CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C. Third Edition Advanced Mathematics and Mechanics Applications Using Howard B. Wilson University of Alabama Louis H. Turcotte Rose-Hulman Institute of Technology David Halpern University of Alabama MATLAB ® © 2003 by Chapman & Hall/CRC This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. 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Visit the CRC Press Web site at www.crcpress.com © 2003 by Chapman & Hall/CRC No claim to original U.S. Government works International Standard Book Number 1-58488-262-X Library of Congress Card Number 2002071267 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper Library of Congress Cataloging-in-Publication Data Wilson, H.B. Advanced mathematics and mechanics applications using MATLAB / Howard B. Wilson, Louis H. Turcotte, David Halpern.—3rd ed. p. cm. ISBN 1-58488-262-X 1. MATLAB. 2. Engineering mathematics—Data processing. 3. Mechanics, Applied—Data processing. I. Turcotte, Louis H. II. Halpern, David. III. Title. TA345 . W55 2002 620 ′ .00151—dc21 2002071267 C262X disclaimer Page 1 Friday, August 2, 2002 11:45 AM © 2003 by Chapman & Hall/CRC For my dear wife, Emma. Howard B. Wilson For my loving wife, Evelyn, our departed cat, Patches, and my parents. Louis H. Turcotte © 2003 by Chapman & Hall/CRC Preface This book uses MATLAB R  to analyze various applications in mathematics and me- chanics. The authors hope to encourage engineers and scientists to consider this modern programming environment as an excellent alternative to languages such as FORTRAN or C++. MATLAB 1 embodies an interactive environment with a high level programming language supporting both numerical and graphical commands for two- and three-dimensional data analysis and presentation. The wealth of intrinsic mathematical commands to handle matrix algebra, Fourier series, differential equa- tions, and complex-valued functions makes simple calculator operations of many tasks previously requiring subroutine libraries with cumbersome argument lists. We analyze problems, drawn from our teaching and research interests, empha- sizing linear and nonlinear differential equation methods. Linear partial differential equations and linear matrix differential equations are analyzed using eigenfunctions and series solutions. Several types of physical problems are considered. Among these are heat conduction, harmonic response of strings, membranes, beams, and trusses, geometrical properties of areas and volumes, ßexure and buckling of inde- terminate beams, elastostatic stress analysis, and multi-dimensional optimization. Numerical integration of matrix differential equations is used in several examples illustrating the utility of such methods as well as essential aspects of numerical ap- proximation. Attention is restricted to the Runge-Kutta method which is adequate to handle most situations. Space limitation led us to omit some interesting MATLAB features concerning predictor-corrector methods, stiff systems, and event locations. This book is not an introductory numerical analysis text. It is most useful as a ref- erence or a supplementary text in computationally oriented courses emphasizing ap- plications. The authors have previously solved many of the examples in FORTRAN. Our MATLAB solutions consume over three hundred pages (over twelve thousand lines). Although few books published recently present this much code, comparable FORTRAN versions would probably be signifcantly longer. In fact, the conciseness of MATLAB was a primary motivation for writing the book. The programs contain many comments and are intended for study as separate en- tities without an additional reference. Consequently, some deliberate redundancy 1 MATLAB is a registered trademark of The MathWorks, Inc. For additional information contact: The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-1500 (508) 647-7000, Fax: (508) 647-7001 Email: info@mathworks.com © 2003 by Chapman & Hall/CRC exists between program comments and text discussions. We also list programs in a style we feel will be helpful to most readers. The source listings show line numbers adjacent to the MATLAB code. MATLAB code does not use line numbers or permit goto statements. We have numbered the lines to aid discussions of particular pro- gram segments. To conserve space, we often place multiple MATLAB statements on the same line when this does not interrupt the logical ßow . All of the programs presented are designed to operate under the 6.x version of MATLAB and Microsoft Windows. Both the text and graphics windows should be simultaneously visible. A windowed environment is essential for using capabilities like animation and interactive manipulation of three dimensional Þgures. The source code for all of the programs in the book is available from the CRC Press website at http://www.crcpress.com. The program collection is organized using an independent subdirectory for each of the thirteen chapters. This third edition incorporates much new material on time dependent solutions of linear partial differential equations. Animation is used whenever seeing the solution evolve in time is helpful. Animation illustrates quite well phenomena like wave propagation in strings and membranes. The interactive zoom and rotation features in MATLAB are also valuable tools for interpreting graphical output. Most programs in the book are academic examples, but some problem solutions are useful as stand-alone analysis tools. Examples include geometrical property cal- culation, differentiation or integration of splines, Gauss integration of arbitrary order, and frequency analysis of trusses and membranes. A chapter on eigenvalue problems presents applications in stress analysis, elastic stability, and linear system dynamics. A chapter on analytic functions shows the efÞciency of MATLAB for applying complex valued functions and the Fast Fourier Transform (FFT) to harmonic and biharmonic functions. Finally, the book concludes with a chapter applying multidimensional search to several nonlinear programming problems. We emphasize that this book is primarily for those concerned with physical appli- cations. A thorough grasp of Euclidean geometry, Newtonian mechanics, and some mathematics beyond calculus is essential to understand most of the topics. Finally, the authors enjoy interacting with students, teachers, and researchers applying ad- vanced mathematics to real world problems.The availability of economical computer hardware and the friendly software interface in MATLAB makes computing increas- ingly attractive to the entire technical community. If we manage to cultivate interest in MATLAB among engineers who only spend part of their time using computers, our primary goal will have been achieved. Howard B. Wilson hwilson@bama.ua.edu Louis H. Turcotte turcotte@rose-hulman.edu David Halpern david.halpern@ua.edu © 2003 by Chapman & Hall/CRC Contents 1Introduction 1.1MATLAB:AToolforEngineeringAnalysis 1.2MATLABCommandsandRelatedReferenceMaterials 1.3ExampleProblemonFinancialAnalysis 1.4ComputerCodeandResults 1.4.1ComputerOutput 1.4.2DiscussionoftheMATLABCode 1.4.3CodeforFinancialProblem 2ElementaryAspectsofMATLABGraphics 2.1Introduction 2.2OverviewofGraphics 2.3ExampleComparingPolynomialandSplineInterpolation 2.4ConformalMappingExample 2.5NonlinearMotionofaDampedPendulum 2.6ALinearVibrationModel 2.7ExampleofWavesinanElasticString 2.8PropertiesofCurvesandSurfaces 2.8.1CurveProperties 2.8.2SurfaceProperties 2.8.3ProgramOutputandCode 3SummaryofConceptsfromLinearAlgebra 3.1Introduction 3.2Vectors,Norms,LinearIndependence,andRank 3.3 Systems of Linear Equations, Consistency, and Least Squares Ap- proximation 3.4ApplicationsofLeastSquaresApproximation 3.4.1AMembraneDeßectionProblem 3.4.2 Mixed Boundary Value Problem for a Function Harmonic InsideaCircularDisk 3.4.3 Using Rational Functions to Conformally Map a Circular DiskontoaSquare 3.5EigenvalueProblems 3.5.1StatementoftheProblem 3.5.2ApplicationtoSolutionofMatrixDifferentialEquations © 2003 by Chapman & Hall/CRC 3.5.3TheStructuralDynamicsEquation 3.6ComputingNaturalFrequenciesforaRectangularMembrane 3.7ColumnSpace,NullSpace,OrthonormalBases,andSVD 3.8ComputationTimetoRunaMATLABProgram 4MethodsforInterpolationandNumericalDifferentiation 4.1ConceptsofInterpolation 4.2Interpolation,Differentiation,andIntegrationbyCubicSplines 4.2.1ComputingtheLengthandAreaBoundedbyaCurve 4.2.2Example:LengthandEnclosedAreaforaSplineCurve 4.2.3GeneralizingtheIntrinsicSplineFunctioninMATLAB 4.2.4Example:ASplineCurvewithSeveralPartsandCorners 4.3NumericalDifferentiationUsingFiniteDifferences 4.3.1Example:ProgramtoDeriveDifferenceFormulas 5GaussIntegrationwithGeometricPropertyApplications 5.1FundamentalConceptsandIntrinsicIntegrationToolsinMATLAB 5.2ConceptsofGaussIntegration 5.3ComparingResultsfromGaussIntegrationandFunctionQUADL 5.4GeometricalPropertiesofAreasandVolumes 5.4.1AreaPropertyProgram 5.4.2ProgramAnalyzingVolumesofRevolution 5.5 Computing Solid Properties Using Triangular Surface Elements and UsingSymbolicMath 5.6NumericalandSymbolicResultsfortheExample 5.7GeometricalPropertiesofaPolyhedron 5.8EvaluatingIntegralsHavingSquareRootTypeSingularities 5.8.1ProgramListing 5.9GaussIntegrationofaMultipleIntegral 5.9.1Example:EvaluatingaMultipleIntegral 6FourierSeriesandtheFastFourierTransform 6.1DeÞnitionsandComputationofFourierCoefÞcients 6.1.1TrigonometricInterpolationandtheFastFourierTransform 6.2SomeApplications 6.2.1UsingtheFFTtoComputeIntegerOrderBesselFunctions 6.2.2DynamicResponseofaMassonanOscillatingFoundation 6.2.3GeneralProgramtoPlotFourierExpansions 7DynamicResponseofLinearSecondOrderSystems 7.1SolvingtheStructuralDynamicsEquationsforPeriodicForces 7.1.1ApplicationtoOscillationsofaVerticallySuspendedCable 7.2DirectIntegrationMethods 7.2.1ExampleonCableResponsebyDirectIntegration © 2003 by Chapman & Hall/CRC 8IntegrationofNonlinearInitialValueProblems 8.1 General Concepts on Numerical Integration of Nonlinear Matrix Dif- ferentialEquations 8.2 Runge-Kutta Methods and the ODE45 Integrator Provided in MAT- LAB 8.3Step-sizeLimitsNecessarytoMaintainNumericalStability 8.4 Discussion of Procedures to Maintain Accuracy by Varying Integra- tionStep-size 8.5ExampleonForcedOscillationsofanInvertedPendulum 8.6DynamicsofaSpinningTop 8.7MotionofaProjectile 8.8ExampleonDynamicsofaChainwithSpeciÞedEndMotion 8.9DynamicsofanElasticChain 9BoundaryValueProblemsforPartialDifferentialEquations 9.1SeveralImportantPartialDifferentialEquations 9.2SolvingtheLaplaceEquationinsideaRectangularRegion 9.3TheVibratingString 9.4ForceMovingonanElasticString 9.4.1ComputerAnalysis 9.5WavesinRectangularorCircularMembranes 9.5.1ComputerFormulation 9.5.2InputDataforProgrammembwave 9.6 Wave Propagation in a Beam with an Impact Moment Applied to OneEnd 9.7ForcedVibrationofaPileEmbeddedinanElasticMedium 9.8TransientHeatConductioninaOne-DimensionalSlab 9.9 Transient Heat Conduction in a Circular Cylinder with Spatially Vary- ingBoundaryTemperature 9.9.1ProblemFormulation 9.9.2ComputerFormulation 9.10TorsionalStressesinaBeamofRectangularCrossSection 10EigenvalueProblemsandApplications 10.1Introduction 10.2ApproximationAccuracyinaSimpleEigenvalueProblem 10.3StressTransformationandPrincipalCoordinates 10.3.1PrincipalStressProgram 10.3.2PrincipalAxesoftheInertiaTensor 10.4VibrationofTrussStructures 10.4.1TrussVibrationProgram 10.5BucklingofAxiallyLoadedColumns 10.5.1ExampleforaLinearlyTaperedCircularCrossSection 10.5.2NumericalResults © 2003 by Chapman & Hall/CRC 10.6 Accuracy Comparison for Euler Beam Natural Frequencies by Finite ElementandFiniteDifferenceMethods 10.6.1MathematicalFormulation 10.6.2DiscussionoftheCode 10.6.3NumericalResults 10.7VibrationModesofanEllipticMembrane 10.7.1AnalyticalFormulation 10.7.2ComputerFormulation 11BendingAnalysisofBeamsofGeneralCrossSection 11.1Introduction 11.1.1AnalyticalFormulation 11.1.2ProgramtoAnalyzeBeamsofGeneralCrossSection 11.1.3ProgramOutputandCode 12ApplicationsofAnalyticFunctions 12.1PropertiesofAnalyticFunctions 12.2DeÞnitionofAnalyticity 12.3SeriesExpansions 12.4IntegralProperties 12.4.1CauchyIntegralFormula 12.4.2ResidueTheorem 12.5PhysicalProblemsLeadingtoAnalyticFunctions 12.5.1Steady-StateHeatConduction 12.5.2IncompressibleInviscidFluidFlow 12.5.3TorsionandFlexureofElasticBeams 12.5.4PlaneElastostatics 12.5.5ElectricFieldIntensity 12.6BranchPointsandMultivaluedBehavior 12.7ConformalMappingandHarmonicFunctions 12.8MappingontotheExteriorortheInteriorofanEllipse 12.8.1ProgramOutputandCode 12.9LinearFractionalTransformations 12.9.1ProgramOutputandCode 12.10Schwarz-ChristoffelMappingontoaSquare 12.10.1ProgramOutputandCode 12.11DeterminingHarmonicFunctionsinaCircularDisk 12.11.1NumericalResults 12.11.2ProgramOutputandCode 12.12InviscidFluidFlowaroundanEllipticCylinder 12.12.1ProgramOutputandCode 12.13TorsionalStressesinaBeamMappedontoaUnitDisk 12.13.1ProgramOutputandCode 12.14StressAnalysisbytheKolosov-MuskhelishviliMethod 12.14.1ProgramOutputandCode © 2003 by Chapman & Hall/CRC 12.14.2StressedPlatewithanEllipticHole 12.14.3ProgramOutputandCode 13NonlinearOptimizationApplications 13.1BasicConcepts 13.2InitialAngleforaProjectile 13.3FittingNonlinearEquationstoData 13.4NonlinearDeßectionsofaCable 13.5QuickestTimeDescentCurve(theBrachistochrone) 13.6DeterminingtheClosestPointsonTwoSurfaces 13.6.1DiscussionoftheComputerCode AListofMATLABRoutineswithDescriptions BSelectedUtilityandApplicationFunctions References © 2003 by Chapman & Hall/CRC [...]... c2=savespnd(T2,t1,0,R,A,I,0 ,1) ; end 98: 99: 10 0: 10 1: 10 2: 10 3: 10 4: 10 5: if t1==0 | t2==0 t=linspace(0,T2,nt)’; else n1=max(2,fix(t1/T2*nt)); n2=max(2,nt-n1) -1; t=[t1/n1*(0:n1),t1+t2/n2* (1: n2)]’; end 10 6: 10 7: 10 8: 10 9: 11 0: 11 1: 11 2: % Solve for the unknown parameter if isnan(q0), q0=(q2-s*c1-p*c2)/c0; elseif isnan(s), s=(q2-q0*c0-p*c2)/c1; elseif isnan(p), p=(q2-q0*c0-s*c1)/c2; else, q2=q0*c0+s*c1+p*c2; end 11 3:... yo=imag(wouter); plot(x,y,’-k’,x (1: end -1, :)’,y (1: end -1, :)’, ’-k’,xo,yo,’-k’) 11 3: 11 4: 11 5: 11 6: 11 7: 11 8: % Add a title and axis labels title([’Mapping of a Square Using a ’, num2str(m),’-term Polynomial’]) xlabel(’x axis’); ylabel(’y axis’) figure(gcf); hold off; 11 9: 12 0: %============================================== 12 1: 12 2: 12 3: 12 4: 12 5: 12 6: 12 7: 12 8: 12 9: 13 0: function range=cubrange(xyz,ovrsiz)... q2=q0*c0+s*c1+p*c2; end 11 3: 11 4: 11 5: 11 6: % Compute results for q(t) q=savespnd(t,t1,q0,R,A,I,s,p); q1=savespnd(t1,t1,q0,R,A,I,s,p); 11 7: 11 8: 11 9: 12 0: 12 1: 12 2: 12 3: 12 4: 12 5: 12 6: 12 7: % Print formatted results b=inline(’blanks(j)’,’j’); B=b(3); d=’%8.3f’; u=[d,B,d,B,d,B,d,B,d,’\n’]; disp(’ ’) disp([b (19 ),’PROGRAM RESULTS’]) disp([’ t1 t2 R’, ’ A I’]) fprintf(u,t1,t2,R,A,I), disp(’ ’) disp([’ q0 q1 q2’, ’ s... shg 15 5: 15 6: %============================================= 15 7: 15 8: 15 9: 16 0: 16 1: function q=savespnd(t,t1,q0,R,A,I,s,p) % % q=savespnd(t,t1,q0,R,A,I,s,p) %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 16 2: 16 3: 16 4: 16 5: 16 6: % % % % This function determines q(t) satisfying q’(t)=r*q+[s*(tt1)* exp(-a*t1)]*exp(a*t), with q(0)=q0, r=(R-I) /10 0; a=(A-I) /10 0 16 7: 16 8: 16 9: 17 0: 17 1: 17 2: 17 3:... 17 3: 17 4: 17 5: r=(R-I) /10 0; a=(A-I) /10 0; c=r-a; T=t-t1; if r~=a q=q0*exp(r*t)+s/c*(exp(r*t)-exp(a*t)) -(p+s*exp(a*t1))/c*(T>0).*( exp(r*T)-exp(a*T)); else % limiting case as a=>r q=q0*exp(r*t)+s*t.*exp(r*t) -(p+s*exp(r*t1)).*T.*(T>0).*exp(r*T); © 2003 by CRC Press LLC 17 6: end 17 7: 17 8: %============================================= 17 9: 18 0: 18 1: 18 2: 18 3: 18 4: 18 5: 18 6: 18 7: 18 8: 18 9: 19 0: 19 1: 19 2:... b=b/sum(b); w=z.*polyval(b(m: -1: 1),z.^4); wouter=zouter.*polyval(b(m: -1: 1),zouter.^4); 10 0: 10 1: 10 2: 10 3: 10 4: 10 5: % Determine square window limits for plotting uu=real([w(:);wouter(:)]); vv=imag([w(:);wouter(:)]); rng=cubrange([uu,vv] ,1. 1); axis(’square’); axis(rng); hold on 10 6: 10 7: 10 8: 10 9: 11 0: 11 1: 11 2: % Plot orthogonal grid lines which represent % the mapping of circles and radial lines x=real(w);... ylabel(’TOTAL SAVINGS IN $K’) 14 0: 14 1: 14 2: 14 3: 14 4: 14 5: 14 6: 14 7: 14 8: 14 9: 15 0: 15 1: 15 2: 15 3: 15 4: % Character label showing data parameters label=char( sprintf(’R = %8.3f’,R), sprintf(’I = %8.3f’,I), sprintf(’A = %8.3f’,A), sprintf(’q0 = %8.3f’,q0), sprintf(’q1 = %8.3f’,q1), sprintf(’q2 = %8.3f’,q2)); w=axis; ymin=w(3); dy=w(4)-w(3); xmin=w (1) ; dx=w(2)-w (1) ; ytop=ymin+.8*dy; Dy=.065*dy;... p’]) fprintf(u,q0,q1,q2,s,p), disp(’ ’), pause (1) 12 8: 12 9: 13 0: % Show results graphically plot(t,q,’k’) © 2003 by CRC Press LLC 13 1: 13 2: 13 3: 13 4: 13 5: 13 6: title([’INFLATION ADJUSTED SAVINGS WHEN ’, ’S = ’,num2str(s),’ AND P = ’,num2str(p)]); titl= [’TOTAL SAVINGS WHEN T1 = ’,num2str(t1), ’, T2 = ’,num2str(t2),’, s = ’,num2str(s), ’, p = ’,num2str(p)]; title(titl) 13 7: 13 8: 13 9: xlabel(’TIME... a tends to r 18 0- 213 Function inputv generalizes the intrinsic function input to read several variables on the same line Inputv is used often throughout this text © 2003 by CRC Press LLC 1. 4.3 Code for Financial Problem Program Þnance 1: 2: 3: 4: 5: 6: 7: 8: 9: 10 : 11 : 12 : 13 : 14 : 15 : 16 : function [q,t,R,A,I,t1,t2,s,p,q0,q1,q2]=finance (R,A,I,t1,t2,s,p,q0,q2) % [q,t,R,A,I,t1,t2,s,p,q0,q1,q2]=finance... polyplot 1: 2: 3: 4: 5: 6: 7: 8: 9: function polyplot % Example: polyplot % ~~~~~~~~~~~~~~~~~~ % This program illustrates polynomial and % spline interpolation methods applied to % approximate the function 1/ (1+ x^2) % % User inline functions used: % cbp, Ylsq, yexact 10 : 11 : 12 : 13 : % Function for Chebyshev data points cbp=inline([’(a+b)/2+(a-b)/2*cos(pi/n*’, ’ (1/ 2:n))’],’a’,’b’,’n’); 14 : 15 : 16 : 17 : 18 : . Finite ElementandFiniteDifferenceMethods 10 .6.1MathematicalFormulation 10 .6.2DiscussionoftheCode 10 .6.3NumericalResults 10 .7VibrationModesofanEllipticMembrane 10 .7.1AnalyticalFormulation 10 .7.2ComputerFormulation 11 BendingAnalysisofBeamsofGeneralCrossSection 11 .1Introduction 11 .1. 1AnalyticalFormulation 11 .1. 2ProgramtoAnalyzeBeamsofGeneralCrossSection 11 .1. 3ProgramOutputandCode 12 ApplicationsofAnalyticFunctions 12 .1PropertiesofAnalyticFunctions 12 .2DeÞnitionofAnalyticity 12 .3SeriesExpansions 12 .4IntegralProperties 12 .4.1CauchyIntegralFormula 12 .4.2ResidueTheorem 12 .5PhysicalProblemsLeadingtoAnalyticFunctions 12 .5.1Steady-StateHeatConduction 12 .5.2IncompressibleInviscidFluidFlow 12 .5.3TorsionandFlexureofElasticBeams 12 .5.4PlaneElastostatics 12 .5.5ElectricFieldIntensity 12 .6BranchPointsandMultivaluedBehavior 12 .7ConformalMappingandHarmonicFunctions 12 .8MappingontotheExteriorortheInteriorofanEllipse 12 .8.1ProgramOutputandCode 12 .9LinearFractionalTransformations 12 .9.1ProgramOutputandCode 12 .10 Schwarz-ChristoffelMappingontoaSquare 12 .10 .1ProgramOutputandCode 12 .11 DeterminingHarmonicFunctionsinaCircularDisk 12 .11 .1NumericalResults 12 .11 .2ProgramOutputandCode 12 .12 InviscidFluidFlowaroundanEllipticCylinder 12 .12 .1ProgramOutputandCode 12 .13 TorsionalStressesinaBeamMappedontoaUnitDisk 12 .13 .1ProgramOutputandCode 12 .14 StressAnalysisbytheKolosov-MuskhelishviliMethod 12 .14 .1ProgramOutputandCode ©. Finite ElementandFiniteDifferenceMethods 10 .6.1MathematicalFormulation 10 .6.2DiscussionoftheCode 10 .6.3NumericalResults 10 .7VibrationModesofanEllipticMembrane 10 .7.1AnalyticalFormulation 10 .7.2ComputerFormulation 11 BendingAnalysisofBeamsofGeneralCrossSection 11 .1Introduction 11 .1. 1AnalyticalFormulation 11 .1. 2ProgramtoAnalyzeBeamsofGeneralCrossSection 11 .1. 3ProgramOutputandCode 12 ApplicationsofAnalyticFunctions 12 .1PropertiesofAnalyticFunctions 12 .2DeÞnitionofAnalyticity 12 .3SeriesExpansions 12 .4IntegralProperties 12 .4.1CauchyIntegralFormula 12 .4.2ResidueTheorem 12 .5PhysicalProblemsLeadingtoAnalyticFunctions 12 .5.1Steady-StateHeatConduction 12 .5.2IncompressibleInviscidFluidFlow 12 .5.3TorsionandFlexureofElasticBeams 12 .5.4PlaneElastostatics 12 .5.5ElectricFieldIntensity 12 .6BranchPointsandMultivaluedBehavior 12 .7ConformalMappingandHarmonicFunctions 12 .8MappingontotheExteriorortheInteriorofanEllipse 12 .8.1ProgramOutputandCode 12 .9LinearFractionalTransformations 12 .9.1ProgramOutputandCode 12 .10 Schwarz-ChristoffelMappingontoaSquare 12 .10 .1ProgramOutputandCode 12 .11 DeterminingHarmonicFunctionsinaCircularDisk 12 .11 .1NumericalResults 12 .11 .2ProgramOutputandCode 12 .12 InviscidFluidFlowaroundanEllipticCylinder 12 .12 .1ProgramOutputandCode 12 .13 TorsionalStressesinaBeamMappedontoaUnitDisk 12 .13 .1ProgramOutputandCode 12 .14 StressAnalysisbytheKolosov-MuskhelishviliMethod 12 .14 .1ProgramOutputandCode ©. Finite ElementandFiniteDifferenceMethods 10 .6.1MathematicalFormulation 10 .6.2DiscussionoftheCode 10 .6.3NumericalResults 10 .7VibrationModesofanEllipticMembrane 10 .7.1AnalyticalFormulation 10 .7.2ComputerFormulation 11 BendingAnalysisofBeamsofGeneralCrossSection 11 .1Introduction 11 .1. 1AnalyticalFormulation 11 .1. 2ProgramtoAnalyzeBeamsofGeneralCrossSection 11 .1. 3ProgramOutputandCode 12 ApplicationsofAnalyticFunctions 12 .1PropertiesofAnalyticFunctions 12 .2DeÞnitionofAnalyticity 12 .3SeriesExpansions 12 .4IntegralProperties 12 .4.1CauchyIntegralFormula 12 .4.2ResidueTheorem 12 .5PhysicalProblemsLeadingtoAnalyticFunctions 12 .5.1Steady-StateHeatConduction 12 .5.2IncompressibleInviscidFluidFlow 12 .5.3TorsionandFlexureofElasticBeams 12 .5.4PlaneElastostatics 12 .5.5ElectricFieldIntensity 12 .6BranchPointsandMultivaluedBehavior 12 .7ConformalMappingandHarmonicFunctions 12 .8MappingontotheExteriorortheInteriorofanEllipse 12 .8.1ProgramOutputandCode 12 .9LinearFractionalTransformations 12 .9.1ProgramOutputandCode 12 .10 Schwarz-ChristoffelMappingontoaSquare 12 .10 .1ProgramOutputandCode 12 .11 DeterminingHarmonicFunctionsinaCircularDisk 12 .11 .1NumericalResults 12 .11 .2ProgramOutputandCode 12 .12 InviscidFluidFlowaroundanEllipticCylinder 12 .12 .1ProgramOutputandCode 12 .13 TorsionalStressesinaBeamMappedontoaUnitDisk 12 .13 .1ProgramOutputandCode 12 .14 StressAnalysisbytheKolosov-MuskhelishviliMethod 12 .14 .1ProgramOutputandCode ©