Special Functions Fourier Transforms Laplace Transforms Chapter 7: SPECIAL FUNCTIONS 1. Gamma Functions 2. Beta Functions 3. Bessel Functions 3.1. Bessel equation for zero order 3.2. Properties of Bessel functions 3.3. Bessel functions of order one (n = 1) 3.4. Relationships between Bessel functions of orders zero and one 3.5. Lammel' s integrals 3.6. Fourier-Bessel expansions of zero order 3.7. Vibration of uniformly stretched membrane 3.8. Application of Bessel functions on conduction of heat 3.9. Modified Bessel function of zero order 3.10. Bessel and Kelvin functions 3.11. Bessel functions of any real order 3.12. Bessel functions of integral order 3.13. Bessel coefficients 3.14. Recurrence formulae 3.15. Bessel function as integrals 3.16. The Bessel functions of order n of the third kind (Hankel functions of order n) 4. Legendre functions 4.1. Alternative definition of Legendre polynomials 4.2. Legendre's recurrence formulae 4.3. Integral properties of Lagendre polynomial 4.4. The associated Legendre functions. 4.5. Applications of Legendre functions 5. Exercises on special functions Chapter 8: Fourier transforms 1. Fourier series and harmonic analysis 2. Fourier theorem 3. Preliminary integrals used in Fourier transforms 4. Determination of the coefficients of the Fourier expansion 5. Examples of Fourier transformations 6. Fourier expansions in cosines only 7. Fourier expansions in sines only 8. Fourier expansions in even harmonics 9. Fourier expansions in odd harmonics 10. Summary of common Fourier transforms 11. Practical Fourier Analysis Chapter 11: Laplace Transforms 1. The Laplace Transformation 2. General Theorems on the Laplace Transformation 2.1. The unit step function 2.2. The second translation or shifting property 2.3. Application of the shift theorem to the solution of difference and differential equations 2.4. The unit impulse function 2.5. The unit doublet 2.6. The behavior of f(s) as s tends to infinity. 2.7. Initial value theorem 2.8. Final value theorem 2.9. Differentiation of transform 2.10. Application of the differentiation of Laplace transform to the solution of linear differential equations with coefficients as polynomials in t. 2.11. Integration of transforms 2.12. Transforms of periodic functions 2.13. The product theorem-Convolution 2.14. Application of the product theorem to the solution of differential and integral equations 2.15. Power series method for the determination of transforms and inverse transforms 2.16. The error function or probability integral 2.17. The sine-integral function Si(t) 2.18. The Cosine -integral function Ci(t) 2.19. The exponential integral function 2.20. Evaluation of definite integrals using the Laplace transformation 2.21. The Heaviside's expansion formulae 2.22. The inversion integral 2.23. Formulae for residues 2.24. Inversion in the case of branch points 2.25. Miscellaneous Examples on Laplace Transform 2.26. Exercises on Laplace Transforms 3. Electrical Applications of the Laplace Transformation 4. Dynamical Applications of Laplace Transforms 5. Structural Applications 5.1. Deflection of beams 5.2. Exercises on Laplace Transform in practical applications 6. Using Laplace Transformation in solving Linear Partial Differential Equations 6.1. Transverse vibrations of a stretched string under gravity 6.2. Longitudinal vibrations of bars 6.3. Partial differential equations of transmission lines 6.4. Conduction of heat 6.5. Exercise on using Laplace Transformation in solving Linear Partial Differential Equations