Preface
Author biography
1 Partial differential equations
Exercise
2 Separation of variables
2.1 Helmholtz equation
2.2 Helmholtz equation in rectangular coordinates
2.3 Helmholtz equation in cylindrical coordinates
2.4 Helmholtz equation in spheri.cal coordinates
2.5 Roadmap: where we are headed
Summary
Exercises
Reference
3 Power-series solutions of ODEs
3.1 Analytic functions and the Frobenius method
3.2 Ordinary points
3.3 Regular singular points
3.4 Wronskian method for obtaining a second solution
3.5 Bessel and Neumann functions
3.6 Legendre polynomials
Summary
Exercises
References
4 Sturm-Liouville theory
4.1 Differential equations as operators
4.2 Sturm-Liouville systems
4.3 The SL eigenvalue problem, L[y]=λwy
4.4 Dirac delta function
4.5 Completeness
4.6 Hilbert space: a brief introduction
Summary
Exercises
References
5 Fourier series and integrals
5.1 Fourier series
5.2 Complex fonll of Fourier series
5.3 General intervals
5.4 Parseval's theorem
5.5 Back to the delta function
5.6 Fourier transform
5.7 Convolution integral
Summary
Exercises
References
6 Spherical harmonics and friends
6.1 Properties of the Legendre polynomials, Pl(x)
6.2 Associated Legendre functions, Pml(x)
6.3 Spherical harmonic functions, Yml(θ, ψ)
6.4 Addition theorem for Yml(θ, ψ)
6.5 Laplace equation in spherical coordinates
Summary
Exercises
References
7 Bessel functions and friends
7.1 Small-argument and asymptotic forms
7.2 Properties of the Bessel functions, J,(x)
7.3 Orthogonality
7.4 Bessel series
7.5 Fourier-Bessel transform
7.6 Spherical Bessel functions
7.7 Expansion of plane waves in spherical coordinates
Summary
Exercises
Reference
Appendices
A Topics in linear algebra
B Vector calculus
C Power series
D Gamma function, F(x)
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