Transcendental and Generalized Functions with Applications to Solids and Fluids

Building on the author’s previous book in the series, Complex Analysis with Applications to Flows and Fields (CRC Press, 2010), Transcendental Representations with Applications to Solids and Fluids focuses on four infinite representations: series expansions, series of fractions for meromorphic functions, infinite products for functions with infinitely many zeros, and continued fractions as alternative representations. This book also continues the application of complex functions to more classes of fields, including incompressible rotational flows, compressible irrotational flows, unsteady flows, rotating flows, surface tension and capillarity, deflection of membranes under load, torsion of rods by torques, plane elasticity, and plane viscous flows. The two books together offer a complete treatment of complex analysis, showing how the elementary transcendental functions and other complex functions are applied to fluid and solid media and force fields mainly in two dimensions.

The mathematical developments appear in odd-numbered chapters while the physical and engineering applications can be found in even-numbered chapters. The last chapter presents a set of detailed examples. Each chapter begins with an introduction and concludes with related topics.

Written by one of the foremost authorities in aeronautical/aerospace engineering, this self-contained book gives the necessary mathematical background and physical principles to build models for technological and scientific purposes. It shows how to formulate problems, justify the solutions, and interpret the results.

Luis Manuel Braga da Costa Campos is the director and founder of the Center for Aeronautical and Space Science and Technology at Lisbon Technical University, where he is also the coordinator of undergraduate and postgraduate degrees in aerospace engineering and coordinator of the applied and aerospace mechanics group in the Department of Mechanical Engineering. Dr. Campos is a member and vice chairman of the Portuguese Academy of Engineering; a fellow of the Royal Aeronautical Society, Astronomical Society, and Cambridge Philosophical Society; and associate fellow of the American Institute of Aeronautics and Astronautics. His research focuses on acoustics, magnetohydrodynamics, special functions, and flight dynamics.

Sequences of Fractions or ProductsPower Series, Singularities, and FunctionsSeries of Fractions for Meromorphic Functions (Mittag-Leffler 1876, 1884)Meromorphic Function as a Ratio of Two Integral FunctionsFactorization with Infinite Number of ZerosInfinite Products for Circular FunctionsRecurrence Formulas and Continued Fractions (Wallis 1656; Euler 1737)Optimal and Doubly Bounded Sharpening ApproximationsTransformation of Series and Products into Fractions (Euler 1785)Continued Fraction for the Ratio of Two Series (Lambert 1770)Conclusion

Compressible and Rotational FlowsSource, Sink, and Vortex in a Compressible FlowPotential Vortex with Rotational Core (Rankine; Hallock and Burnham 1997)Minimum Energy (Thomson 1849) and Intrinsic Equations of MotionLaplace/Poisson Equations in Complex Conjugate CoordinatesSecond Forces/Moment and Circle TheoremsCylinder in a Unidirectional Shear FlowMonopole Interactions and Equilibrium PositionsCylinder in a Stream with Two Trailing Monopoles (Föppl 1913)Reciprocity Theorem (Green 1828) and Path Function (Routh 1881)Conclusion

Exponential and Logarithmic FunctionsDerivation Property, Series, and Rational LimitContinued Fractions and Computation of the Number eTransformation of Sums to Products and PowersLimits, Period, and Absence of ZerosLogarithm as the Function Inverse to ExponentialSeries Expansions and Continued FractionsExponential and Logarithm with Complex BaseTables of Natural (Napier 1614) and Decimal (Briggs 1624) LogarithmsGaussian and Related Hypergeometric FunctionsConclusion

Plane Elasticity and Multiharmonic FunctionsDisplacement Vector and Deformation and Strain TensorsStress Vector, Tensor, and FunctionElastic Energy and Moduli of a Material (Hooke 1678; Poisson 1829a; Lamé 1852)Momentum Equation for Isotropic ElasticityCavities and Static and Rotating CylindersMultiharmonic Equation and Fluid Loading on a DamForces and Moments on a WedgeElastic Potential and Stresses in an Infinite MediumDriven Loaded Wheel with Traction or BrakingConclusion

Circular and Hyperbolic FunctionsSine/Cosine Representations on the Ellipse/HyperbolaSecant, Cosecant, Tangent, and CotangentFormulas of Addition of Several VariablesFormulas for Multiple, Double, and Half VariablePowers, Products, and Sums of the FunctionsChebychev (1859) Polynomials of Two KindsOrthogonal and Normalized Trigonometric FunctionsRelations between Complex and Real FunctionsPeriods, Symmetries, Values, and LimitsConclusion

Membranes, Capillarity, and TorsionLinear and Nonlinear Deflection of a MembraneLarge Deflection of a Membrane by Weight or PressureBoundary Condition with Surface TensionWetting Angle and Capillary RiseWarping, Stress, and Displacement FunctionsTorsional Stiffness of a Multiply Connected SectionHollow Elliptical or Thin or Cut Cross SectionsTorsion of Prisms with Triangular Cross Section (Saint-Venant 1885; Campos and Cunha 2010)Trajectories of Fluid Particles in a Rotating VesselConclusion

Infinite and Cyclometric RepresentationsPower Series and Euler (1755)/Bernoulli (1713) NumbersBranch Points and Branch Cuts for Cyclometric FunctionsDerivatives/Primitives of Direct/Inverse FunctionsPower Series for Cyclometric FunctionsSlopes at Zeros and Residues at SingularitiesSeries of Fractions for Meromorphic FunctionsRelation with Factorization in Infinite ProductsContinued Fractions for Direct/Inverse FunctionsGregory, Leibnitz, Brouncker, and Wallis QuadraturesSummary

Confined and Unsteady FlowsCylinder Moving in Large CavityTwo Cylinders in Relative MotionEccentric, Biconcave/Biconvex, and Semi-Recessed CylindersRamp/Step in a Wall and Thick Pointed/Blunt PlateChannel with Contraction/Expansion and a Thick PlateFlat Plate with Partially Separated FlowAirfoil with Plain/Slotted Flap/SlatFlow Past Tandem Airfoils and CascadesInstability of a Plane Vortex Sheet (Helmholtz 1868; Kelvin 1871)Conclusion

Infinite Processes and SummabilityBounds for Integrals of Monotonic FunctionsGenus of an Infinite Canonical Product (Weierstrass 1876)Convergent and Periodic Continued FractionsDerangement of Conditionally Convergent SeriesSummation of Series of Rational FunctionsExtension of Convergence to Summability (Euler 1755; Césaro 1890)Cardinals of Enumerable, Continuum, and Discontinuous (Cantor 1874, 1878)Transfinite Cardinal and Ordinal Numbers (Cantor 1883a,b; Hardy 1903)Three Antinomies and the Axiom of Selection (Burali-Forti 1897; Russell 1903; Zermelo 1908)Conclusion

Twenty ExamplesExamples 10.1 through 10.20Conclusion

Bibliography
Index