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The Sommerfeld construction of a multiple-valued function, both for solutions of Laplace’s equation and the equation of wave propagation, leads to a contour integral with part of the contour at infinity. Subsequently, it is necessary to verify that the function derived satisfies the latter equations; this requires what is often called “differentiation under the integral sign.” To justify such a process, it is not enough that the integral converge as part of the contour recedes to infinity; to ensure that differentiation can take place, it is sufficient that the integral converge uniformly.
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