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Exploring the Complex Plane: Green's Functions, Hilbert Transforms, and Analytic Continuation
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The complex plane has an important role in modern physics, and a great variety of problems in physics--both conceptual and technical--can be explored by using it. Of particular importance in condensed-matter, nuclear, and particle physics are Green's functions and Hilbert transforms. In the theory of a many-body system, a convenient way to characterize the system's eigenenergies and eigenfunctions is through a Green's function, from which quantities closely related to experimental parameters such as densities of states can be obtained. In quantum scattering theory, the Green's function usually characterizes the response to a point source (configuration-space Green's function) or to a unit impulse (time-domain Green's function). Modern texts on mathematical physics provide introductions to complex variables, Green's function, and Hilbert transforms.
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