Diffraction by a Right-Angled No-Contrast Penetrable Wedge: Analytical Continuation of Spectral Functions

Summary

We study the problem of diffraction by a right-angled no-contrast penetrable wedge by means of a two-complex-variable Wiener–Hopf approach. Specifically, the analyticity properties of the unknown (spectral) functions of the two-complex-variable Wiener–Hopf equation are studied. We show that these spectral functions can be analytically continued onto a two-complex dimensional manifold, and unveil their singularities in $C2$⁠. To do so, integral representation formulae for the spectral functions are given and thoroughly used. It is shown that the novel concept of additive crossing holds for the penetrable wedge diffraction problem, and that we can reformulate the physical diffraction problem as a functional problem using this concept.

1 Introduction

For over a century, the canonical problem of diffraction by a penetrable wedge has attracted a great deal of attention in search for a clear analytical solution, which remains an open and challenging problem. Beyond their importance to mathematical physics as one of the building blocks of the geometrical theory of diffraction [1], wedge diffraction problems also have applications to climate change modelling, as they are related to the scattering of light waves by atmospheric particles such as ice crystals, which is one of the big uncertainties when calculating the Earth’s radiation budget (see [2–5]).

An important parameter when studying the diffraction by a penetrable wedge is the contrast parameter λ which is defined as the ratio of either the electric permittivities $ε1,2$⁠, magnetic permeabilities $μ1,2$⁠, or densities $ρ1,2$ corresponding to the material inside and outside the wedge, respectively, depending on the physical context, cf. section 2. The case of $λ≪1$ (high contrast) is, for instance, studied in [6] and [7]. In [6], Lyalinov adapts the Sommerfeld–Malyuzhinets technique to penetrable scatterers and concludes with a far-field approximation taking the geometrical optics components and the diffracted cylindrical waves into account whilst neglecting the lateral waves’ contribution. More recently, in [7] Nethercote et al. provide a method to accurately and rapidly compute the far-field for high-contrast penetrable wedge diffraction taking the lateral waves’ contribution into account, by using a combination of the Wiener–Hopf and Sommerfeld–Malyuzhinets technique. The case of general contrast parameter λ is for example considered by Daniele and Lombardi in [8], which is based on adapting the classical, one-complex-variable Wiener–Hopf technique to penetrable scatterers, and Salem, Kamel and Osipov in [9], which is based on an adaptation of the Kontorovich–Lebedev transform to penetrable scatterers. When the wedge has very small opening angle, Budaev and Bogy obtain a convergent Neumann series by using the Sommerfeld–Malyuzhinets technique [10]. All of these papers offer different ways to numerically compute the total wave field. Another, more theoretically oriented perspective on penetrable wedge diffraction when the wedge is right angled is provided by Meister, Penzel, Speck and Teixeira in [11], which follows an operator-theoretic approach and takes different interface conditions on the two faces of the wedge into account.

The present article studies the case of a no-contrast penetrable wedge. That is, we set λ = 1. Moreover, we assume that the wedge is right angled. Previous work on this special case includes that of Radlow [12], Kraut and Lehmann [13] and Rawlins [14]. References [12] and [13] are based on a two-complex-variable Wiener–Hopf approach, whereas [14]’s approach is based on Green’s functions. In [12], Radlow gives a closed-form solution, but it was deemed erroneous by Kraut and Lehmann [13] as it led to the wrong corner asymptotics. Kraut and Lehmann assume that the wavenumbers inside and outside of the wedge are of similar size, and in [14], Rawlins extends their work by generalising [13]’s scheme to arbitrary opening angles. A description of the diffraction coefficient is given in the right-angled case, in addition to the near-field description provided in [13]. Moreover, in [15], Rawlins obtains the diffraction coefficient for penetrable wedges with arbitrary angles. Both [14] and [15] require the wavenumbers inside and outside of the wedge to be of similar size. Another approach on the right-angled no-contrast wedge, that is based on physical optics approximations (also referred to as Kirchhoff approximation), is presented in [16] and [17]. These papers modify the ansatz posed in [18], which extends classical physical optics [19] from perfect to penetrable scatterers.

Recently, a correction term missing in Radlow’s work was given by the authors in [20]. This correction term includes an unknown spectral function and thus, the no-contrast right-angled penetrable wedge diffraction problem remains unsolved. The present work is part of an ongoing effort to apply multidimensional complex analysis to diffraction theory [20–26]. Another approach, also exploiting interesting ideas of multidimensional analysis in the context of wedge diffraction, is given in the monograph [27] that also contains an excellent review of wedge diffraction problems.

We first reformulate the diffraction problem as a two-complex-variable functional problem in the spirit of [21] and prove that this functional formulation is indeed equivalent to the physical problem. Therefore, solving the functional problem would directly solve the diffraction problem at hand, which immediately motivates further study of the former. Specifically, we will endeavour to study the analytical continuation of the unknown (spectral) functions of the two-complex-variable Wiener–Hopf equation (2.14). Indeed, not only is the knowledge of the spectral functions’ domains of analyticity crucial for completing the classical (one-complex-variable) Wiener–Hopf technique (cf. [28]), but by the recent work of Assier et al. [22], we know that knowledge of the spectral functions’ singularities allows for computation of the physical fields’ far-field asymptotics. Specifically, to obtain closed-form far-field asymptotics of the physical fields, which are represented as inverse double Fourier integrals as given in (2.20) and (2.21), we need to answer the following questions:

1. What are the spectral functions’ singularities in $C2$?

2. How can we represent the spectral functions in the vicinity of these singularities?

Addressing these questions, and thereby building the framework that allows us to make further progress, is the main endeavour of the present article. Note that, at this point, it is not clear how the two-complex-variable Wiener–Hopf equation can be solved and generalising the Wiener–Hopf technique to two or more dimensions remains a challenging practical and theoretical task. We refer to the introduction of [21] for a comprehensive overview of the Wiener–Hopf technique and the difficulties with its generalisation to two-complex-variables.

The content of the present article is organised as follows. After formulating the physical problem in section 2.1 and rewriting it as a two-complex-variable functional problem involving the unknown spectral functions ‘$Ψ++$’ and ‘$Φ3/4$’ in section 2.2, the equivalence of these two formulations is proved in Theorem 2.5. Thereafter, in section 3, we will study the analytical continuation of the spectral functions in the spirit of [21]. Using integral representation formulae given in section 3.2, we unveil the spectral functions’ singularities in $C2$⁠, as well as their local behaviour near those singularities, in sections 3.4 and 3.5. Throughout sections 2–3.5, we assume positive imaginary part of the wavenumbers k₁ and k₂, and in section 3.6, we discuss the spectral functions’ singularities on $R2$ in the limit $Im(k1,2)→0$⁠. Finally, in section 4, we show that the novel additive crossing property (introduced in [21]) holds for the spectral function $Φ3/4$ at intersections of its branch sets. This property is critical to obtaining the correct far-field asymptotics (see [22]) and will lead to the final spectral reformulation of the physical problem, as given in section 4.2.

2 The functional problem for the penetrable wedge

2.1 Formulation of the physical problem

In the electromagnetic setting, this assumption would correspond to either $μ1=μ2$ (electric polarisation) or $ϵ1=ϵ2$ (magnetic polarisation) where μ₁ and μ₂ (resp. ϵ₁ and ϵ₂) are the magnetic permeability of the media in $R2∖PW$ and PW (resp. the electric permittivities of the media in $R2∖PW$ and $PW$⁠). In the acoustic setting, this assumption corresponds to $ρ1=ρ2$⁠, where ρ₁ and ρ₂ are the densities of the media (at rest) in $R2∖PW$ and $PW$⁠, respectively.

Let $k1=|k1|=c1/ω$ and $k2=c2/ω$ denote the wavenumbers inside and outside PW, respectively, where c₁ and c₂ are the wave speeds relative to the media in $R2∖PW$ and PW, respectively. In the electromagnetic setting, $cj,j=1,2$ corresponds to the speed of light whereas in the acoustic setting, c_j corresponds to the speed of sound. Although λ = 1, the wavenumbers are different (⁠$k1≠k2$⁠) since the other media properties defining the speeds of light (in the electromagnetic setting) or sound (in the acoustic setting) are different. We refer to [20], Section 2.1 for a more detailed discussion of the physical context.

In the electromagnetic setting, $ϕ$ and ψ correspond either to the electric or magnetic field (depending on the polarisation of the incident wave, cf. [12, 13]) in $R2∖PW$ and PW, respectively, whereas in the acoustic setting, $ϕ$ and ψ represent the total pressure in $R2∖PW$ and PW, respectively.

This condition on the incident angle is rather restrictive since it says that the field cannot produce secondary reflected and transmitted waves as the incident wave is coming from the Q₃-region (see Fig. 1, middle and right). However, we will work around this restriction in section 3.6 as long as $ϕin$ is incident from within $Q2∪Q3∪Q4$⁠. For an incident wave coming from within $Q1=PW$⁠, the following analysis has to be repeated separately.

We refer to [20] Section 2.1 for a more detailed discussion. Note that (2.8) and (2.9) are only valid when λ = 1, and we refer to [7] for the general case. Finally, we note that specifying the behaviour of the fields near the wedge’s tip and at infinity is required to guarantee unique solvability of the problem described by (2.1)–(2.4), see [29].

2.2 Formulation as functional problem

Here, δ is as in (2.7) and since we chose $Im(k1)=Im(k2)$⁠, we have, in fact, $ε=δ/2$⁠.

 

2.2.1 1/4-based and 3/4-based functions

The 2D Wiener–Hopf equation (2.14) contains two unknown ‘spectral’ functions, $Ψ++$ and $Φ3/4$⁠. In the spirit of [21], our aim is to convert the physical problem discussed in section 2.1 into a formulation in 2D Fourier space, similar to the traditional Wiener–Hopf procedure. For this, the properties of the Wiener–Hopf equation’s unknowns are of fundamental importance. Following [21], we call these properties 1/4-basedness and 3/4-basedness.

 

In [20], it was shown that $Ψ++$ is analytic in $UHP(−2ε)×UHP(−2ε)$⁠, where $ε>0$ is as in (2.16). This is indeed a criterion for 1/4-basedness, that is if a function is analytic in $UHP×UHP$⁠, then it is 1/4-based, see [21]. However, although a function analytic in $LHP×LHP$ is 3/4-based, the unknown function $Φ3/4$ is, in general, not analytic in $LHP×LHP$ (see [21] and section 3.5) and therefore this does not seem to be a criterion useful to diffraction theory. Instead, the correct criterion for the quarter-plane involves the novel concept of additive crossing, see [21], and analysing this phenomenon for the penetrable wedge diffraction problem is one of the main endeavours of the present article.

 

2.2.2 Asymptotic behaviour of spectral functions

Before we can reformulate the physical problem of section 2.1 as a functional problem similar to the 1D Wiener–Hopf procedure, we require information about the asymptotic behaviour of the unknowns $Ψ++$ and $Φ3/4$⁠. This will not only be crucial to recover the Meixner conditions, but it will also be of fundamental importance for all of sections 3 and 4.

Similarly, the function $Φ3/4$ satisfies the growth estimates (2.17)–(2.19) as $|α1,2|→∞$ in $S×S$⁠.

2.2.3 Reformulation of the physical problem

Using the results above, we can rewrite the physical problem given by (2.1)–(2.4) as the following functional problem.

 

1. $−K(α)Ψ++(α)=Φ3/4(α)+P++(α)$for all $α∈S×S;$

2. $Ψ++$is analytic in $UHP(−ε)×UHP(−ε)$;

3. $Φ3/4$is 3/4-based;

4. $Ψ++$and $Φ3/4$satisfy the ‘spectral edge conditions’ given in section 2.2.2.

The importance of Definition 2.4 stems from the following theorem, which proves the equivalence of the penetrable wedge functional problem and the physical problem discussed in section 2.1.

 

satisfy the penetrable wedge problem described by (2.1)–(2.4) for an incident wave given by $ϕin=exp(−i(a1x1+a2x2))$⁠.

 

3 Analytical continuation of spectral functions

 

3.1 Some useful functions

Let $z→$ denote the square root with branch cut on the positive real axis, and with branch determined by $1→=1$ (that is $arg(z)∈[0,2π)$⁠). In particular, $Im(z→)≥0$ for all z and $Im(z→)=0$ if, and only if, $z∈(0,∞)$⁠.

3.2 Primary formulae for analytical continuation

Using the kernel’s factorisation given in section 3.1, we have the following analytical continuation formulae:

 

Here, and for the remainder of the article, $a1$and $a2$are given by (2.6).

The theorem’s proof is the exact same as the proof of the analytical continuation formulae for the quarter-plane problem (cf. (33)–(34) and Appendix A in [21]) and hence omitted. Indeed, we can write $Φ3/4=Φ−++Φ−−+Φ+−$ for functions $Φ−+, Φ−−$ and $Φ+−$ analytic in $LHP(ε)×UHP(−ε), LHP(ε)×LHP(ε)$ and $UHP(−ε)×LHP(ε)$⁠, respectively (see [20] eq. $(2.21)$⁠), and the analyticity properties of $Φ−+, Φ−−$ and $Φ+−$ in these domains as well as the analyticity of $Ψ++$ in $UHP(−ε)×UHP(−ε)$ are the only key points to finding (3.5) and (3.6), and these domains agree with those of the quarter-plane problem. The only difference of (3.5)–(3.6) and the corresponding analytical continuation formulae in the quarter-plane problem is due to the difference of the kernel K, as discussed in Remark 2.1, and its factorisation.

Observe that the variable on the LHS of (3.5)–(3.6) is $α∈S×S$⁠. Since all terms involving $α$ on the equations’ RHS are known explicitly, we can choose $α$ to belong to a domain much larger than $S×S$⁠, thus providing an analytical continuation of $Ψ++$⁠. This procedure will be discussed in the following sections.

 

3.3 Domains for analytical continuation

Moreover, we set $H−=H−(0),H+=H+(0)$⁠, see Fig. 2 for an illustration. By the properties of $z→$⁠, and due to the positive imaginary part of k₁ and k₂ we indeed have $hj+∈UHP$ for j = 1, 2 and consequently $hj−∈LHP j=1,2$⁠; see [21]. Moreover, the following holds:

Lemma 3.3 ([21], Lemma 3.2). If $z∈C∖(h1−∪h2−∪h1+∪h2+)$, then $kj2−z2→∈H+$for j = 1, 2.

Now, define the contours P₁ and P₂ as the boundaries of $C∖h1−$ and $C∖h2−$⁠, see Fig. 2. That is, for j = 1, 2, P_j is the contour ‘starting at $−i∞$’ and moving up along $hj−$’s left side, up to $−kj$⁠, and then moving back towards $−i∞$ along $hj−$’s right side. Intuitively, P_j is just $hj−$ but ‘keeps track’ of which side $hj−$ was approached from. Set $P=P1∪P2$⁠.

 

3.4 First step of analytical continuation: analyticity properties of $Ψ++$

Since, as already pointed out, the only difference between the present work and [21] is the structure of the kernel K and, therefore, the structure of the domains H⁺ and H^–, most of the following discussion very closely follows [21], and we will just sketch most of the details. The reader familiar with [21] may wish to skip to Theorem 3.11.

Let us first analyse the integral term in (3.5). As in [21], by an application of Stokes’ theorem, which tells us that it is possible to deform the surface of integration continuously without changing the value of the integral as long as no singularity is hit during the deformation (cf. [23]), it is possible to show that:

 

Henceforth, until specified otherwise, J denotes the function given by (3.8).

   

Corollary 3.9. The 1/4-based spectral function $Ψ++$satisfying the penetrable wedge functional problem 2.4 can be analytically continued onto $H−∖{a1}×UHP$. It can moreover be analytically continued onto the boundary elements $R×UHP, H−∖{a1}×R, P∖{−k1,−k2}×UHP$, and continuously continued onto $(P×R)∖{(α1,−k22−α12→)| α1∈P2}$⁠.

Repeating the above procedure but using (3.6) instead (again, see [21] for the technical details involved), we obtain:

Corollary 3.10. $Ψ++$can be analytically continued onto $UHP×H−∖{a2}$and onto the boundary elements $UHP×R, R×H−∖{a2},UHP×P∖{−k1,−k2}$, and continuously continued onto $(R×P)∖{(−k22−α22→,α2)| α2∈P2}$⁠.

Therefore:

 

Naturally, we ask whether a formula can be found for $Ψ++$ in the ‘missing’ parts of Fig. 4 that is, whether we can find a formula for $Ψ++$ in $LHP×LHP$⁠. Moreover, from [21], we anticipate that the study of $Ψ++$ in $LHP×LHP$ is directly linked to finding a criterion for 3/4-basedness of $Φ3/4$⁠. This will be the topic of the following sections.

3.5 Second step of analytical continuation: analyticity properties of $Φ3/4$

Our proof is slightly different from [21], as the integral in (3.12) is not improper.

The remainder of the proof is identical to [21]; that is, compute the residue using formula (3.6) where only the external additive term in (3.6) contributes to the residue, and identify it as the minus-part of a Cauchy sum-split which can be explicitly computed by pole removal. Use the resulting formula for J in (3.5) to obtain (3.12). ■

Similarly, changing the roles of (3.5) and (3.6) in the previous proof, we find:

 

where the RHS of (3.15) is defined for $α∈UHP×(H−(−b0)∖{a2})$⁠.

We are now ready to prove this section’s main result.

 

This residue has the required continuity as can be seen from (3.6), so we can safely let $α2→α2⋆$ for any $α2⋆∈h1−∪h2−$⁠, which gives the sought continuation. The residue of $Φ$ at the pole $α1=a1$ can be computed explicitly from (3.18) since only the external additive term is singular at $α1=a1$⁠. Similarly, the residue of $Φ$ at $α2=a2$ is computed. ■

3.6 Singularities of spectral functions

Then, just as before we find that (3.26) is satisfied for all $Re(α2)∈[−k2,k2]$ and $Re(α1)≤0$⁠.

The real traces of the singularities are shown in Fig. 7.

Change of incident angle. Let us now consider the case $ϑ0∈(π/2,π)$⁠. Due to symmetry, the case $ϑ0∈(3π/2,2π)$ can be dealt with similarly. We now treat $ϑ0$ as a parameter within the formulae for analytic continuation (3.5), (3.6), (3.12) and (3.15) of $Ψ++$⁠. This yields formulae for $Ψ++$ when $ϑ0∈(π/2,π)$⁠. We then obtain new singularities within these formulae for analytic continuation. Namely, the external additive term in (3.5) becomes singular at ${α1≡−k22−a22→}$⁠. This procedure therefore yields a new singularity of $Ψ++$ and $Φ3/4$ within $LHP×C$⁠. The real traces of the spectral functions’ singularities in this case are shown in Fig. 8. Note that we may not allow $ϑ0∈(0,π/2)$⁠. This is because such change of incident angle changes the incident wave’s wavenumber from k₁ to k₂, and therefore such change cannot be assumed to be continuous.

 

4 The additive crossing property

We want to investigate the behaviour of $Φ3/4$ on P × P. In particular, we wish to investigate whether the additive crossing property introduced in [21] is satisfied with respect to the points $(−k1,−k1),(−k2,−k2),(−k1,−k2)$ and $(−k2,−k1)$ which are the points at which the branch sets are ‘crossing’, see Fig. 7. Other than yielding a criterion for 3/4-basedness in the quarter-plane problem (cf. [21]), this property was also crucial to solving the simplified quarter-plane functional problem corresponding to having a source located at the quarter-plane’s tip, see [24]. It also emerged in the different context of analytical continuation of real wave fields defined on a Sommerfeld surface, see [25]. Therefore, it seems that the property of additive crossing is strongly related to the physical behaviour of the corresponding wave fields. Indeed, the additive crossing property is crucial to obtaining the correct far-field asymptotics as it prohibits the existence of unphysical waves, as shown in [22].

Let us investigate what happens when we change the domain of integration from $R×R$ to P × P in (4.1). Due to the asymptotic behaviour of $Φ3/4$ (cf. section 2.2.2), we will not obtain any ‘boundary terms at infinity’.

 

4.1 Three quarter-basedness and additive crossing

4.2 Reformulation of the functional problem

Finally, we obtain the following reformulation of Theorem 2.5.

Theorem 4.2. Let $P++$ and K be as in (2.13). Let $Ψ++$ satisfy the following properties:

1. $Ψ++$is analytic in

$(UHP×C∖(h1−∪h2−∪{a2}))∪(C∖(h1−∪h2−∪{a1})×UHP),$

1. There exists an $ε>0$such that the function $Φ3/4$defined by $Φ3/4=−KΨ++−P++$is analytic in

$(H−(ε)∖{a1})×(H−(ε)∖{a2})$

with simple poles at $α1=a1$and $α2=a2$⁠,

1. The residues of $−Φ3/4−P++$at the poles $α1=a1$and $α2=a2$are given by (3.16) and (3.17),

2. $Φ3/4$is continuous on P × P and satisfies the additive crossing property for each of the following points: $(−k1,−k1), (−k1,−k2), (−k2,−k2)$and $(−k2,−k2)$⁠,

3. The functions $Ψ++$and $Φ3/4$have the asymptotic behaviour (2.17)–(2.19).

The theorem’s proof is immediate since, according to sections 3–4, the conditions 1–5 of Theorem 4.2 imply that $Ψ++$ and $Φ3/4$ satisfy the penetrable wedge functional problem and therefore Theorem 2.5 holds for $ϕsc$ and ψ.

5 Concluding remarks

We have shown that the novel additive crossing property, which was introduced in [21] in the context of diffraction by a quarter-plane, holds for the problem of diffraction by a penetrable wedge. Indeed, in similarity to the 1D Wiener–Hopf technique, the spectral functions’ singularities within $C2$ solely depend on the kernel K and the forcing $P++$⁠, and therefore the techniques developed by Assier and Shanin in [21] could be adapted to the penetrable wedge diffraction problem, once equivalence of the penetrable wedge functional problem and the physical problem was shown in section 2.2.3. However, as in [21], we cannot apply Liouville’s theorem since, according to Theorem 4.2, the domains of analyticity of the unknowns $Ψ++$ and $Φ3/4$ span all of $C2$minus some set of singularities.

Nonetheless, using the in section 3.6 established real traces of the spectral functions’ singularities, we expect to be able to obtain far-field asymptotics of the physical fields using the framework developed in [22]. In particular, as in [26], we expect the diffraction coefficient in $R2∖PW$ (resp. PW) to be proportional to $Ψ++$ (resp. $Φ$⁠) evaluated at a given point. Moreover, we expect that a similar phenomenon holds for the lateral waves. That is, we expect that the results of the present article and [22] allow us to represent the lateral waves such that their decay and phase are explicitly known, whereas their coefficients are proportional to, say, $Ψ++$⁠, evaluated at a given point. Thus, we expect to be able to use the results of [20] to accurately approximate the far field in the spirit of [33]. We moreover plan to test far-field accuracy of Radlow’s erroneous ansatz (which was given in [12]).

Finally, we note that Liouville’s theorem is not only applicable to functions in $C2$ but also to functions defined on suitably ‘nice’ complex manifolds, see [34, 35]. Therefore, gaining a better understanding of the complex manifold on which $Ψ++$ and $Φ3/4$ are defined could be crucial to completing the 2D Wiener–Hopf technique. Note that this final step is presumably easier for the penetrable wedge than for the quarter-plane since in the latter, the real trace of the complexified circle is a branch set (see [21]) which could drastically change the topology of the sought complex manifold.

Footnotes

A Uniqueness theorems

 

Then $f1=f2=f3=f4≡0$⁠.

Now, since $fj(x)=O(xν)$⁠, by the Abelian theorem (cf. [31]), we find that for every $j=1,2,3,4 Fj(α)=O(1/αν+1)$ as $|α|→∞$ in $UHP(0)$⁠. In particular, it implies $Fj(α)→0$ as $|α|→∞$ in $UHP(0)$⁠.

Since $F2(α1)→0$ as $|α1|→∞$⁠, we find $F1(0)=F3(0)=0$ and therefore $F2(α1)≡0$⁠.

Again, since $F1(α2)→0$ as $|α2|→∞$ we find $F4(0)=0$ and therefore $F1(α2)≡0$⁠.

As before, since $F3→0$ as $|α2|→∞$⁠, we find $F4(α1⋆)=0$ and therefore $F3(α2)≡0$⁠. Similarly, we find $F4(α1)≡0$⁠. By inverse Laplace transform, we find $f1=f2=f3=f4≡0$⁠. ■

The following is a direct generalisation of [21] Theorem C.1 (and the techniques used for its proof are almost identical).

 

then $f≡0$on $γ1∪γ2$⁠.

Proof.

Here, for j = 1, 2, $yj(τr)$ (resp. $yj(τl)$⁠) refers to the limiting value of $yj(τ)$ as τ approaches γ_j from the right (resp. left), as illustrated in Fig. A.1. We now show that y₁ and y₂ are in fact continuous everywhere in $C$ thus proving the theorem. Continue γ₁ (resp. γ₂) towards 0 via curves s₁ (resp. s₂) and set $f≡0$ on s₁ (resp. s₂), cf.Fig. A.1 top right. As γ₁ and γ₂ are completely within the sector $φ2<arg(z)<φ1$ (and because they have finite length within each finite disk), these curves can be chosen to lie completely within this sector as well. Denote the curve $γ1·s1$ (resp. $γ2·s2$⁠) by $γ˜1$ (resp. $γ˜2$⁠).

Then, using Theorem A.2, we find:

The proof is identical to [21], and omitted for brevity. Moreover, these results can be directly generalised to the case of $n∈N$ curves $γ1∪…∪γn$ by following the construction given in Theorem A.2’s proof.

B Single integral analytical continuation formulae

Here, we show how (3.5) and (3.6) can be simplified, as mentioned in Remark 3.2. Thereafter, we give analogous simplifications of formulae (3.12) and (3.15), thereby simplifying all formulae for analytic continuation derived in the present article. We discuss this for rewriting (3.5) only, as the procedure for rewriting (3.6), (3.12) and (3.15) is analogous.

Again, these formulae are valid for $α∈S×S$⁠, but can be used for analytical continuation similar to the procedure outlined in section 3. Specifically, following the discussion of section 3.4, we find that (B.4) yields analyticity of $Ψ++$ within $(H−∖{a1})×UHP$ whereas (B.5) yields analyticity of $Ψ++$ within $UHP×(H−∖{a2})$⁠.

Acknowledgements

The authors would like to acknowledge funding by EPSRC (EP/W018381/1 and EP/N013719/1) for R. C. Assier and a University of Manchester Dean’s scholarship award for V. D. Kunz.

References