Conformal Mapping from the Half-Plane Onto a Circular Polygon with Cusps

2021 ◽  
Vol 65 (6) ◽  
pp. 8-20
Author(s):  
I. A. Kolesnikov
Keyword(s):  
Conformal Mapping ◽  
Half Plane ◽  
Circular Polygon

Discussion on the conformal mapping of a half-plane onto a unit disk in anisotropic elasticity and related applications

2020 ◽  
Vol 26 (1) ◽  
pp. 110-117
Author(s):  
Ming Dai ◽  
Jian Hua
Keyword(s):  
Thin Films ◽  
Conformal Mapping ◽  
Unit Disk ◽  
Half Plane ◽  
Complex Variable ◽  
Plane Deformation ◽  
Anisotropic Elasticity ◽  
Two Dimensional ◽  
Plane Shear ◽  
Anisotropic Elastic

The conformal mapping, which transforms a half-plane into a unit disk, has been used widely in studies involving an isotropic elastic half-plane under anti-plane shear or plane deformation. However, very little attention has been paid to the possibility of utilizing this mapping in the study of an anisotropic elastic half-plane under the same deformation. In this paper, we discuss a general case of an arbitrarily located anisotropic elastic half-plane that corresponds to several affine counterparts (resulting from corresponding complex variable formalism). We show that this mapping is indeed applicable to each of the affine half-planes if and only if the key parameters in the mapping satisfy simple geometrical conditions. In addition, we introduce the application of this mapping with the corresponding geometrical conditions to the related study of anisotropic thin films under two-dimensional deformation.

A Point Dislocation Interacting with an Elliptical Hole Located at a Bi-Material Interface

2012 ◽  
Vol 151 ◽  
pp. 75-79 ◽  
Author(s):  
Xian Feng Wang ◽  
Feng Xing ◽  
Norio Hasebe ◽  
P.B.N. Prasad
Keyword(s):  
Conformal Mapping ◽  
Unit Circle ◽  
Half Plane ◽  
Elliptical Hole ◽  
Mapping Function ◽  
Complex Stress ◽  
Complex Variables ◽  
Material Interface ◽  
Rational Mapping ◽  
Point Dislocation

The problem of a point dislocation interacting with an elliptical hole at the interface of two bonded half-planes is studied. Complex stress potentials are obtained by applying the methods of complex variables and conformal mapping. A rational mapping function that maps a half plane with a semi-elliptical notch onto a unit circle is used for mapping the bonded half-planes. The solution derived can serve as Green’s function to study internal cracks interacting with an elliptical interfacial cavity.

scholarly journals On Tangential Principal Cluster Sets of Normal Meromorphic Functions

1970 ◽  
Vol 40 ◽  
pp. 133-137 ◽  
Author(s):  
Theodore A. Vessey
Keyword(s):  
Meromorphic Function ◽  
Conformal Mapping ◽  
Finite Order ◽  
Real Axis ◽  
Half Plane ◽  
Meromorphic Functions ◽  
The Family ◽  
Normal Meromorphic Function ◽  
Upper Half Plane ◽  
Principal Cluster

Let w = f(z) be a normal meromorphic function defined in the upper half plane U = {Im(z) >0}. We recall that a meromorphic function f(z) is normal in U if the family {f(S(z))}, where z′ = S(z) is an arbitrary one-one conformal mapping of U onto U, is normal in the sense of Montel. It is the purpose of this paper to state some results on the behavior of f(z) on curves which approach a point x0 on the real axis R with a fixed (finite) order of contact q at x0.

scholarly journals NEW INTEGRALS OF MOTION AND NON-CANONICAL HAMILTONIAN STRUCTURE FOR 2D HYDRODYNAMICS WITH FREE SURFACE

2019 ◽  
Vol 47 (1) ◽  
pp. 51-54
Author(s):  
A.I. Dyachenko ◽  
S.A. Dyachenko ◽  
P.M. Lushnikov ◽  
V.E. Zakharov
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Conformal Mapping ◽  
Poisson Bracket ◽  
Euler Equations ◽  
Half Plane ◽  
Hamiltonian Structure ◽  
Integrals Of Motion ◽  
Complex Half Plane ◽  
Constants Of Motion

We consider a potential motion of ideal incompressible fluid with a free surface and infinite depth in two dimensions with gravity forces and surface tension. A time-dependent conformal mapping z(w, t) of the lower complex half-plane of the variable w into the area filled with fluid is performed with the real line of w mapped into the free fluid’s surface. We study the dynamics of singularities of both z(w, t) and the complex fluid potential Π(w, t) in the upper complex half-plane of w. We reformulate the exact Eulerian dynamics through a non-canonical nonlocal Hamiltonian structure for a pair of the Hamiltonian variables (Dyachenko et al., submitted), the imaginary part of z(w, t) and the real part at Π(w, t) (both evaluated of fluid’s free surface). The corresponding Poisson bracket is non-degenerate, i.e. it does not have any Casimir invariant. Any two functionals of the conformal mapping commute with respect to the Poisson bracket. New Hamiltonian structure is a generalization of the canonical Hamiltonian structure of (Zakharov, 1968) (valid only for solutions for which the natural surface parametrization is single valued, i.e. each value of the horizontal coordinate corresponds only to a single point on the free surface). In contrast, new non-canonical Hamiltonian equations are valid for arbitrary nonlinear solutions (including multiple-valued natural surface parametrization) and are equivalent to Euler equations. We also consider a generalized hydrodynamics with the additional physical terms in the Hamiltonian beyond the Euler equations as in (Lushnikov and Zubarev, 2018) with the powerful reductions which allowed to find general classes of particular solutions. In Eulerian case we show the existence of solutions with an arbitrary finite number N of complex poles in zw(w, t) and Πw(w, t) which are the derivatives of z(w, t) and Π(w, t) over w (Dyachenko et al., submitted). These solutions are not purely rational because they generally have branch points at other positions of the upper complex halfplane with generally the infinite number of sheets of the Riemann surface for z(w, t) and Π(w, t) (Lushnikov, 2016). The order of poles is arbitrary for zero surface tension while all orders are even for nonzero surface tension. We find that the residues of zw(w, t) at these N points are new, previously unknown constants of motion. These constants of motion commute with each other with respect to the Poisson bracket. There are more integrals of motion beyond these residues. If all poles are simple then the number of independent real integrals of motion is 4N for zero gravity and 4N-1 for nonzero gravity. For higher order poles the number of the integrals is increasing. These nontrivial constants of motion provides an argument in support of the conjecture of complete integrability of free surface hydrodynamics. Work of A. Dyachenko, P. Lushnikov and V. Zakharov was supported by state assignment «Dynamics of the complex materials».

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