1, …, sp) be the sheets of S accessible by turning around Pj. Whatever sheet u is on the initial outer circle, after p rotations, it will have ‘visited’ all sheets s1, …, sp once. The same is true for the inner circle. Hence, the overall contribution of the circles to (5.5) can be written as $$\begin{array}{rl}& {\int }_{A_1}\left\{\right[\frac{\mathrm{\partial }G}{\mathrm{\partial }{n}^{\prime }}(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime };h_1)u({\mathbf{\text{r}}}^{\prime };s_1)-\frac{\mathrm{\partial }u}{\mathrm{\partial }{n}^{\prime }}({\mathbf{\text{r}}}^{\prime };s_1)G(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime };h_1)]\\ & +\cdots +[\frac{\mathrm{\partial }G}{\mathrm{\partial }{n}^{\prime }}(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime };h_p)u({\mathbf{\text{r}}}^{\prime };s_p)-\frac{\mathrm{\partial }u}{\mathrm{\partial }{n}^{\prime }}({\mathbf{\text{r}}}^{\prime };s_p)G(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime };h_p)]\}\text{d}{l}^{\prime }\\ -& {\int }_{A_1}\left\{\right[\frac{\mathrm{\partial }G}{\mathrm{\partial }{n}^{\prime }}(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime };h_1^{\prime })u({\mathbf{\text{r}}}^{\prime };s_1)-\frac{\mathrm{\partial }u}{\mathrm{\partial }{n}^{\prime }}({\mathbf{\text{r}}}^{\prime };s_1)G(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime };h_1^{\prime })]\\ & +\cdots +[\frac{\mathrm{\partial }G}{\mathrm{\partial }{n}^{\prime }}(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime };h_p^{\prime })u({\mathbf{\text{r}}}^{\prime };s_p)-\frac{\mathrm{\partial }u}{\mathrm{\partial }{n}^{\prime }}({\mathbf{\text{r}}}^{\prime };s_p)G(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime };h_p^{\prime })]\}\text{d}{l}^{\prime },\end{array}$$ 5.7 for some (h1, …, hp) and (h1′, …, hp′) corresponding to given sheets of G(r, · ), where the ; notation in the argument of a function specifies which sheet it is on, and where ${\int }_{A_1}$ specifies integration along a small circle encircling A1. Now we can group the terms for which u lies on the same sheet. For each such pair, we can use the reasoning used below (5.1) in the proof of theorem 5.1 to show that even if G may be on a different sheet for each element of the pair, the logarithmic singularities cancel out. Hence one can safely ‘shrink’ the circles to A1 without leading to any contribution to (5.5).
(iii)	Finally, for the straight lines, one can check that due to the identity (5.4), the branch of $H_0^{(1)}$ (and hence of G) does not matter. Namely, let C1, C2, C3, C4 be some points of δΓ projected onto a single point $C\in {\mathbb{R}}^2$. These points are shown in figure 5, but for clarity they are shown close to each other, not above one another. Consider the points C1 and C4. They belong to the same sheet of S, but the branches of $H_0^{(1)}$ are different at these points. Namely, if the argument of $H_0^{(1)}$ is equal to z at C1, then it is equal to z e−iπ at C4 (see again appendix C). Since the contours have different directions at C1 and C4, the contribution of the two outer lines to the integral over δΓ is of the form $${\int }_{\delta {\mathit{\Gamma }}^{\prime }}[\frac{\mathrm{\partial }{G}^{\prime }}{\mathrm{\partial }{n}^{\prime }}(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime })u({\mathbf{\text{r}}}^{\prime };s)-\frac{\mathrm{\partial }u}{\mathrm{\partial }{n}^{\prime }}({\mathbf{\text{r}}}^{\prime };s)G^{\prime }(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime })]\hspace{0.17em}\text{d}{l}^{\prime },$$ 5.8 for some s ∈ (s1, …, sp), where $G^{\prime }(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime })=-\frac{i}{4}(H_0^{(1)}(\mathcal{k}r(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime }))-H_0^{(1)}(e^{-i\pi }\mathcal{k}r(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime }))$ and where δΓ′ is a straight line between A1 and Pj. Perform a bypass σ′. In the course of this bypass, the points C1 and C4 encircle A1 in the positive direction. Thus, the corresponding part of the integral along δΓ(1) contains $H_0^{(1)}(z{e}^{i\pi })-H_0^{(1)}(z)$ and its derivative. However, due to (5.4), $$H_0^{(1)}(z\hspace{0.17em}{e}^{i\pi })-H_0^{(1)}(z)=H_0^{(1)}(z)-H_0^{(1)}(z\hspace{0.17em}{e}^{-i\pi }),$$ 5.9 and hence the value of G′ is unaffected by such a bypass. As before, after p bypasses, u would have ‘visited’ all the sheets (s1, …, sp) and the overall contribution of the two outer lines to (5.5) can be written $${\int }_{\delta {\mathit{\Gamma }}^{\prime }}[\frac{\mathrm{\partial }{G}^{\prime }}{\mathrm{\partial }{n}^{\prime }}(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime })u^{\prime }({\mathbf{\text{r}}}^{\prime })-\frac{\mathrm{\partial }{u}^{\prime }}{\mathrm{\partial }{n}^{\prime }}({\mathbf{\text{r}}}^{\prime })G^{\prime }(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime })]\hspace{0.17em}\text{d}{l}^{\prime },$$ 5.10 where u′( · ) = u( · ;s1) + · · · + u( · ;sp). The same consideration can be conducted for the points C2 and C3 on the inner lines to show an overall contribution equal to minus that of (5.10). Hence the overall contribution of the straight lines to (5.5) is also zero.

We have therefore proved that the equality (5.5) is correct, and hence that $L_1^{(j)}$ is a branch 2-line of order p. The case of σ′ bypassing $L_2^{(j)}$ can be considered in a similar way. Note that this time, if σ′ bypasses $L_2^{(j)}$ once in the positive direction (and no other branch 2-line), then the corresponding A₂ (τ) bypasses P_j once in the negative direction, and A₁ (τ) does not bypass any branch point. ▪

Theorem 5.3.

Consider two points $A,B\in ({\mathbb{C}}^2\mathrm{\setminus }T)$, and let σ₁ and σ₂ be piecewise-smooth paths in $({\mathbb{C}}^2\mathrm{\setminus }T)$ connecting A with B. Assume that it is possible to continue homotopically σ₁ to σ₂ in $({\mathbb{C}}^2\mathrm{\setminus }T)$. Let u_c(A) be some branch of u_c in some neighbourhood of A. Then the branches of u_c(B) obtained by continuation of u_c(A) along σ₁ and along σ₂ coincide:

$$u_{\text{c}}(B;{\sigma }_1)=u_{\text{c}}(B;{\sigma }_2).$$

This theorem follows naturally from the principle of analytical continuation, which remains valid in two-dimensional complex analysis. In the following theorem, we show that any closed contour in ${\mathbb{C}}^2\mathrm{\setminus }T$ can be deformed to a concatenation of canonical building blocks.

Theorem 5.4.

Let A be a point in ${\mathbb{C}}^2\mathrm{\setminus }T$, and let σ be a closed path in ${\mathbb{C}}^2\mathrm{\setminus }T$ starting and ending at A. Then, within ${\mathbb{C}}^2\mathrm{\setminus }T$, σ can be homotopically transformed into a concatenation contour σ′ of the form

$${\sigma }^{\prime }={\sigma }_{{\alpha }_1}^{m_1}{\sigma }_{{\alpha }_2}^{m_2}\dots {\sigma }_{{\alpha }_H}^{m_H},$$ 5.11

for some positive integer H. For h ∈ {1, …, H}, the powers m_h belong to $\mathbb{Z}$ and the elementary contours ${\sigma }_{{\alpha }_h}$ are fixed for each α_h. The indices α_h are pairs of the type (ℓ, j), where ℓ ∈ {1, 2} and j ∈ {1, …, N}. These contours can themselves be represented as concatenations

$${\sigma }_{{\alpha }_h}={\gamma }_{{\alpha }_h}{\sigma }_{{\alpha }_h}^{\ast }{\gamma }_{{\alpha }_h}^{-1},$$ 5.12

where each contour ${\gamma }_{{\alpha }_h}$ goes from A to some point near $L_{\ell }^{(j)}$, and ${\sigma }_{{\alpha }_h}^{\ast }$ is a local contour encircling $L_{\ell }^{(j)}$ once. On each contour σ_1,j the value x₁ − ix₂ is constant. On each contour σ_2,j the value x₁ + ix₂ is constant.

Examples of such contours can be found in figure 6.

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Proof.

Consider a small neighbourhood $U\subset {\mathbb{C}}^2\mathrm{\setminus }T$ of some point A. By the principles of analytic continuation, all possible branches u_c(A) are obtained by continuation along loops ending and starting at A that cover all possible combinations of the homotopy classes of ${\mathbb{C}}^2\mathrm{\setminus }T$. Each ${\sigma }_{\alpha }$ represents one of the homotopy classes, and hence by allowing σ′ to take the form (5.11), all possible combinations of such classes are covered. Finally since any closed contour σ can be written as a combination of homotopy classes, then we can in principle deform any closed contour σ to one akin to σ′. ▪

Note that the elementary paths σ_α have the following property. Since either x₁ − i x₂ or x₁ + i x₂ is constant on such a path, either A₂ or A₁ remains constant as the path σ_α is passed.

In the next theorem, we formulate the main general result of the paper, which is that there exists a finite basis of elementary functions such that any branch of the analytical continuation u_c is a linear combination of such functions with integer coefficients.

Theorem 5.5.

For a neighbourhood $U\subset {\mathbb{C}}^2\mathrm{\setminus }T$ of a given point A, one can find a finite set of basis functions g₁ (A), …, g_Q (A), A ∈ U, which are analytical solutions of the complex Helmholtz equation (3.2) in U, and such that any branch u_c(A ;σ), can be written as a linear combination

$$u_{\text{c}}(A;\sigma )=\sum _{q=1}^Q{b}_q(\sigma )g_q(A),$$ 5.13

where b_q (σ) are integer coefficients that are constant with respect to A. The dimension of the basis, Q, can be defined from the topology of S.

Proof.

According to theorem 4.2, theorem 5.1, theorem 5.2 and the basic principles of analytical continuation, it is enough to prove the statement of the theorem for an arbitrary small neighbourhood U.

Indeed, if the theorem is true for such a neighbourhood, then the elements of the basis can be expressed as linear combinations of Q linearly independent branches of u_c. The coefficients of the combination are constant, therefore the elements of the basis have singularities only at T, and the type of branching is the same as that of u_c. Any other neighbourhood U′ can be connected with U with a path in ${\mathbb{C}}^2\mathrm{\setminus }T$, and the formula (5.13) can be continued along this path.

Hence, below, we prove the theorem for a point A, for which the mutual location of the points A_1,2 (the real points associated with A) and P_j (the branch points on S) is convenient in some sense.

Choose the ‘convenient’ neighbourhood U as follows. Let us assume that the point A ≡ (x₁, x₂) is far enough from the branch points P_j, is close to ${\mathbb{R}}^2$ but does not belong to ${\mathbb{R}}^2$. This results in a configuration akin to that illustrated in figure 7.

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Define the basis functions g_q as follows. Let g₁ be defined by the integral (4.6) with the contour Γ₀ shown in figure 7. Such a function g₁ is equal to u_c obtained by analytical continuation from a small neighbourhood of a point belonging to S as per theorem 4.1.

All the other basis functions are constructed as follows. Let δΓ_ℓ,j be a double-eight contour based on the points A_ℓ, ℓ ∈ {1, 2}, and P_j, j ∈ {1, …, N}, as illustrated in figure 7. Consider all preimages ψ⁻¹(δΓ_ℓ,j) on S. Denote them by $\delta {\mathit{\Gamma }}_{\ell ,j}^{(1)},\dots ,\delta {\mathit{\Gamma }}_{\ell ,j}^{(M)}$, where we remind the reader that M is the finite number of sheets of S. Some of them are linearly independent. The functions g₂, …, g_Q are the integrals of the form (4.6) taken with all linearly independent contours from the set $\{\delta {\mathit{\Gamma }}_{\ell ,j}^{(1)},\dots ,\delta {\mathit{\Gamma }}_{\ell ,j}^{(M)}\}$.

To see this, let us continue the function u_c from A along some closed path σ. Deform the path σ into a path σ′ as in theorem 5.4. Each building block of σ′, σ_ℓ,j, can be analysed by the procedure described in the proof of theorem 5.2. A local contour ${\sigma }_{\ell ,j}^{\ast }$ produces several local double-eight loops, and the path ${\gamma }_{\ell ,j}^{-1}$ stretches the loops into those shown in figure 7. ▪

As mentioned in the statement of theorem 5.5, the exact value of the dimension Q of the basis depends on the topology of S and hence on the specific diffraction problem considered. In the next section we show (among other results) that in the case of the Dirichlet strip problem, we have Q = 4.

6. The strip problem

Everything so far has been done for a generic scattering problem described in the Introduction. We will here deal with the specific problem of diffraction by a finite strip ( − a < x₁ < a, x₂ = 0) . The canonical problem of diffraction by a strip has attracted a lot of attention since the beginning of the twentieth century, and various innovative mathematical methods have been designed and implemented to solve it: Schwarzschild’s series [20], Mathieu functions expansion [21], modified Wiener–Hopf technique [22,23], embedding and reduction to ODEs [24,25]. It has important applications, including in aero- and hydro-acoustics, see [26] for example.

The problem can be formulated as follows: find the total field u satisfying the Helmholtz equation and Dirichlet boundary conditions (u = 0) on the strip, resulting from an incident plane wave. The scattered field (the difference between the total and the incident field) should satisfy the Sommerfeld radiation condition, and the total field should satisfy the Meixner conditions at the edges of the strip.

Here we assume that this physical field u is known and we consider the associated Sommerfeld surface S. It has two branch points P₁ = (a, 0) and P₂ = ( − a, 0), so that N = 2. They are each of order 2, and the Sommerfeld surface S has two sheets (M = 2). The surface S is shown in figure 1b. Let us assume that sheet 1 is the physical sheet, while sheet 2 is its mirror reflection.

We will now apply the general theory developed in the paper in order to unveil the analytical continuation u_c.

Let $A\equiv (x_1,x_2)\in {\mathbb{C}}^2\mathrm{\setminus }(T\cup {\mathbb{R}}^2)$ be some point near ${\mathbb{R}}^2$. Consider all possible continuations of u_c(A) along closed paths. According to theorem 5.4, any such path can be represented as a concatenation of elementary paths σ_ℓ,j for ℓ ∈ {1, 2} and j ∈ {1, 2}. The elementary paths can be chosen in a such a way that corresponding trajectories of the points A₁ and A₂ are as shown in figure 8.

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As it follows from the considerations in previous sections, each path σ_ℓ,j is an elementary bypass about the branch 2-line $L_{\ell }^{(j)}$. The 2-lines $L_1^{(j)}$ are bypassed in the positive direction, while the 2-lines $L_2^{(j)}$ are bypassed in the negative direction.

Now let us choose the basis functions g_j of theorem 5.5. For this, consider the contours Γ₀ and the double-eight loops δΓ_ℓ,j for ℓ, j ∈ {1, 2} shown in figure 9. For definiteness, we assume that the points marked by a small black circle on the contours belong to the physical sheet of S.

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Formally, since the branch points P_j are of order 2, there should exist two double-eight loops for each pair of indexes ℓ, j: δΓ_ℓ,j and $\delta {\mathit{\Gamma }}_{\ell ,j}^{(1)}$. However, due to the identity (5.6), the loops $\delta {\mathit{\Gamma }}_{\ell ,j}^{(1)}$ need not be included in the basis, reducing the number of candidates for the basis functions from nine to five.

Moreover, using contour deformation and cancellation, one can check the following contour identity

by showing that each side of (6.1) is equal to 2Γ₀. Therefore, one of the double-eight loops, for example δΓ_2,2, needs not be included in the basis since it depends linearly on the other three contours. This effectively reduces the number of basis functions to four, as required and these basis functions can be explicitly written as where the […] integrand is the same as in (4.6).

Upon introducing the vector of functions

the analytic continuation of u_c is fully described by the following theorem.

Theorem 6.1.

The analytical continuation along each closed fundamental path σ_ℓ,j affects the vector of basis functions as follows:

$$\text{W}\stackrel{{\sigma }_{\ell ,j}}{⟶}{\text{M}}_{\ell ,j}\text{W},$$ 6.5

where the 4 × 4 constant matrices M_ℓ,j are given by

$${\text{M}}_{1,1}=\left(\begin{array}{cccc}1& -1& 0& 0\\ 0& -1& 0& 0\\ 0& -2& 1& 0\\ 0& 0& 0& 1\end{array}\right),{\text{M}}_{1,2}=\left(\begin{array}{cccc}1& 0& -1& 0\\ 0& 1& -2& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right),$$ 6.6

$${\text{M}}_{2,1}=\left(\begin{array}{cccc}1& 0& 0& -1\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right),{\text{M}}_{2,2}=\left(\begin{array}{cccc}1& -1& 1& -1\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& -2& 2& -1\end{array}\right).$$ 6.7

The statement of the theorem can be checked directly by studying how the double-eight contours are transformed under the action of each σ_ℓ,j, though it is omitted here for brevity.

Nevertheless, the following theorem and its proof can be considered as a check that the matrices given in theorem 6.1 are indeed correct.

Theorem 6.2.

Let j, k, ℓ ∈ {1, 2}. The following statements are correct:

(a)	Each branch 2-line $L_{\ell }^{(j)}$ has order 2, i.e.: $$u_{\text{c}}(A;{({\sigma }_{\ell ,j})}^2)=u_{\text{c}}(A).$$ 6.8
(b)	The bypasses σ1,j and σ2,k commute for any pair j, k: $$u_{\text{c}}(A;{\sigma }_{1,j}{\sigma }_{2,k})=u_{\text{c}}(A;{\sigma }_{2,k}{\sigma }_{1,j}).$$ 6.9
(c)	The intersecting branch 2-lines $L_1^{(j)}$ and $L_2^{(k)}$ have the additive crossing property: $$u_{\text{c}}(A)+u_{\text{c}}(A;{\sigma }_{1,j}{\sigma }_{2,k})=u_{\text{c}}(A;{\sigma }_{1,j})+u_{\text{c}}(A;{\sigma }_{2,k}).$$ 6.10

Proof.

To prove (a) it is enough to show that for all j, ℓ ∈ {1, 2}, we have

$${\text{M}}_{\ell ,j}^2={\text{I}}_4,$$

where I₄ is the 4 × 4 identity matrix, while to prove (b) one just needs to show that for all j, k ∈ {1, 2}, we have

$${\text{M}}_{1,j}{\text{M}}_{2,k}={\text{M}}_{2,k}{\text{M}}_{1,j}.$$

Finally, to prove (c), it is enough to show that for all j, k ∈ {1, 2}, we have

$${\text{I}}_4+{\text{M}}_{1,j}{\text{M}}_{2,k}={\text{M}}_{1,j}+{\text{M}}_{2,k}.$$

All this can be shown to be true directly for the matrices (6.6) and (6.7). ▪

Let us now discuss the proven relations. The relation (6.8) can be considered as an alternative check of theorem 5.2. Indeed, since P₁ and P₂ are branch points of order 2 on S, the branch 2-lines $L_{\ell }^{(j)}$ have order 2.

The second relation (6.9) follows from a fundamental property of multi-dimensional complex analysis discussed at the end of §3: the bypasses about $L_1^{(j)}$ and $L_2^{(k)}$ commute.

Finally, the relation (6.10) is discussed in detail in [7]. This relation means that the function u_c can be represented locally as a sum of two functions having branch 2-lines, separately, at $L_1^{(j)}$ and at $L_2^{(k)}$. As shown in [7,8] the additive crossing property plays a fundamental role in the process of integration of functions of several complex variables.

7. Link between finite basis and coordinate equations

Here we continue to study the Dirichlet strip problem from the previous section.

The property (6.5) of the vector W is reminiscent of the behaviour of a Fuchsian ordinary differential equation on a plane of a single complex variable. Namely, the poles of the coefficients of a Fuchsian ODE are branch points of its solution, and the vector composed of linearly independent solutions is multiplied by a constant monodromy matrix as the argument bypasses a branch point. Indeed, here the M_ℓ,j play the role of such monodromy matrices. It is also well known that, conversely, if a vector of linearly independent functions of a single variable has this behaviour, then there exists a Fuchsian equation obeyed by them (this is Hilbert’s 21st problem). This can be shown using the concept of fundamental matrices and their determinants (the Wronskian), see [27].

Here the situation is more complicated. There are two independent complex variables instead of one. Moreover, the behaviour of the components of W at infinity are not explored. However, we can still prove some important statements.

Throughout the paper, it is implicitly assumed that the field u, and, thus, its continuation basis W, depend on the angle of incidence φⁱⁿ. Let us now consider four different incidence angles ${\phi }_{\text{I}}^{\text{in}}$, ${\phi }_{\text{II}}^{\text{in}}$, ${\phi }_{\text{III}}^{\text{in}}$ and ${\phi }_{\text{IV}}^{\text{in}}$. It hence leads to four different wave fields, and to four basis vectors, all defined by (6.4): W_I, W_II, W_III, W_IV. Let us construct the 4 × 4 square matrix function V made of these vectors, defined by

We claim here (without proof) that the matrix V is non-singular almost everywhere (in ${\mathbb{C}}^2$ minus a set having complex codimension 1), so that we can freely write V⁻¹.

Note that the whole matrix is only branching at T, and that the equations (6.5) are valid for the matrix V as a whole:

This allows us to formulate the following theorem, linking the theory of differential equations to the strip diffraction problem:

Theorem 7.1.

There exist two 4 × 4 matrix functions Z₁(x₁, x₂) and Z₂(x₁, x₂), meromorphic in ${\mathbb{C}}^2$, such that

(a)	the matrix function V satisfies the following differential equations $${\mathrm{\partial }}_{x_1}\text{V}=\text{V}\hspace{0.17em}{\text{Z}}_1\text{ and }{\mathrm{\partial }}_{x_2}\text{V}=\text{V}\hspace{0.17em}{\text{Z}}_2;$$ 7.3
(b)	these matrix functions obey the consistency relation: $${\text{Z}}_1\hspace{0.17em}{\text{Z}}_2-{\text{Z}}_2\hspace{0.17em}{\text{Z}}_1={\mathrm{\partial }}_{x_2}{\text{Z}}_1-{\mathrm{\partial }}_{x_1}{\text{Z}}_2,$$ 7.4

where ∂x_ℓ for ℓ ∈ {1, 2} are the complex derivatives defined in (3.2).

Proof.

For (a) assume that V is known and that V⁻¹ exists almost everywhere (in ${\mathbb{C}}^2$ minus a set of complex codimension 1). In this case, the coefficients Z₁ and Z₂ are simply given by

$${\text{Z}}_1={\text{V}}^{-1}{\mathrm{\partial }}_{x_1}\text{V}\text{ and }{\text{Z}}_2={\text{V}}^{-1}{\mathrm{\partial }}_{x_2}\text{V}.$$ 7.5

Let us show that the matrices Z₁ and Z₂ are single-valued in ${\mathbb{C}}^2$. The only sets at which one can expect branching are the branch 2-lines of V, i.e. $L_{\ell }^{(j)}$. Make a bypass σ_ℓ,j about a 2-line $L_{\ell }^{(j)}$ and study the change of Z₁ and Z₂ as the result of this bypass:

$${\text{Z}}_k\stackrel{{\sigma }_{\ell ,j}}{⟶}{\text{V}}^{-1}{\text{M}}_{\ell ,j}^{-1}{\mathrm{\partial }}_{x_k}({\text{M}}_{\ell ,j}\text{V})={\text{Z}}_k,k\in \{1,2\},$$ 7.6

because M_ℓ,j are constant matrices. Thus, the coefficients Z₁ and Z₂ are not changing at the branch 2-lines of V, and, therefore, they are single-valued in ${\mathbb{C}}^2$. A detailed study shows that they have simple polar sets at the lines $L_{\ell }^{(j)}$, and, possibly, polar sets at the zeros of det(V) though we omit this discussion here for brevity.

To prove (b), differentiate the first equation of (7.3) with respect to x₂, and the second equation with respect to x₁. The expressions on the left are equal, and we get

$$({\mathrm{\partial }}_{x_2}\text{V}){\text{Z}}_1+\text{V}{\mathrm{\partial }}_{x_2}{\text{Z}}_1=({\mathrm{\partial }}_{x_1}\text{V}){\text{Z}}_2+\text{V}{\mathrm{\partial }}_{x_1}{\text{Z}}_2.$$

Applying (7.3) and multiplying by V⁻¹, we obtain (7.4).

As it follows from Frobenius theorem, this relation guarantees the solvability of the system (7.3). ▪

A detailed form of the coefficients Z₁ and Z₂ can be found in [1–3]. Here our aim was just to demonstrate that the existence of the coordinate equations is connected with the structure of analytical continuation of the solution. It has to be stressed that one can use the coordinate equation to evaluate the physical wave field (see e.g. [1]) and hence, our method, by providing a systematic way of obtaining coordinate equations, can in principle be used to recover the physical field.

Finally, before concluding the paper, it is interesting to note that the idea of considering a set of different incident angles and trying to link the solutions to each other by means of differential equations is somewhat reminiscent of Biggs’ interpretation of embedding formulae [28].

8. Conclusion

We have provided an explicit method to analytically continue two-dimensional wave fields emanating from a broad range of diffraction problems and described the singular sets (in ${\mathbb{C}}^2$) of their analytical continuation. We have shown that, even though the analytical continuation may have potentially infinitely many branches, each branch can be expressed as a linear combination of finitely many basis functions. Such basis functions are expressed as Green’s integrals over a real double-eight contour. The effectiveness of the general theory was illustrated via the example of diffraction by an ideal strip, for which we proved that only four basis functions were needed. Using these, we were able to completely describe the analytical continuation and study its branching behaviour. Finally, we have shown that this finite basis property was directly related to the existence of the so-called coordinate equation for the strip problem.

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Authors' contributions

A.V.S. came up with the initial idea. Both authors co-developed the mathematics and contributed equally to the manuscript. Both authors approved the final version and agree to be accountable for all aspects of the work.

Competing interests

We declare we have no competing interests.

Funding

R.C.A. would like to acknowledge the support by UK EPSRC (grant no. EP/N013719/1). Both authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme ‘Bringing pure and applied analysis together via the Wiener–Hopf technique, its generalizations and applications’ where some work on this paper was undertaken. This work was supported by EPSRC (grant no. EP/R014604/1) and, in the case of A.V.S., the Simons foundation. Both authors are also grateful to the Manchester Institute for Mathematical Sciences for its financial support.