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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.

**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)

Conclusion

**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)

Conclusion

**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

Conclusion

**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

Conclusion

**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

Conclusion

**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

Conclusion

**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

Summary

**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)

Conclusion

**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)

Conclusion

**Twenty Examples**Examples 10.1 through 10.20

Conclusion

**Bibliography**

**Index**

**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.

"… 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*