Applications of Complex Numbers to Geometry

1932 ◽  
Vol 25 (4) ◽  
pp. 215-226
Author(s):  
Allen A. Shaw
Keyword(s):  
Complex Variable ◽  
Geometric Reasoning ◽  
Present Writer ◽  
Complex Numbers ◽  
Two Dimensional ◽  
Single Variable ◽  
Argand Diagram

Introduction. It was with a real pleasure that the present writer read the two excellent articles by Professors L. L. Smail and A. A. Schelkunoff on geometric applications of the complex variable.1 Both papers are important for the doctrine they expound and for the good training they give the reader in rigorous geometric reasoning on the Argand diagram. While Smail prefers the use of the complex variable in the two-dimensional form, x+iy, Schelkunoff employs and recommends the usage of the single variable z to prove the same theorems and obtains very simple and elegant demonstrations.

scholarly journals The Electrostatic Energy of a Two-Dimensional System

1959 ◽  
Vol 42 ◽  
pp. 1-2
Author(s):  
LL. G. Chambers
Keyword(s):  
Complex Variable ◽  
Complex Potential ◽  
Unit Length ◽  
Electrostatic Energy ◽  
Dimensional System ◽  
Two Dimensional ◽  
Image System ◽  
Actual System ◽  
Image Domain ◽  
Two Dimensional System

The use of the complex variable z( = x + iy) and the complex potential W(= U + iV) for two-dimensional electrostatic systems is well known and the actual system in the (x, y) plane has an image system in the (U, V) plane. It does not seem to have been noticed previously that the electrostatic energy per unit length of the actual system is simply related to the area of the image domain in the (U, V) plane.

FOURIER ANALYSIS OVER HYPERBOLIC ALGEBRA, PSEUDO-DIFFERENTIAL OPERATORS, AND HYPERBOLIC DEFORMATION OF CLASSICAL MECHANICS

2007 ◽  
Vol 10 (03) ◽  
pp. 421-438 ◽  
Author(s):  
ANDREI KHRENNIKOV
Keyword(s):  
Quantum Mechanics ◽  
Fourier Analysis ◽  
Poisson Bracket ◽  
Commutative Algebra ◽  
Differential Operators ◽  
Classical Mechanics ◽  
Complex Numbers ◽  
Two Dimensional ◽  
Quantum Deformation ◽  
Pseudo Differential Operators

We develop Fourier analysis over hyperbolic algebra (the two-dimensional commutative algebra with the basis e1 = 1, e2 = j, where j2 = 1). We demonstrated that classical mechanics has, besides the well-known quantum deformation over complex numbers, another deformation — so-called hyperbolic quantum mechanics. The classical Poisson bracket can be obtained as the limit h → 0 not only of the ordinary Moyal bracket, but also a hyperbolic analogue of the Moyal bracket.

A Mixed Boundary-Value Problem of Elasticity With Parabolic Boundary

1957 ◽  
Vol 24 (1) ◽  
pp. 122-124
Author(s):  
Gunadhar Paria
Keyword(s):  
Stress Distribution ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Complex Variable ◽  
Hilbert Problem ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Two Dimensional ◽  
Mixed Boundary Value Problem ◽  
Parabolic Boundary

Abstract The problem of finding the stress distribution in a two-dimensional elastic body with parabolic boundary, subject to mixed boundary conditions, has been reduced to the solution of the nonhomogeneous Hilbert problem following the method of complex variable. The result has been compared with that for a straight boundary.

Potential flow over a submerged rectangular obstacle: Consequences for initiation of boulder motion

2019 ◽  
Vol 31 (4) ◽  
pp. 646-681 ◽  
Author(s):  
J. G. HERTERICH ◽  
F. DIAS
Keyword(s):  
Fluid Flow ◽  
Complex Variable ◽  
Potential Flow ◽  
Two Dimensional ◽  
Wake Region ◽  
Local Flow ◽  
Slowing Down ◽  
Complex Variable Methods ◽  
Drag And Lift ◽  
Rectangular Obstacle

AbstractSteady two-dimensional fluid flow over an obstacle is solved using complex variable methods. We consider the cases of rectangular obstacles, such as large boulders, submerged in a potential flow. These may arise in geophysics, marine and civil engineering. Our models are applicable to initiation of motion that may result in subsequent transport. The local flow depends on the obstacle shape, slowing down in confining corners and speeding up in expanding corners. The flow generates hydrodynamic forces, drag and lift, and their associated moments, which differ around each face. Our model replaces the need for ill-defined drag and lift coefficients with geometry-dependent functions. We predict smaller flow velocities to initiate motion. We show how a joint-bound boulder can be transported against gravity, and analyse the influence of a wake region behind an isolated boulder.

On two dimensional electrohydrostatic stability

1976 ◽  
Vol 349 (1657) ◽  
pp. 231-243 ◽  
Keyword(s):  
Approximate Solution ◽  
Numerical Solution ◽  
Edge Effects ◽  
Complex Variable ◽  
Equilibrium Configuration ◽  
Quantitative Agreement ◽  
Two Dimensional ◽  
Equilibrium Conditions ◽  
Conducting Membranes ◽  
Special Case

Complex variable techniques are used for the study of the electrohydrostatic stability of two dimensional charged conducting membranes, which are assumed to be fixed along their edges. The formulation of the problem is quite general, but the numerical solution presented refers to the case when the membranes are symmetrical with respect to the plane bisecting their width and carry equal and opposite charges. It is found, as expected, that for a given set of data the equilibrium configuration breaks down if the membranes are sufficiently charged. When the membranes are sufficiently apart the breakdown occurs at their edges and is manifested as inability of the system to satisfy the equilibrium conditions there. When the membranes are sufficiently close together and are charged to a certain level, they touch at their mid-points and the equilibrium breaks down. Our results are compared with an approximate solution of this problem, presented by two other authors. The approximate solution ignores the edge effects of the membranes and overestimates the amount of charge that the membranes can carry before breakdown occurs. In the special case when the gap between the membranes is much less than their width, our results are in quantitative agreement with the approximate solution but as the gap between the membranes increases, the accuracy of the approximate solution decreases.

scholarly journals A non-embeddable composite of embeddable functions

1968 ◽  
Vol 8 (1) ◽  
pp. 109-113 ◽  
Author(s):  
A. Ran
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Complex Variable ◽  
Complex Parameter ◽  
Complex Numbers ◽  
Group Operation ◽  
Image Position ◽  
Formal Power

Let Ω be the group of the functions ƒ(z) of the complex variable z, analytic in some neighborhood of z = 0, with ƒ(0) = 0, ƒ′(0) = 1, where the group operation is the composition g[f(z)](g(z), f(z) ∈ Ω). For every function f(z) ∈ Ω there exists [4] a unique formal power series where the coefficients ƒq(s) are polynomials of the complex parameter s, with ƒ1(s) = 1, such that and, for any two complex numbers s and t, the formal law of composition is valid.

scholarly journals Complex potentials in two-dimensional elasticity

1945 ◽  
Vol 184 (997) ◽  
pp. 129-179 ◽  
Keyword(s):  
Complex Variable ◽  
Body Force ◽  
Displacement Function ◽  
The Body ◽  
Complex Variable Method ◽  
Complex Potentials ◽  
Variable Method ◽  
Two Dimensional ◽  
Special Cases ◽  
Airy’S Stress Function

This paper gives an approach to two-dimensional isotropic elastic theory (plane strain and generalized plane stress) by means of the complex variable resulting in a very marked economy of effort in the investigation of such problems as contrasted with the usual method by means of Airy’s stress function and the allied displacement function. This is effected (i) by considering especially the transformation of two-dimensional stress; it emerges that the combinations xx + yy , xx — yy + 2 ixy are all-important in the treatment in terms of complex variables; (ii) by the introduction of two complex potentials Ω( z ), ω( z ) each a function of a single complex variable in terms of . which the displacements and stresses can be very simply expressed. Transformation of the cartesian combinations u + iv , xx + yy , xx — yy + 2 ixy to the orthogonal curvilinear combinations u ξ + iu n , ξξ + ηη, ξξ - ηη + 2iξη is simple and speedy. The nature of "the complex potentials is discussed, and the conditions that the solution for the displacements shall be physically admissible, i.e. single-valued or at most of the possible dislocational types, is found to relate the cyclic functions of the complex potentials. Formulae are found for the force and couple resultants at the origin z = 0 equivalent to the stresses round a closed circuit in the elastic material, and these also are found to relate the cyclic functions of the complex potentials. The body force has bhen supposed derivable from a particular body force potential which includes as special cases (i) the usual gravitational body force, (ii) the reversed mass accelerations or so-called ‘centrifugal’ body forces of steady rotation. The power of the complex variable method is exhibited by finding the appropriate complex potentials for a very wide variety of problems, and whilst the main object of the present paper has been to extend the wellknown usefulness of the complex variable method in non-viscous hydrodynamical theory to two-dimensional elasticity, solutions have been given to a number of new problems and corrections made to certain other previous solutions.

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