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
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