## Abstract

A generalization of the Schwarz–Christoffel mapping to multiply connected polygonal domains is obtained by making a combined use of two preimage domains, namely, a rectilinear slit domain and a bounded circular domain. The conformal mapping from the circular domain to the polygonal region is written as an indefinite integral whose integrand consists of a product of powers of the Schottky-Klein prime functions, which is the same irrespective of the preimage slit domain, and a prefactor function that depends on the choice of the rectilinear slit domain. A detailed derivation of the mapping formula is given for the case where the preimage slit domain is the upper half-plane with radial slits. Representation formulae for other canonical slit domains are also obtained but they are more cumbersome in that the prefactor function contains arbitrary parameters in the interior of the circular domain.

## 1. Introduction

The Schwarz–Christoffel mapping to polygonal domains is an important result in the theory of complex-valued functions and one that finds numerous applications in applied mathematics, physics and engineering [1]. The applicability of Schwarz–Christoffel formula is nonetheless limited by the fact that it pertains only to simply connected polygonal domains. Therefore, there has long been considerable interest in extending the Schwarz–Christoffel transformation to multiply connected polygonal domains.

Recently, explicit formulae for generalized Schwarz–Christoffel mappings from a circular domain to both bounded and unbounded multiply connected polygonal regions have been obtained using two equivalent approaches. Using reflection arguments, DeLillo and co-workers [2,3] derived infinite product formulae for the generalized Schwarz–Christoffel mappings, whereas Crowdy [4,5] used a function-theoretical approach to obtain a mapping formula in terms of the Schottky-Klein prime function associated with the circular domain. (The equivalence of the two methods was established in [3].) It is important to point out that both these formulations entail an implicit choice of a multiply connected rectilinear slit domain in an auxiliary preimage plane, whereby each rectilinear boundary segment of this preimage domain is mapped to a polygonal boundary (see below). The introduction of a preimage domain with rectilinear boundaries makes it possible to write down an explicit expression for the derivative of the desired conformal mapping, which can then be integrated to yield a generalized Schwarz–Christoffel formula. It is thus clear that different choices of rectilinear slit domains in the auxiliary plane will result in different formulae for the generalized Schwarz–Christoffel mapping. Therefore, a derivation of alternative representation formulae is of some interest.

In this paper, a general formalism to construct conformal mappings of multiply connected polygonal domains is presented. The method applies to both bounded and unbounded polygonal domains but emphasis will be given to the bounded case as formulated in §2. A key ingredient in the approach described herein is the introduction of a multiply connected rectilinear slit domain in an auxiliary λ-plane, such that each rectilinear boundary in the λ-plane is mapped to a polygonal boundary in the *z*-plane. It is advantageous to have rectilinear boundaries in the preimage domain because the derivative of the respective conformal mapping has constant argument on each of domain’s boundaries. To derive an expression for the mapping derivative, it is expedient to introduce a slit map from a bounded multiply connected circular domain in an auxiliary *ζ*-plane to the rectilinear slit domain in the λ-plane. Using the properties of the Schottky-Klein prime function associated with the circular domain, as summarized in §§3 and 4, it is possible to obtain an explicit expression for the derivative of the mapping from the circular domain to the polygonal region. The desired conformal mapping is then written in final form as an indefinite integral whose integrand consists of (i) a product of powers of the Schottky-Klein prime functions and (ii) a prefactor function, *S*(*ζ*), that depends on the choice of the rectilinear slit domain in the λ-plane.

A detailed derivation of the prefactor *S*(*ζ*) is given in §5 for the case where the rectilinear slit domain consists of the upper half-plane cut along radial segments. This preimage domain turns out to be the most convenient one not only in that it naturally extends to multiply connected domains the usual treatment of simply connected polygonal regions but most importantly because it yields a simpler formula for the generalized Schwarz–Christoffel mapping. As further illustration of the method presented here, the corresponding formulae for two other canonical slit domains (including the case considered by Crowdy [4]) are derived in §6. These alternative representations have, however, the inconvenience of containing certain arbitrary parameters in the interior of the circular domain. For completeness, the case of unbounded polygonal domains is briefly discussed in §7, after which our main results and conclusions are summarized in §8.

## 2. Mathematical formulation

### (a) Bounded polygonal regions

Let us consider a target region *D*_{z} in the *z*-plane that is a bounded (*M*+1)-connected polygonal domain. The outer boundary polygon is denoted by *P*_{0} and the *M* inner polygons by *P*_{j}, *j*=1,…,*M* (figure 1*b*). Let the vertices at polygon *P*_{j}, *j*=0,1,…,*M*, be denoted by *k*=1,…,*n*_{j}, and let

### (b) Radial slit domains in the upper half-plane

Let us now consider a domain *D*_{λ} in a subsidiary λ-plane consisting of the upper half-plane with *M* slits pointing towards the origin. Denote by *L*_{0}, the real axis in the λ-plane and by *L*_{j}, *j*=1,…,*M*, the *M* radial slits (figure 1*a*). We seek a conformal mapping *z*(λ) from the radial slit domain *D*_{λ} in the upper half-λ-plane to the polygonal domain *D*_{z} in the *z*-plane, where the real λ-axis is mapped to the outer polygon *P*_{0} and each radial slit *L*_{j} is mapped to an inner polygon *P*_{j}.

If we denote by *P*_{j}, *j*=0,1,…,*M*, then *z*(λ) must have a branch point at *z*/dλ must have constant arguments on each boundary segment in the λ-plane, that is,
*z*-planes. In the case of simply connected polygonal regions, conditions (2.3) and (2.4) can be easily satisfied by writing d*z*/dλ as a product of monomials of the type *z*/dλ, for no longer is it obvious how to construct a function that has constant argument on the multiple rectilinear boundaries of the domain *D*_{λ}. This obstacle can however be overcome by introducing a conformal mapping, λ(*ζ*), from a circular domain *D*_{ζ} in an auxiliary *ζ*-plane to the slit domain *D*_{λ}. This allows us to write an explicit expression for the derivative d*z*/dλ in terms of the *ζ* variable, as will be seen in §5.

### (c) Bounded circular domains

Let *D*_{ζ} be a circular domain in the *ζ*-plane consisting of the unit disc with *M* smaller non-overlapping discs excised from it. Label the unit circle by *C*_{0} and the *M* inner circular boundaries by *C*_{1},…,*C*_{M}, and denote the centre and radius of the circle *C*_{j} by *δ*_{j} and *q*_{j}, respectively. For convenience, we introduce the notation *δ*_{0}=0 and *q*_{0}=1 for the unit circle. A schematic of *D*_{ζ} is shown in figure 1*c*.

Now, let λ(*ζ*) be a conformal mapping from the circular domain *D*_{ζ} to the domain *D*_{λ} defined above, where the unit circle, is mapped to the real axis in the λ-plane and the interior circles *C*_{j} map to the slits *L*_{j}, *j*=1,…,*M*. Furthermore, let the point *ζ*=1 map to the origin in the λ-plane and the point *ζ*=−1 map to infinity (figure 1*a*,*c*). In an abuse of notation, we shall write *z*(*ζ*)≡*z*(λ(*ζ*)) for the associated conformal mapping from *D*_{ζ} to the polygonal region *D*_{z}, where the unit circle *C*_{0} maps to the outer polygonal boundary *P*_{0} and the interior circles *C*_{j} map to the inner polygons *P*_{j} (figure 1*b*,*c*). If we denote by *ζ*-plane of the vertices *P*_{j}, then condition (2.3) can be recast as
*z*_{λ}(*ζ*) denotes the derivative d*z*/dλ expressed in terms of the *ζ*-variable. Similarly, the requirement (2.4) becomes
*z*_{λ}(*ζ*) that satisfies conditions (2.5) and (2.6) by exploiting the properties of the Schottky-Klein prime function associated with the circular domain *D*_{ζ}. Once an expression for *z*_{λ}(*ζ*) is obtained, a corresponding expression for *z*_{ζ}(*ζ*) follows from the chain rule which can then be integrated yielding the desired conformal mapping *z*(*ζ*), as will be shown in §5. Prior to this, however, a brief overview of the Schottky-Klein prime function is in order.

## 3. Schottky groups and the Schottky-Klein prime function

Consider the bounded circular domain *D*_{ζ} defined above (figure 1*c*). Introduce the following Möbius maps:
*ζ*∈*C*_{j}, it is easy to establish the following relations:

Now let *C*′_{j}, *j*=1,…,*M*, denote the reflection of the circle *C*_{j} in the unit circle *C*_{0}. One can readily verify that every *θ*_{j}(*ζ*) maps the interior of the circle *C*_{j}′ onto the exterior of the circle *C*_{j}. Thus, the set *Θ* consisting of all compositions of the maps *θ*_{j}(*ζ*), *j*=1,…,*M*, and their inverses defines a classical Schottky group [7]. The region in the complex plane exterior to all of the circles *C*_{j} and *C*_{j}′ is called the *fundamental region*, *F*, of the Schottky group *Θ*. Given a Schottky group *Θ*, we can associate a Schottky-Klein prime function, denoted by *ω*(*ζ*,*α*), for any two distinct points *ζ* and *α* in the fundamental region *F*.

The Schottky-Klein prime function has the following infinite product representation [7]:
*Θ*′′⊂*Θ* consists of all compositions of the maps *θ*_{j}(*ζ*) and *Θ*′′, then

Let us now recall some useful functional identities involving the Schottky-Klein prime function. Firstly, note that by definition the Schottky-Klein prime function is antisymmetric in its two arguments:
*D*_{ζ}, the following functional relation
*ζ*_{1}, *ζ*_{2}, *γ*_{1} and *γ*_{2} be four points in *F*, then we have
*ω*(*θ*_{j}(*ζ*),*γ*)/*ω*(*ζ*,*γ*); see also related discussion in the monograph by Hejhal [9]. Here, the functions {*v*_{j}(*ζ*)|*j*=1,…,*M*} are the *M* integrals of the first kind of the Riemann surface associated with the domain *D*_{ζ}. These are analytic but not single-valued functions everywhere in *F*. (They can be made single valued by inserting cuts connecting each pair of circles *C*_{j} and *C*_{j}′[7].) Each function *v*_{j}(*ζ*) has constant imaginary part on the circles *C*_{1},…,*C*_{M}, and zero imaginary part on the unit circle *C*_{0}, that is,
*Q*_{jl} is a real constant, with *Q*_{j0}=0 [10].

As particular cases of (3.6), we have
*ζ*_{1},*ζ*_{2}∈*C*_{l}, *l*=0,1,…,*M*, the function *β*_{j}(*ζ*,*γ*) satisfies the following relations:
*ζ*∈*C*_{0} and everywhere else in *F* by analytic continuation. On the other hand, for *ζ*∈*C*_{l}, *l*=1,…,*M*, relation (3.8) implies that
*j*=0, if we define

## 4. Radial slit maps

In this section, two general classes of functions are defined as ratios of Schottky-Klein prime functions (or of products thereof) in such a way that they have constant arguments on the circles *C*_{j}, *j*=0,1,…,*M*. Because of this property, which will be extensively used in §5, these functions represent radial slit maps defined on the circular domain *D*_{ζ}. Here, there are two cases to consider depending on whether the image radial slit domain is bounded or unbounded.

### (a) Bounded radial slit domains

First, define the functions
*C*_{l}, *l*=0,1,…,*M*. To see this, note that for *ζ*∈*C*_{l}, one has

As *F*_{j}(*ζ*;*ζ*_{1},*ζ*_{2}) has constant argument on *C*_{l}, *l*=0,1,…,*M*, a simple zero at *ζ*=*ζ*_{1}∈*C*_{j} and a simple pole at *ζ*=*ζ*_{2}∈*C*_{j}, it immediately follows that this function maps the circular domain *D*_{ζ} onto a half-plane punctured along *M* radial segments, where the circle *C*_{j} is mapped to the axis containing the origin whose direction has argument *C*_{l}, *l*≠*j*, are mapped to the slits. An alternative formula for this mapping in terms of an infinite product was obtained by DeLillo & Kropf [11].

From the preceding discussion, it is then clear that the conformal mapping defined by
*D*_{ζ} onto the upper half-λ-plane with *M* radial slits excised from it, where the points *ζ*=1 and *ζ*=−1 are, respectively, mapped to the origin and infinity in the λ-plane, the unit circle maps to the real axis and the inner circles map to the radial slits (figure 1*a*,*c*). For later use, we quote here the derivative of mapping (4.4)
*ω*_{ζ}(*ζ*,*α*) denotes the derivative of *ω*(*ζ*,*α*) with respect to its first argument.

### (b) Unbounded radial slit domains

Now, consider a second class of functions defined by the following ratio:
*α* and *β* are two arbitrary points in *D*_{ζ}. Using arguments analogous to those employed in the §4*a*, it is easy to verify [4,10] that
*D*_{ζ} to the entire λ-plane cut along *M*+1 radial slits, where the point *ζ*=*α* is mapped to the origin in the λ-plane, the point *ζ*=*β* is mapped to infinity and each circle *C*_{j}, *j*=0,1,…,*M*, is mapped to a radial slit in the λ-plane.

The functions *F*_{j}(*ζ*;*ζ*_{1},*ζ*_{2}) and *Q*(*ζ*;*α*,*β*) defined above play an important role in constructing conformal mappings to multiply connected polygonal domains, as will become evident in §5.

## 5. Conformal mappings to bounded polygonal domains

In this section, we construct an explicit formula for the conformal mappings, *z*(*ζ*), from the bounded circular domain *D*_{ζ} to a bounded multiply connected polygonal domain *D*_{z}, using as an auxiliary tool the slit map λ(*ζ*) from *D*_{ζ} to the upper half-λ-plane with *M* radial slits. As explained in §2*c*, we first need to obtain an expression for the derivative *z*_{λ}(*ζ*) such that it has: (i) the appropriate branch point at the prevertices *C*_{j}.

To this end, let *C*_{j}. Using (2.2), (4.1) and (4.3), it is not difficult to show that the functions
*C*_{l}, *l*=0,1,…,*M*. Let us then write
*R*(*ζ*) is a correction function to be determined later. Because *ω*(*ζ*,*γ*) has a simple zero in *ζ*=*γ*, it is clear that *z*_{λ}(*ζ*) has the right branch point at *C*_{j}, so long as the function *R*(*ζ*) does so. This requirement, together with additional constraints on *z*_{λ}(*ζ*) concerning the location of its zeros and poles, dictates the specific form of *R*(*ζ*), as shown next.

First, note that the points *ζ*-plane of the end points of the respective slits in the λ-plane. This follows from the fact that dλ/d*ζ* vanishes at the preimages of the slit end points, whereas d*z*/dλ does not; hence, *z*_{λ}(*ζ*) must have simple poles at these points. More specifically, the points *j*=1,…,*M*, are obtained by computing the roots (on the circle *C*_{j}) of the following equation:
*ζ*; see (4.5). Note, furthermore, that as *z*_{λ}(*ζ*) is non-zero, the terms containing these points in the numerator of (5.1) must be cancelled out by an appropriate choice of the function *R*(*ζ*). In addition, *R*(*ζ*) must also produce a double zero for *z*_{λ}(*ζ*) at *ζ*=−1, because dλ/d*ζ* has a double pole at this point; see (4.5).

These requirements can be satisfied by choosing *R*(*ζ*) of the form
*C*_{j}; see (4.3). Inserting (5.3) into (5.1) yields
*i*) has been absorbed into

Upon integrating (5.5), one then finds that the desired mapping *z*(*ζ*) is given by
*S*(*ζ*) in (5.6) are to be determined by solving equation (5.2), which in turn depends on the conformal moduli {*δ*_{j},*q*_{j}|*j*=1,…,*M*} of the domain *D*_{ζ}. For a given polygonal domain *D*_{z}, the conformal moduli of the domain *D*_{ζ} are not known *a priori* and must be determined simultaneously with the other parameters appearing in (5.5), namely, the prevertices *parameter problem* [1] in general is a difficult task.

Fortunately, in many applications, the specific details of the target polygonal domain (e.g. the areas and centroids of the polygons *P*_{j}, and the lengths of their respective edges) need not be known *a priori*. In such cases, one can freely specify the domain *D*_{ζ}, all prevertices on the unit circle and all but two prevertices on each inner circle *C*_{j}, and then solve the reduced parameter problem associated with the orientation of the various polygons and the univalence of the mapping function *z*(*ζ*). For instance, formulae (5.6) and (5.7) were recently used by Green & Vasconcelos [12] to construct a conformal mapping from the circular domain *D*_{ζ} to a degenerate polygonal domain consisting of a horizontal strip with *M* vertical slits in its interior (this conformal mapping corresponds to the complex potential for multiple steady bubbles in a Hele–Shaw channel).

## 6. Representation formulae using other canonical slit domains

The procedure described in the previous section can be readily extended to other rectilinear slit domains *D*_{λ} in the subsidiary λ-plane, so long as the corresponding slit map λ(*ζ*) is known explicitly. For each choice of domain *D*_{λ}, a specific formula results for the prefactor *S*(*ζ*) appearing in conformal mapping (5.7). As an illustration of our procedure, we derive below the respective expressions for the function *S*(*ζ*) associated with two of the five canonical slit domains listed in the book of Nehari [13], namely: (i) a circular disc with *M* concentric circular-arc slits and (ii) an unbounded radial slit domain obtained by excising from the entire plane *M*+1 rectilinear slits pointing towards the origin. (Other canonical rectilinear slit domains can be treated in a similar manner.)

Let us first discuss the case of an auxiliary slit domain consisting of a disc with concentric circular-arc slits, as originally considered by Crowdy [4]. Here, the function
*α*∈*D*_{ζ}, maps the circular domain *D*_{ζ} onto the unit disc with *M* concentric circular slits, where the point *ζ*=*α* maps to the origin in the *η*-plane [4]. Thus, the logarithmic transformation
*ζ*=*α* to *ζ*=1, maps *D*_{ζ} to a domain *D*_{λ} in the λ-plane consisting of a semi-infinite strip bounded from the right by the line Re[λ]=0 and containing *M* vertical slits in its interior, where the unit circle *C*_{0} is mapped to the vertical edge of the strip and the circles *C*_{j}, *j*=1,…,*M*, are mapped to the vertical slits. As before, the points *ζ*-plane of the end points of the slits in the λ-plane, so that *z*_{λ}(*ζ*) has simple poles at these points. As the point *ζ*=*α* is a logarithmic singularity of the slit map λ(*ζ*), then *z*_{λ}(*ζ*) must have a simple zero at this point.

Starting from (5.1) and in the light of the preceding discussion, one readily concludes that in this case the correction function *R*(*ζ*) can be chosen as
*C*_{j}, as follows from (4.7) and from the fact that *z*_{ζ}(*ζ*) can be rewritten as in (5.5), where the prefactor function *S*(*ζ*) now reads

As a further illustration of the generality of our approach, consider now the case where the domain *D*_{λ} consists of the entire λ-plane with *M*+1 radial slits, denoted by *L*_{j}, *j*=0,1,…*M*. Recall that the corresponding slit map λ(*ζ*) in this case is given by (4.8) which has a simple pole at *ζ*=*β*; hence, *z*_{λ}(*ζ*) must have a double zero at this point. Note furthermore that the points *L*_{j}. This implies, in particular, that the prime functions containing the points *z*_{λ}(*ζ*) must now have simple poles at these two points, as well as at *j*=1,…,*M*. This can be accomplished with an appropriate choice of the function *R*(*ζ*), which must also produce the required double zero at *ζ*=*β*. Indeed, these requirements can be satisfied by choosing
*S*(*ζ*) for this case is given by
*S*(*ζ*) pertaining to other canonical slit domains can be readily obtained, but further details will not be presented here.

It is to be emphasized that, in contrast to formula (5.6) for the upper half-plane with radial slits, the prefactor functions *S*(*ζ*) obtained for other canonical slit domains have arbitrary parameters, e.g. the point *α* in (6.5) and the points *α* and *β* in (6.7), in the interior of the domain *D*_{ζ}. To avoid this extra unnecessary complication, the formulation given in §5 should be preferred in applications; see §8 for further discussion on this point.

## 7. Conformal mappings to unbounded polygonal domains

In this section, we consider, for completeness, the problem of conformal mappings from the bounded circular domain *D*_{ζ} to unbounded multiply connected polygonal regions, using the upper half-λ-plane with *M* radial slits as our auxiliary rectilinear slit domain.

Let the target domain *D*_{z} in the *z*-plane be the unbounded region exterior to *M*+1 non-overlapping polygons *P*_{j}, *j*=0,1,…,*M*. We shall adopt the same notation as in §2 to designate the vertices of the polygonal boundaries and the corresponding turning angles. Note, however, that now we have
*z*(*ζ*), from a bounded circular domain *D*_{ζ} to the unbounded polygonal region *D*_{z}, where each circle *C*_{j}, *j*=0,1,…,*M*, is mapped to a polygonal boundary *P*_{j} and the point

The starting point for constructing the desired mapping is the equation
*ζ*-plane of the end points of the *M* slits in the λ-plane, whereas *C*_{0} at our disposal.

It is also clear from previous discussions that *z*_{λ}(*ζ*) must have a double pole at *ζ*=−1. These requirements can be enforced by choosing
*S*(*ζ*) as given in (5.6).

After integrating (7.5), one finds that the conformal mapping *z*(*ζ*) from *D*_{ζ} to an unbounded multiply connected polygonal region is given by the same integral expression (5.7) obtained for the case of bounded polygonal domains, the only difference being that the prefactor is now given by the function *D*_{ζ} to an unbounded polygonal region by implicitly considering an auxiliary rectilinear slit domain consisting of a semi-infinite strip with vertical slits. As shown above, relation (7.6) holds irrespective of the choice of the rectilinear slit domain used to construct the corresponding mapping formulae for bounded and unbounded multiply connected polygonal domains.

## 8. Discussion

A general framework has been presented for constructing conformal mappings from a bounded circular domain *D*_{ζ} to a multiply connected polygonal region *D*_{z} (either bounded or unbounded). A key ingredient in our scheme is the introduction of a conformal mapping from *D*_{ζ} to a rectilinear slit domain *D*_{λ} in a subsidiary λ-plane. This allows us to write an explicit formula for the derivative *z*_{λ}(*ζ*), and hence for *z*_{ζ}(*ζ*), in terms of the Schottky-Klein prime function associated with the domain *D*_{ζ}. After integration, the desired conformal mapping *z*(*ζ*) is then obtained as an indefinite integral whose integrand consists of a product of powers of the Schottky-Klein prime functions and a prefactor function *S*(*ζ*) that depends on the choice of the rectilinear slit domain *D*_{λ}.

An explicit formula for *S*(*ζ*) was derived by first considering the case where the rectilinear slit domain *D*_{λ} consists of the upper half-plane with radial slits. The generality of our approach was subsequently demonstrated by obtaining alternative formulae for the prefactor function *S*(*ζ*) pertaining to other canonical slit domains. For a given polygonal domain *D*_{z}, these various formulae (once their associated parameters have been determined) provide different representations of the same conformal mapping *z*(*ζ*).

It is to be noted, however, that the formula for *S*(*ζ*) obtained by considering the upper half-plane with radial slits is arguably the simplest one, in the sense that the only unknown parameters are the zeros of the slit map λ(*ζ*), which can be numerically computed once the domain *D*_{ζ} is specified. By contrast, the corresponding formulae for *S*(*ζ*) obtained for other canonical slit domains have, in addition, one or more arbitrary parameters inside the domain *D*_{ζ}. Although the function *S*(*ζ*) does not ultimately depend on the values of these parameters (except for an overall factor independent of *ζ* [14]), the existence of arbitrary parameters inside *D*_{ζ} may present an additional (and unnecessary) source of complication. This is particularly true in the case that a given target polygonal domain *D*_{z} is specified, for here the conformal moduli of the domain *D*_{ζ} are not known *a priori* and hence the arbitrary parameters cannot be fixed beforehand.

In the light of the foregoing discussion, it can be argued that the mapping formula derived in §5 using the upper half-plane with radial slits should be viewed as the natural extension to multiply connected polygonal domains of the standard Schwarz–Christoffel mapping from the upper half-plane to a simply connected polygonal region. It should also be preferable in applications because of its simplicity. In this context, it is worth noting that this mapping formula was recently employed by Green & Vasconcelos [12] to construct exact solutions for multiple bubbles steadily moving in a Hele–Shaw channel. It is thus hoped that other problems involving multiply connected polygonal domains may be conveniently tackled with the formalism presented here.

## Funding statement

Financial support from a scholarship from the Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (Brazil) for a sabbatical stay at ICL is acknowledged.

## Acknowledgements

The author wishes to thank C. C. Green and D. G. Crowdy for helpful discussions. He is also appreciative of the hospitality of the Department of Mathematics at Imperial College London (ICL), where this research was carried out.

- Received December 20, 2013.
- Accepted March 3, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.