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This textbook accounts for two seemingly unrelated mathematical topics drawn from two separate areas of mathematics that have no evident points of contiguity. Green's function is a topic in partial differential equations and covered in most standard texts, while infinite products are used in mathematical analysis. For the two-dimensional Laplace equation, Green's functions are conventionally constructed by either the method of images, conformal mapping, or the eigenfunction expansion. The present text focuses on the construction of Green's functions for a wide range of boundary-value problems. Green's Functions and Infinite Products provides a thorough introduction to the classical subjects of the construction of Green's functions for the two-dimensional Laplace equation and the infinite product representation of elementary functions.  Every chapter begins with a review guide, outlining the basic concepts covered. A set of carefully designed challenging exercises is available at the end of each chapter to provide the reader with the opportunity to explore the concepts in more detail. Hints, comments, and answers to most of those exercises can be found at the end of the text. In addition, several illustrative examples are offered at the end of most sections. This text is intended for an elective graduate course or seminar within the scope of either pure or applied mathematics.

Green's functions --- Products, Infinite --- Conformal mapping --- Eigenfunction expansions --- Engineering & Applied Sciences --- Physics --- Physical Sciences & Mathematics --- Applied Mathematics --- Atomic Physics --- Green's functions. --- Products, Infinite. --- Infinite products --- Functions, Green's --- Functions, Induction --- Functions, Source --- Green functions --- Induction functions --- Source functions --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Differential equations. --- Partial differential equations. --- Applied mathematics. --- Engineering mathematics. --- Analysis. --- Ordinary Differential Equations. --- Partial Differential Equations. --- Applications of Mathematics. --- Algebra --- Processes, Infinite --- Differential equations --- Potential theory (Mathematics) --- Global analysis (Mathematics). --- Differential Equations. --- Differential equations, partial. --- Math --- Science --- Partial differential equations --- 517.91 Differential equations --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Conformal mapping. --- Eigenfunction expansions. --- Engineering --- Engineering analysis --- Mathematical analysis --- 517.1 Mathematical analysis --- Mathematics --- classical Euler representations --- Hilbert's theorem --- method of images --- method of variation

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A general principle, discovered by Robert Langlands and named by him the "functoriality principle," predicts relations between automorphic forms on arithmetic subgroups of different reductive groups. Langlands functoriality relates the eigenvalues of Hecke operators acting on the automorphic forms on two groups (or the local factors of the "automorphic representations" generated by them). In the few instances where such relations have been probed, they have led to deep arithmetic consequences. This book studies one of the simplest general problems in the theory, that of relating automorphic forms on arithmetic subgroups of GL(n,E) and GL(n,F) when E/F is a cyclic extension of number fields. (This is known as the base change problem for GL(n).) The problem is attacked and solved by means of the trace formula. The book relies on deep and technical results obtained by several authors during the last twenty years. It could not serve as an introduction to them, but, by giving complete references to the published literature, the authors have made the work useful to a reader who does not know all the aspects of the theory of automorphic forms.

511.33 --- Analytical and multiplicative number theory. Asymptotics. Sieves etc. --- 511.33 Analytical and multiplicative number theory. Asymptotics. Sieves etc. --- Automorfe vormen --- Automorphic forms --- Formes automorphes --- Representation des groupes --- Representations of groups --- Trace formulas --- Vertegenwoordiging van groepen --- Formulas, Trace --- Discontinuous groups --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Automorphic functions --- Forms (Mathematics) --- Analytical and multiplicative number theory. Asymptotics. Sieves etc --- Representations of groups. --- Trace formulas. --- Automorphic forms. --- 0E. --- Addition. --- Admissible representation. --- Algebraic group. --- Algebraic number field. --- Approximation. --- Archimedean property. --- Automorphic form. --- Automorphism. --- Base change. --- Big O notation. --- Binomial coefficient. --- Canonical map. --- Cartan subalgebra. --- Cartan subgroup. --- Central simple algebra. --- Characteristic polynomial. --- Closure (mathematics). --- Combination. --- Computation. --- Conjecture. --- Conjugacy class. --- Connected component (graph theory). --- Continuous function. --- Contradiction. --- Corollary. --- Counting. --- Coxeter element. --- Cusp form. --- Cyclic permutation. --- Dense set. --- Density theorem. --- Determinant. --- Diagram (category theory). --- Discrete series representation. --- Discrete spectrum. --- Division algebra. --- Eigenvalues and eigenvectors. --- Eisenstein series. --- Exact sequence. --- Existential quantification. --- Field extension. --- Finite group. --- Finite set. --- Fourier transform. --- Functor. --- Fundamental lemma (Langlands program). --- Galois extension. --- Galois group. --- Global field. --- Grothendieck group. --- Group representation. --- Haar measure. --- Harmonic analysis. --- Hecke algebra. --- Hilbert's Theorem 90. --- Identity component. --- Induced representation. --- Infinite product. --- Infinitesimal character. --- Invariant measure. --- Irreducibility (mathematics). --- Irreducible representation. --- L-function. --- Langlands classification. --- Laurent series. --- Lie algebra. --- Lie group. --- Linear algebraic group. --- Local field. --- Mathematical induction. --- Maximal compact subgroup. --- Multiplicative group. --- Nilpotent group. --- Orbital integral. --- P-adic number. --- Paley–Wiener theorem. --- Parameter. --- Parametrization. --- Permutation. --- Poisson summation formula. --- Real number. --- Reciprocal lattice. --- Reductive group. --- Root of unity. --- Scientific notation. --- Semidirect product. --- Special case. --- Spherical harmonics. --- Subgroup. --- Subset. --- Summation. --- Support (mathematics). --- Tensor product. --- Theorem. --- Trace formula. --- Unitary representation. --- Weil group. --- Weyl group. --- Zero of a function.