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Euge
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For ##c > 0## and ##0 \le x \le 1##, find the complex integral $$\int_{c - \infty i}^{c + \infty i} \frac{x^s}{s}\, ds$$
I excluded it intensionally.Euge said:Hi @Paul Colby, you missed the ##x = 1## case!
Integration over a line in the complex plane is a mathematical process of finding the area under a curve along a specific path in the complex plane. It involves calculating the integral of a complex-valued function along a line segment.
The purpose of integration over a line in the complex plane is to solve complex-valued integrals, which cannot be solved using traditional methods. It allows us to find the area under a curve in the complex plane, which has many applications in physics, engineering, and other fields.
Integration over a line in the complex plane differs from integration in the real plane in that it involves calculating the integral along a specific path instead of over a region. This path can be a straight line, a curve, or a combination of both. Additionally, the integrand in the complex plane is a complex-valued function, while in the real plane it is a real-valued function.
Some common techniques for integration over a line in the complex plane include the method of residues, contour integration, and Cauchy's integral theorem. These methods use properties of complex functions, such as analyticity and Cauchy-Riemann equations, to evaluate the integral along a given path.
Integration over a line in the complex plane has many real-world applications, including in electrical engineering, signal processing, and quantum mechanics. It is used to calculate the electric field around a charged wire, analyze the frequency response of a system, and solve Schrödinger's equation in quantum mechanics, among others.