Topics in mathematical physics
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Diverse Topics In Theoretical And Mathematical Physics: Lectures By Roman Jackiw
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Institutional Subscription. Instructor Ancillary Support Materials. Free Shipping Free global shipping No minimum order. Brings together the most important aspects and recent advances in theoretical and computational chemistry Covers computational methods for small molecules, density-functional methods, and computational chemistry on personal and quantum computers Presents cutting-edge developments in theoretical and computational chemistry that are applicable to graduate students and research professionals in chemistry, physics, materials science and biochemistry.
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Powered by. You are connected as. Connect with:. Evaluation of definite integrals using contour integration - Contour integration. Mittag-Leffler expansions of meromorphic functions - Causal, linear, retarded response. Dynamic susceptibility.
Symmetry properties of the dynamic susceptibility. Dispersion relations. Hilbert transform pairs - Subtracted dispersion relations.
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Admittance of an LCR circuit. Discrete and continuous relaxation spectra - Analytic continuation and the gamma function:Definition of the gamma function and its analytic continuation. Analytic properties - Connection with gaussian integrals. Mittag-Leffler expansion of the gamma function. Logarithmic derivative of the gamma function. Infinite product representation for the gamma function. The beta function. Reflection and duplication formulas for the gamma function;Mobius transformations:Conformal mapping.
Definition of a Mobius transformation and its basic properties - Fixed points of a Mobius transformation. The cross-ratio. Normal form of a Mobius transformation.
Iterates of a Mobius transformation - Classification of Mobius transformations. The isometric circle. Group properties; the Mobius group. The Mobius group over the reals. The modular group. The invariance group of the unit circle. Connection with the pseudo-unitary group SU 1,1.
The group of cross-ratios. Multivalued functions; integral representations:Branch points and branch cuts. Algebraic and logarithmic branch points, winding point. Riemann sheets - Contour integrals in the presence of branch points. An integral involving a class of rational functions. Contour integral representation for the gamma function - Contour integral representations for the beta function and the Riemann zeta function. Connection with Bernoulli numbers. Zeroes of the zeta function. Statement of the Riemann hypothesis - Contour integral representations of the Legendre functions of the first and second kinds.
Singularities of functions defined by integrals. End-point and pinch singularities, examples. Singularities of the Legendre functions. Dispersion relations for the Legendre functions;Laplace transforms:Definition of the Laplace transform.
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The convolution theorem. Laplace transforms of derivatives. The inverse transform, Mellin's formula. The LCR series circuit. Laplace transform of the Bessel and modified Bessel functions of the first kind. Laplace transforms and random processes: the Poisson process - Laplace transforms and random processes: biased random walk on a linear lattice and on a d-dimensional lattice;Fourier transforms:Fourier integrals.
Parseval's formula for Fourier transforms. Fourier transform of the delta function. Generalization of Parseval's formula. Iterates of the Fourier transform operator - Unitarity of the Fourier transformation. Eigenvalues and eigenfunctions of the Fourier transform operator - The Fourier transform in d dimensions. The Poisson summation formula.
Research topics of the Mathematical Physics group, Theoretical Physics, Heidelberg
Some illustrative examples. Generalization to higher dimensions;Fundamental Green function for the laplacian operator:Green functions. Poisson's equation. The fundamental Green function for the laplacian operator. Solution of Poisson's equation for a spherically symmetric source - The Coulomb potential in d dimensions. Ultraspherical coordinates. A divergence problem.