Fymm II a, list of contents AD 2001

1. Fourier-transforms revisited

The Fourier integral representation. The Delta-distribution. Cauchys singular integral. Laplace- and Mellin-transforms. Application: Eulers $\Gamma$-function. Convolution of transforms. The Parseval formula. L^p- ja L¹-theory of Fourier-transforms. Weierstrass singular integral.

2. On partial differential equations

Generalities on partial differential equations. Classification of second order partial differential equations; elliptic, hyperbolic, and parabolic equations. Examples: The Laplace and Poisson equations in finite domains. Boundary conditions of the Dirichlet-, v. Neumann- and mixed type. The wave equation and Cauchys initial value problem. The equation of heat transfer. Correctly posed problems. Two-dimensional elliptic equations, connection with the Cauchy-Riemann equations. The general solution of the two-dimensional wave equation. Separation of variables in the Laplace equation using various coordinate systems (Cartesian-, spherical-, cylindrical). Separation of variables in the Schrödinger equation. The stationary Schrödinger equation and its eigenvalue spectrum. Boundary conditions for bound state solutions and scattering solutions. The Schrödinger initial value problem. Separation of variables in the spherically symmetric stationary Schrödinger equation. Spherical harmonics. Relation to the Lie algebra of the rotation group.

3. On second order linear differential equations in one dependent variable

Generalities, homogeneous and inhomogeneous equations. Linearly independent solutions and the general solution. Wronskis determinant. Method of variation of constants. Construction of a second lineqarly independent solution from a known solution using Wronskis determinant. Classification of irregular points in the coefficient functions of the differential equations (including $\infty$). The series solution method (Frobenius method), the characteristic equation and its roots. The cases when the roots of the characteristic equation differ by an integer or zero. The case of double roots. The series solution around a regular point.

4. The special functions

4.1. Legendre polynomials and -functions.

The Legendre differential equation and the Legendre polynomials P_l(z). Series solutions around z = 0 and z = 1 The Rodrigues formula. Orthogonality prpoerties of P_l(z). The represenation of an arbitrary polynomial as a linear combination of Legendre polynomials. On the representaion of functions defined on the interval [- 1, 1] as series of Legendre polynomials. Convergence conditions. The Schläfli integral reperesentation for the polynomials P_l(z). The generating function and recursion formulas for Legendre polynomials. On the associated Legendre plynomials and their orthogonality properties. Relation with spherical harmonidcs. Legendre functions of the first- and second kind.

4.2. Bessel functions

Bessels differential equation. Series solutions around z = 0. Bessel functions of the first kind when the index $\nu$ is not a half-integer. The special case 2$\nu$ = 2n + 1 where n is an integer. The linear independence of the Bessel functions J_±$\nu$. The special case $\nu$ = n, where n is an integer. The Neumann functions N_$\nu$(z) and Hankel functions H_$\nu$¹(z) ja H_$\nu$²(z). The general recursion formulae for the functions J_$\nu$. The generating function for J_$\nu$ with applications. The Poisson integral representation for the functions J_$\nu$(z). Expression of the functions J_{±(n + 1/2}(z) in closed form in terms of trigonometric functions. The spherical Bessel and Hankel functions. The expansion of a plane wave in terms of spherical waves. Asymptotic expansions of Bessel functions.

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