Transcendental Representations with Applications to Solids and Fluids

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    • Provides mathematical models of physical phenomena and engineering processes particularly relevant in aerospace and mechanical engineering
    • Unifies interdisciplinary topics of physics, mathematics, and engineering
    • Explores the interplay between physical laws and mathematical methods as a basis for modeling natural phenomena and engineering devices
    • Includes examples of applications with interpretation of results and discussion of assumptions and their consequences
    • Enables readers to construct mathematical-physical models suited to new observations or novel engineering devices
    • Contains many illustrations, tables, and diagrams that clarify the links between topics


      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.

      Table of Contents

      Sequences of Fractions or Products
      Power Series, Singularities, and Functions
      Series of Fractions for Meromorphic Functions (Mittag-Leffler 1876, 1884)
      Meromorphic Function as a Ratio of Two Integral Functions
      Factorization with Infinite Number of Zeros
      Infinite Products for Circular Functions
      Recurrence Formulas and Continued Fractions (Wallis 1656; Euler 1737)
      Optimal and Doubly Bounded Sharpening Approximations
      Transformation of Series and Products into Fractions (Euler 1785)
      Continued Fraction for the Ratio of Two Series (Lambert 1770)

      Compressible and Rotational Flows
      Source, Sink, and Vortex in a Compressible Flow
      Potential Vortex with Rotational Core (Rankine; Hallock and Burnham 1997)
      Minimum Energy (Thomson 1849) and Intrinsic Equations of Motion
      Laplace/Poisson Equations in Complex Conjugate Coordinates
      Second Forces/Moment and Circle Theorems
      Cylinder in a Unidirectional Shear Flow
      Monopole Interactions and Equilibrium Positions
      Cylinder in a Stream with Two Trailing Monopoles (Föppl 1913)
      Reciprocity Theorem (Green 1828) and Path Function (Routh 1881)

      Exponential and Logarithmic Functions
      Derivation Property, Series, and Rational Limit
      Continued Fractions and Computation of the Number e
      Transformation of Sums to Products and Powers
      Limits, Period, and Absence of Zeros
      Logarithm as the Function Inverse to Exponential
      Series Expansions and Continued Fractions
      Exponential and Logarithm with Complex Base
      Tables of Natural (Napier 1614) and Decimal (Briggs 1624) Logarithms
      Gaussian and Related Hypergeometric Functions

      Plane Elasticity and Multiharmonic Functions
      Displacement Vector and Deformation and Strain Tensors
      Stress Vector, Tensor, and Function
      Elastic Energy and Moduli of a Material (Hooke 1678; Poisson 1829a; Lamé 1852)
      Momentum Equation for Isotropic Elasticity
      Cavities and Static and Rotating Cylinders
      Multiharmonic Equation and Fluid Loading on a Dam
      Forces and Moments on a Wedge
      Elastic Potential and Stresses in an Infinite Medium
      Driven Loaded Wheel with Traction or Braking

      Circular and Hyperbolic Functions
      Sine/Cosine Representations on the Ellipse/Hyperbola
      Secant, Cosecant, Tangent, and Cotangent
      Formulas of Addition of Several Variables
      Formulas for Multiple, Double, and Half Variable
      Powers, Products, and Sums of the Functions
      Chebychev (1859) Polynomials of Two Kinds
      Orthogonal and Normalized Trigonometric Functions
      Relations between Complex and Real Functions
      Periods, Symmetries, Values, and Limits

      Membranes, Capillarity, and Torsion
      Linear and Nonlinear Deflection of a Membrane
      Large Deflection of a Membrane by Weight or Pressure
      Boundary Condition with Surface Tension
      Wetting Angle and Capillary Rise
      Warping, Stress, and Displacement Functions
      Torsional Stiffness of a Multiply Connected Section
      Hollow Elliptical or Thin or Cut Cross Sections
      Torsion of Prisms with Triangular Cross Section (Saint-Venant 1885; Campos and Cunha 2010)
      Trajectories of Fluid Particles in a Rotating Vessel

      Infinite and Cyclometric Representations
      Power Series and Euler (1755)/Bernoulli (1713) Numbers
      Branch Points and Branch Cuts for Cyclometric Functions
      Derivatives/Primitives of Direct/Inverse Functions
      Power Series for Cyclometric Functions
      Slopes at Zeros and Residues at Singularities
      Series of Fractions for Meromorphic Functions
      Relation with Factorization in Infinite Products
      Continued Fractions for Direct/Inverse Functions
      Gregory, Leibnitz, Brouncker, and Wallis Quadratures

      Confined and Unsteady Flows
      Cylinder Moving in Large Cavity
      Two Cylinders in Relative Motion
      Eccentric, Biconcave/Biconvex, and Semi-Recessed Cylinders
      Ramp/Step in a Wall and Thick Pointed/Blunt Plate
      Channel with Contraction/Expansion and a Thick Plate
      Flat Plate with Partially Separated Flow
      Airfoil with Plain/Slotted Flap/Slat
      Flow Past Tandem Airfoils and Cascades
      Instability of a Plane Vortex Sheet (Helmholtz 1868; Kelvin 1871)

      Infinite Processes and Summability
      Bounds for Integrals of Monotonic Functions
      Genus of an Infinite Canonical Product (Weierstrass 1876)
      Convergent and Periodic Continued Fractions
      Derangement of Conditionally Convergent Series
      Summation of Series of Rational Functions
      Extension 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)

      Twenty Examples
      Examples 10.1 through 10.20



      Author Bio(s)

      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.

      Editorial Reviews

      "… the book will be useful to engineers who do not want to learn so much on mathematical results but who are mostly interested by the resolution of concrete mechanical problems. The length of the book is not a problem as the many tables may help the reader to and quickly the solution of the problem he is looking for."
      — Alain Brillard in Zentralblatt MATH