## Abstract

A general framework for analytic evaluation of singular integral equations with a Cauchy kernel is developed for higher order line elements of curvilinear geometry. This extends existing theory which relies on numerical integration of Cauchy integrals since analytic evaluation is currently published only for straight lines, and circular and hyperbolic arcs. Analytic evaluation of Cauchy integrals along straight elements is presented to establish a context coalescing new developments within the existing body of knowledge. Curvilinear boundaries are partitioned into sectionally holomorphic elements that are conformally mapped from a local curvilinear *Z*-plane to a straight line in the -plane. Cauchy integrals are evaluated in these planes to achieve a simple representation of the complex potential using Chebyshev polynomials and a Taylor series expansion of the conformal mapping. Bell polynomials and the Faà di Bruno formula provide this Taylor series for mappings expressed as inverse mappings and/or compositions. Examples illustrate application of the general framework to boundary-value problems with boundaries of natural coordinates, Bezier curves and B-splines. Strings formed by the union of adjacent curvilinear elements form a large class of geometries along which Dirichlet and/or Neumann conditions may be applied. This provides a framework applicable to a wide range of fields of study including groundwater flow, electricity and magnetism, acoustic radiation, elasticity, fluid flow, air flow and heat flow.

## 1. Introduction

Vector fields generated by two-dimensional objects with idealized geometry are important in many fields of study. Seminal investigations include Rankine (1864) for flow past ship-shaped curves, Maxwell (1881) for electricity and magnetism, Lamb (1945) for fluid flow, Carslaw & Jaeger (1959) for heat flow and Hess & Smith (1967) for aerofoils. These studies used combinations of objects with geometries formed by the natural coordinates of coordinate systems where the Laplace equation separates (Morse & Feshbach 1953). Exact solutions for a broader range of geometries are provided by conformal mapping. Moon & Spencer (1961) formulated such mappings using power, exponential, logarithmic, hyperbolic and elliptic functions. Solutions based upon these mappings have been widely used, for example in groundwater and petroleum applications by Muskat (1937), Polubarinova-Kochina (1962), Pilatovskii (1966) and Bear (1972). Craster (1997) summarized other applications where free boundaries were mapped to the upper half plane.

Cauchy integrals have been used extensively to develop approximate solutions to Dirichlet and Neumann boundary-value problems with boundaries formed through combinations of natural coordinates and conformal mappings. Strack (1989) developed methods in the groundwater field to study jumps in aquifer properties, drainage through cracks and faults and groundwater/surface water interactions. Distributions of charge and polarization in electromagnetic theory were studied by Stratton (1941). Schenck (1968) studied acoustic radiation of pressure fields produced or perturbed by objects in an acoustic medium. Stresses and displacement in an elastic medium were studied by Crouch (1983) near excavations and by Aparicio & Atkinson (1997) near cracks. Saffman (1992) discussed a wide range of applications in fluid mechanics, aerodynamics and turbulence. A summary of application to slender bodies (ships, submarines, marine animal propulsion, bird flight, aircraft manoeuvring and missile guidance) was presented by Chadwick (2005). Chang *et al*. (1973) discussed heat conduction applications.

Evaluation of Cauchy integrals commonly employs numerical integration techniques (Burton & Miller 1971; Jaswon & Symm 1977; Theocaris & Iokimidis 1979). A review of Cauchy integrals in the Boundary Element Method was presented by Guiggiani (1991), who noted that analytic integration can be used only for low-order (i.e. small *M* in equation (2.8)) straight elements. The need for numerical integration techniques is iterated by Rosen & Cormack (1995): ‘because analytical evaluation of singular and near-singular integrals on curved surfaces is in general not possible, it is essential to have efficient and systematic techniques for their numerical evaluation’.

Recent advances in the Analytic Element Method (AEM) have produced formulations using analytic, closed-form integration techniques for Cauchy integrals of higher order strength (large *M*) along straight lines (Janković & Barnes 1999; Strack 2003; Steward *et al*. 2005). The AEM has also been applied to boundary-value problems with geometries of circular arcs (Strack 1989), hyperbolic arcs (Strack 2003), Bezier curves (Le Grand 1999) and B-splines (Le Grand 2003).

A general framework for analytic evaluation of Cauchy integrals of higher order strength and curvilinear geometry is developed. First, a method for analytic evaluation of Cauchy integrals along straight elements is presented within a framework coalescing the new development of curvilinear elements within the existing body of knowledge. Cauchy integrals along curvilinear elements are then formulated along boundaries generated through conformal mappings. This approach brings to bear a unique application of conformal mappings, Taylor series, Chebyshev polynomials, Bell (1934) polynomials and the Faà di Bruno (1855) formula. Examples illustrate the ability to accurately reproduce the near-field behaviour along smoothly varying curvilinear elements while preserving small-scale singular behaviour near corners (at the intersection of non-parallel line segments) and tips (at the ends of line segments).

## 2. Cauchy integrals

A two-dimensional irrotational, divergence-free vector field with components *v*_{x} and *v*_{y} may be represented using the complex potential(2.1)where the vector field is directed towards decreasing gradient of potential, *Φ*, and tangent to streamlines of constant Lagrange (1781) stream function, *Ψ*. The vector field may be obtained from the complex discharge(2.2)formed by the complex conjugate of the derivative of *Ω* with respect to *z*=*x*+i*y*. Cauchy's integral formula provides a means of computing the complex potential at a point *z* within a closed domain *D* with a simple, smooth boundary ∂*D*,(2.3)where *Ω*(*δ*) takes on the values of potential and stream function along the boundary.

Cauchy integrals may be developed along strings formed by the union of a finite number of adjacent smooth elements which do not intersect one another (Muskhelishvili 1953, p. 33). For example, a singular integral equation with Cauchy kernel and endpoints *z*_{j} and *z*_{j+1} is given by(2.4a)where *Ω*(*δ*)=*μ*(*δ*)+i*ν*(*δ*) is assumed to satisfy the Hölder condition (Muskhelishvili 1953, p. 11) along the element. The first integral creates a jump in potential *μ* across the element and is called a double layer (Muskhelishvili 1953) or line-doublet (Strack 1989). The second integral creates a jump in stream function *ν* across the element and is called a line-dipole (Strack 1989). Integration of (2.4*a*) by parts (Muskhelishvili 1953, p. 24) leads to Fredholm integral equations of the first kind,(2.4b)using *γ*=d*μ*/d*δ* and *σ*=−d*ν*/d*δ*. The first integral contains a distribution of vortex filaments (Helmholtz 1858; Tait 1867) with a vortex density *γ* and is called a line-vortex (Saffman 1992). The second integral removes a net flux per unit length *σ* and is called a simple layer (Muskhelishvili 1953), single layer (e.g. Burton & Miller 1971) or line-sink (Strack 1989).

### (a) Straight elements

This section formulates Cauchy integrals along straight elements in the context of the AEM and establishes a framework coalescing the new development of curvilinear elements within the existing body of knowledge. To simplify mathematical expressions, each straight element (e.g. between endpoints *z*_{j} and *z*_{j+1}) is translated, rotated and scaled to a standardized *Z*-plane, with endpoints at *Z*=−1 and 1, using (Strack 1989, p. 284)(2.5)The complex potential is formulated such that *Ω* has the same values at the same relative positions in the *z*- and *Z*-planes, which gives(2.6)where *W*=*V*_{X}+i*V*_{Y} is the vector field in local coordinates. The line integrals (2.4*a*) and (2.4*b*) are expressed in local coordinates, following Strack (1989, p. 292) for a line-dipole,(2.7)and the strengths *μ*, *ν*, *γ* and *σ* are real along the straight line where .

The functional form of the strength of the line integrals will be approximated here using an orthogonal Chebyshev expansion following Janković & Barnes (1999) and Kacimov (2001),(2.8)where *μ*_{m}, *ν*_{m}, *γ*_{m} and *σ*_{m} are *a priori* unknown strength parameters that will be adjusted to match boundary conditions along the element and *M* is the order of the straight element. The *m*th Chebyshev polynomial of the first kind may be written as(2.9a)where the coefficients *T*_{mn} are found using the recursive relations (Abramowitz & Stegun 1972)(2.9b)The line integrals for straight elements with strengths of the form of Chebyshev polynomials are now written as(2.10)Note that straight elements with strengths expressed as commonly used power series result if *T*_{mn} is set equal to the identity matrix.

Closed-form expressions for the kernel functions , , and are obtained here using integrals adapted from Gröbner & Hofreiter (1975)(2.11a)and(2.11b)Integrating using *a*=−1 and *b*=*Z*, substituting the limits of integration and rearranging the variable of summation gives(2.12)These equations are a generalization of Strack's (1989) eqn (41.24).

The kernel functions take on simpler forms outside a neighbourhood of the element. Following Strack (1989), the logarithm term is expanded in a series (Abramowitz & Stegun 1972)(2.13)Substituting this into the kernel functions and cancelling higher order terms give Laurent series,(2.14)where the number of significant terms in these series decreases at larger distances from the element. It should be noted that *Ω* is analytic everywhere except the ends of the element where *Z*=±1 where ln((*Z*+1)/(*Z*−1)) is singular.

Closed-form expressions may also be developed for the vector field associated with the kernel functions(2.15)where use has been made of(2.16)The vector field also has a simpler form outside a neighbourhood of the element,(2.17)

### (b) Solving strength coefficients to match boundary conditions

The complex potential associated with fields containing line-doublets, line-dipoles, line-vortexes and line-sinks is obtained by summing the contributions over all *j* elements,(2.18)The notation adopted here for a function *a* with indices indicates that *a* is evaluated at the *k*th point on the *i*th element for the *m*th strength coefficient of the *j*th line-doublet, line-dipole, line-vortex or line-sink.

Control points where boundary conditions are to be applied are distributed along these straight elements following the overspecification principle of Janković & Barnes (1999). For example, a set of *K* control points uniformly distributed along the *i*th straight element are located at(2.19)The value of the complex potential at each control point is given by(2.20)and the value of the vector field at each control point is given by(2.21)

Following Strack (1989), a system of equations is developed to determine strength coefficients that satisfy boundary conditions (e.g. Dirichlet, Neumann or combinations of Dirichlet and Neumann) specified in terms of *Ω* and/or *w*. A set of matrices that represent the contribution from the strength of the *j*th element (, , or ,) at control points on the *i*th element is given by(2.22)where takes on values associated with , , or (for boundary conditions specified in terms of *Ω*) or values associated with , , or (for conditions specified in terms of *w*). The unknown strengths ( for line-doublets, for line-dipoles, for line-vortexes or for line-sinks) are determined such that take on the specified boundary condition at .

Following Janković & Barnes (1999), the number of collocation points is specified to be larger than the number of unknown strength coefficients, and each element has a portion that is overspecified (i.e. *k*>*m* in ). The *i*th element may also have a set of fixed conditions,(2.23)that are satisfied exactly (e.g. the net flux removed by a line-sink may be specified; Steward & Jin 2003). The following objective function satisfies the collocation conditions in a least squares sense and the fixed condition exactly(2.24)where are Lagrange multipliers. The objective function is minimal where (where *k* takes on values from 1 to *m*) and . Together, these conditions yield a linear system of equations where the number of equations is equal to the number of unknown strengths,(2.25)This system of equations may be solved directly, for example via Gaussian elimination with pivoting of rows and columns. It may also be solved iteratively (Janković & Barnes 1999) by successively solving for the strengths of the *j*th element.

## 3. Curvilinear elements

Modern implementation of the AEM is capable of handling thousands of straight elements with strengths of order *M*=100 (Craig *et al*. 2006; Bandilla *et al*. 2007). These models enable accurate computation of local detail (singularities in the vicinity of corners and tips) within the context of large regional models. Regional examples in the groundwater discipline include the National Groundwater Model of The Netherlands (de Lange 1996) and the Yucca Mountain model encompassing the State of Nevada (Bakker *et al*. 1999).

The standard approach to modelling boundary segments with curvilinear geometry is to subdivide the segment into a number of straight elements (Strack 1989). While the curvature of a curvilinear segment is more closely approximated as the number of straight elements increases, a singularity in the vector field is introduced at intersections of adjacent, non-parallel straight line segments.

This section develops mathematical methods for curvilinear elements that accurately reproduce the near-field behaviour along smoothly varying curves while preserving the local detail of small-scale singular behaviour. Development occurs within the presented framework for straight elements and provides a general approach to incorporating curvilinear elements with straight elements in large regional models.

### (a) Conformal mappings

Development of Cauchy integrals for curvilinear geometry is facilitated using the set of transformations illustrated in figure 1 that conformally map a curvilinear element to a straight line along the real axis. First, a continuous line segment is mapped onto the real axis in the *ζ*-plane using the conformal mapping *ζ*(*z*). This segment is then partitioned into sectionally holomorphic elements (Muskhelishvili 1953) that are individually mapped to a curvilinear line between *Z*=−1 and 1, using (2.5), and to a straight line between and , using(3.1)where these coordinate transformations are specified for the *j*th element. This partitioning subdivides a segment into a set of elements whereby *Ω*(*z*) is holomorphic in a finite neighbourhood about each element (Muskhelishvili 1953). Later, the complex potential will be formulated in the standardized *Z*-plane and *μ*, *ν*, *γ* and *σ* in (2.4*a*) and (2.4*b*), which are real along line integrals, shall be specified as functions of providing real values along the line.

The conformal mapping and coordinate transformations together provide a means to map from *Z*-coordinates (where the line integral is formulated) to -coordinates (where *μ*, *ν*, *γ* and *σ* are specified);(3.2)Since *ζ*(*z*) is analytic in a neighbourhood of the element ( in figure 1), it may be expanded in this neighbourhood as a Taylor series about *Z*=0,(3.3a)and the coefficients *a*_{p} may be expressed in terms of *ζ*(*z*) giving(3.3b)The conformal mapping *ζ*(*z*) may have singularities, for example a pole exists at *z*=*S*_{1} in figure 1 (Strack 1989, p. 477). A line segment is partitioned into sectionally holomorphic elements that lie within an analytic neighbourhood in the *Z*-plane; for example, partitioning the segment in figure 1 into two elements, one between *z*_{j} and *z*_{j+1} and the other between *z*_{j+1} and *z*_{j+2}, results in sectionally holomorphic elements.

### (b) Complex potential

The Cauchy integrals (2.7) are written along a curvilinear element as(3.4)where *Δ* is the distance along the element in the *Z*-plane. Following Muskhelishvili (1953), and are added to the numerator in the first two integrals, and integration gives(3.5)Strack (1989, p. 293) proved that, when *μ* and *ν* are specified as a polynomial function of *Z*, integration of the last terms gives polynomial functions *p* and *q*,(3.6)The functional form of the strength of curvilinear elements are approximated using an orthogonal Chebyshev expansion, (2.9*a*) and (2.9*b*), in the -plane giving real functions along the element where ;(3.7)Truncating the conformal mapping in (3.3*a*) at the *p*th term and substituting gives(3.8)The number of terms *P* in the truncating Taylor series, which is uniformly convergent in the neighbourhood *Z*<*R*, is chosen such that the approximated geometry of the curvilinear element very closely matches the idealized geometry given by (3.2). The terms associated with Chebyshev polynomials and the conformal mapping are collected in *c*_{mn}(3.9)where *c*_{mn} are obtained by gathering terms in the Chebyshev series with common factors of *Z* using(3.10)This gives expressions in the same form as (2.10). The coefficients *d*_{kp} are obtained through aid of Bell polynomials, *B*_{nk}; equation (A 6) gives(3.11)and the binomial formula enables recursive evaluation of the *k* th power from lower order terms, gathered in the matrix *d*_{kp},(3.12)This method of series expansion numerically reproduces results from the recursive relation presented by Henrici (1956) and Knuth (1998).

While the kernel functions in (2.12) take on the same form for curvilinear and straight elements, the location of the branch cut in the logarithm term differs. For straight elements, the branch cut of the logarithm term in *Ω*_{n}, (2.12), was chosen by Strack (1989) to lie along the *x*-axis, as illustrated in figure 2*a*. Here, equipotentials (dark lines) jump from [(i/2*π*)ln(*Z*+1/*Z*−1)]=−0.5 to 0.5 for points immediately beneath to immediately above the element, while streamlines (light lines) are continuous across the element. Thus, by specifying *μ* as a function of *Z* for straight lines, *μ*(*X*) is real along the element and represents the jump in *Φ* across the element. For curvilinear elements, the branch cut in *Ω*_{n} is chosen to occur across the element as illustrated in figure 2*b*, where is real and *μ*()=*μ*(*Ξ*) is the jump in *Φ* across the element.

### (c) Conformal mappings expressed as an inverse or composition

Development of curvilinear elements presented here uses the derivatives in (3.3*a*) and (3.3*b*). For many important problems, a conformal mapping is expressed as an inverse function, *z*(*ζ*), or as a composition of more than one conformal mapping, , in which case methods are needed to develop the previous derivatives.

Development of the Taylor series for the inverse function is based upon the Lagrange inversion formula (Lagrange 1768),(3.13)A recursive relation to evaluate the *p*th derivative was developed by Apostol (2000),(3.14)where are given by(3.15)This relationship is represented using Bell polynomials, *B*_{nk}, described in appendix A, (A 7*a*) and (A 7*b*), as(3.16)where [−*p*]_{k} are the falling factorials, (A 8).

A conformal mapping may be expressed as a composition of more than one conformal mapping, for example maps *z* to the auxiliary plane, which is then mapped to the *ζ*-plane. In this case, the derivatives in (3.3*a*) and (3.3*b*) may be obtained using Faà di Bruno's formula (1855). Given the Taylor series expansions(3.17)derivatives in the composition are represented using Bell polynomials, (A 5*a*) and (A 5*b*), as(3.18)The composition is sectionally holomorphic if and are both analytic in a neighbourhood of the element (Churchill & Brown 1984; Constantine & Savits 1996). The terms in (3.3*a*) and (3.3*b*) are obtained by evaluating all derivatives at locations corresponding to *Z*=0; *Z*_{c}=(*Z*_{j+1}+*Z*_{j})/2, and . Note that the term associated with *p*=0 in (3.17) is zero, which restricts to mappings where .

## 4. Illustrative examples

Examples are presented next to illustrate the efficacy of the general framework for Cauchy integrals with curvilinear geometry for both Dirichlet and Neumann boundary conditions. First examples are presented for boundaries formed by combinations of natural coordinates, which illustrate application of the general framework to conformal mappings, inverse mappings and compositions. Examples are then presented for B-splines and Bezier curves, geometries widely used to delineate physical boundaries in real-world applications. To aid the reader interested in reproducing the set of example figures, the order of terms *Ω*_{n} in (2.12) is PM between 60 and 90 (the orders of series in the conformal mapping and Chebyshev expansion), with an overspecification factor *I*=4(PM+1), and control points were distributed uniformly along elements in the -plane. The analytic framework has been fully implemented in the open source platform Scilab, which was used to generate figures.

### (a) Curvilinear elements with the geometry of natural coordinates

A straight line segment between *z*_{j} and *z*_{j+1} is mapped to the real axis between *ζ*_{j}=−1 and *ζ*_{j+1}=1 by(4.1)Evaluating *ζ*(*z*) and its derivatives at *z*_{c}=(*z*_{j+1}+*z*_{j})/2 and substituting into (3.3*a*) give the Taylor series coefficients(4.2)which, as expected, gives the mapping(4.3)In this case, the coefficients *c*_{mn} in (2.12) are equal to the Chebyshev coefficients *T*_{mn} and the general framework reproduces previously published results for straight line segments (Janković & Barnes 1999; Strack 2003; Steward *et al*. 2005).

A regular hyperbolic arc connecting *z*_{j} to *z*_{j+1} is mapped to the straight line between *ζ*_{j}=−1 and *ζ*_{j+1}=1 by(4.4)which may be expressed as (*Z*) using (3.2) with *ζ*=, giving(4.5)The complex parameter *c* may be chosen, for example, to match prescribed angles −arg(d/d*Z*)=*θ*_{j} and *θ*_{j+1} at the ends of the element, giving(4.6)

The complex potential and the vector field are illustrated in figure 3*a* using two straight elements and one curvilinear element with hyperbolic geometry. This geometry represents a surface roughness along a straight line in an ambient uniform flow from left to right, and is important in the study of small irregularities placed on an underlying smooth surface (Sarkar & Prosperetti 1995; Smith & Walton 1998). Neumann boundary conditions are satisfied by setting the normal component of the vector field equal to zero at control points,(4.7)where *α* is the direction tangent to the element.

A parabolic arc connecting *z*_{a} to *z*_{b} is mapped to the straight line between *ζ*_{a}=−1 and *ζ*_{b}=1 by(4.8)The value of *z* and the inverse derivatives(4.9)are used to obtain in (3.16). An example of surface roughness of parabolic geometry is illustrated in figure 3*b*. Note that singularities in the conformal mapping may require the segment to be subdivided into sectionally holomorphic elements. For example, the parabolic arc was subdivided into six elements in figure 3*b* to give elements with holomorphic neighbourhoods that avoided the singularity in d*ζ*/d*z* and the zero in d*z*/d*ζ* at *ζ*=−1/*c*.

A circle is mapped on the real axis through use of the bilinear transformation(4.10)with derivatives(4.11)where *z*_{∞}, *z*_{0} and *z*_{a} are three points on the circle that map to *ζ*(*z*_{∞})=∞, *ζ*(*z*_{0})=0 and *ζ*(*z*_{a})=*ζ*_{a}, respectively. The coefficients in the Taylor series in (3.3*a*) for the element between *ζ*_{j}=ζ(*z*_{j}) and *ζ*_{j+1}=*ζ*(*z*_{j+1}) are(4.12)An example of surface roughness with circular geometry is illustrated in figure 3*c*. Note that this mapping coincides with figure 1.

An ellipse centred at *z*_{e} with minor axis of length *Rρ* and major axis of length *R*+*ρ* oriented at angle *α* is mapped to a circle in the plane of radius *R* centred at by(4.13)with inverse(4.14)This maps the centre of an element to the origin, , when(4.15)and the ± sign is positive if where . This circle is mapped to the real axis using(4.16)where , and are three points on the circle and , and . The derivatives required for the Taylor series in (3.3*a*) and (3.3*b*) are obtained by first evaluating using(4.17)in the inversion formula (3.16), and using this with(4.18)in the composition formula (3.18). An example of a potential field near an elliptical surface roughness is illustrated in figure 3*d*, where the elliptical curve was subdivided into seven sectionally holomorphic elements.

### (b) Curvilinear elements with the geometry of Bezier curves and B-splines

A Bezier curve (Bezier 1986; Le Grand 1999) lying on the *ζ*-axis between 0 and 1 is given by the Bernstein polynomials,(4.19)where *z*_{n} are the *K*+1 vertices of the characteristic polygon. This is represented as a Taylor series using the binomial formula and grouping terms with common powers of *ζ*,(4.20)and has derivatives(4.21)The inverse derivatives in (3.16) are obtained by evaluating these derivatives at *ζ*_{c}, which may be obtained iteratively using *z*_{c} and its derivative in Newton's method,(4.22)An example of a potential field associated with a Bezier curve is illustrated in figure 4*a*, with a Dirichlet boundary condition of uniform potential. This string has the behaviour of a point sink as *z*→∞ with strength equal to the cumulative jump in stream function along all elements (Strack 1989, p. 297).

A central basis spline with knots at *ζ*−*m*={−*n*/2, −*n*/2+1, …, *n*/2} is given by (Schoenberg 1946, 1973)(4.23)where *x*_{+}=max(0, *x*) is the one-sided power function. The coefficients *c*_{m} for a spline connecting *z*_{0}, …, *z*_{M} may be obtained from the value of *z* and its derivatives at the knots. For example, a cubic B-spline (*n*=4) with known values of and the first derivatives at the ends *ζ*=0 and *M* gives (Schoenberg 1973, p. 79)(4.24)

A line segment for the portion of a B-spline lying between knots *k* and *k*+1 may be expressed without the one-sided power function as follows for even *n*(4.25a)with derivatives(4.25b)using *M*_{n}(*ζ*)=0 in (4.23) for *ζ*≤−*n*/2 and *ζ*≤*n*/2. The conformal mapping for an element with ends *ζ*_{j} and *ζ*_{j+1} between two knots is obtained by computing *z*_{j} and *z*_{j+1} using (4.25*a*), obtaining *ζ*_{c} from *z*_{c}=(*z*_{j+1}+*z*_{j})/2 using Newton's method (4.22) and using derivatives of the inverse mapping, (4.25*b*), in (3.16). The potential field associated with a B-spline is illustrated in figure 4*b*.

## 5. Discussion

The examples illustrate that this analytic formulation accurately reproduces the near-field behaviour along smoothly varying curvilinear elements. The stream function is uniform along the impermeable Neumann boundary condition in figure 3, the potential is uniform along the Dirichlet boundary condition in figure 4, and the vector field is continuous across adjacent elements on a curvilinear line segment. These examples also illustrate singular behaviour in the vicinity of corners and tips.

An exact solution for an impermeable corner which goes from angle *α* to *α*+*β* when *β*>0 is adapted from Polubarinova-Kochina (1962, p. 39),(5.1)where *a*_{1} and *a*_{2} are real and *b*_{1} and *b*_{2} are complex. This exact solution is plotted along with a close-up view in figure 5, with good agreement in both complex potential and vector field. A stagnation point is formed above the point where adjacent line segments join and the vector field is singular below this point.

An exact solution for a tip element with uniform potential is obtained by multiplying (5.1) by −i as *β*→*π*. This exact solution is plotted in figure 5, and also shows good agreement in both complex potential and vector field with a close-up view of the analytic formulation.

## 6. Conclusions

A general framework is presented for Cauchy integrals along boundaries composed of adjacent segments with curvilinear geometry. Each segment is conformally mapped from physical *z*-coordinates to a straight line along the real axis in the *ζ*-plane using *ζ*(*z*) as illustrated in figure 1. A segment is decomposed into a set of sectionally holomorphic elements mapping from a local curvilinear *Z*-plane to a straight line in the -plane with equation (3.2) where the ends of the element lie at −1 and +1 in each plane.

Cauchy integrals are formulated in terms of *Z*, (3.4), and the jump in potential or stream function across the element is specified as a real function of , (3.7). Development of the complex potential, *Ω* in (2.12), uses Chebyshev polynomials, (2.9*a*) and (2.9*b*), and a Taylor series of the conformal mapping, (3.3*a*) and (3.3*b*), to achieve a simple formula for *Ω*. Bell polynomials are used to obtain this Taylor series for conformal mappings specified as the inverse mapping *z*(*ζ*) in (3.16), and Faà di Bruno's formula, (3.18), is used for conformal mapping specified as a composition.

Examples illustrate application to Neumann & Dirichlet conditions along boundaries formed by natural coordinates (figure 3) and Bezier curves and B-splines (figure 4). This analytic framework very accurately satisfies boundary conditions along smooth curvilinear segments while reproducing small-scale singular behaviour near corners and tips as illustrated in figure 5. Strings of line segments may be joined to generate the complex potential and vector fields for a very broad range of geometries.

This formulation extends analytic evaluation of Cauchy integrals to boundaries of curvilinear geometry formed through conformal mapping. It represents a unique application of conformal mappings, Taylor series, Chebyshev polynomials, Bell polynomials and Faà di Bruno's formula. This provides a framework to study singular integral equations with a Cauchy kernel along boundaries with curvilinear geometry for a wide range of fields of study including groundwater flow, electricity and magnetism, acoustic radiation, elasticity, fluid flow, air flow and heat flow.

## Acknowledgments

The authors gratefully acknowledge financial support provided by The Netherland's National Institute for Integrated Water Management and Waste Water Treatment (RIZA), the Provost Office's Targeted Excellence Program at Kansas State University, the National Science Foundation (grant EPS0553722) and the United States Department of Agriculture/Agriculture Research Service (Cooperative Agreement 58-6209-3-018). The very constructive referee comments by Anvar R. Kacimov clarified and strengthened contributions.

## Footnotes

- Received July 20, 2007.
- Accepted October 4, 2007.

- © 2007 The Royal Society