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Publication# Adjusted complex position of extra sources in the image sources method

Abstract

Using a series development of the integral solution, a formal backing of the presence of image sources in geometrical acoustics methods has been shown. Furthermore, the existence of "invisible" sources is suggested, especially in the vicinity of obtuse angles. These extra sources bring useful information but need to be adjusted in order to avoid divergences. By more closely comparing the terms of the series development and the corresponding image source, the position of each extra source can be adjusted so as to minimize the distance between term and source contribution. In other words, a correction factor must be applied to the pure geometrical position of the "invisible" sources. In order to obtain the best results, this position may need to be defined as a complex number. This approach could have a companion idea in electromagnetism, where a function is identified with the objective of optimizing the position of the image sources.

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Ontological neighbourhood

Error function

In mathematics, the error function (also called the Gauss error function), often denoted by erf, is a complex function of a complex variable defined as: Some authors define without the factor of . This nonelementary integral is a sigmoid function that occurs often in probability, statistics, and partial differential equations. In many of these applications, the function argument is a real number. If the function argument is real, then the function value is also real.

Gamma function

In mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n, Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles.

Complex plane

In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the x-axis, called the real axis, is formed by the real numbers, and the y-axis, called the imaginary axis, is formed by the imaginary numbers. The complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors.

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