Transforms | Harmonic functions
The Kelvin transform is a device used in classical potential theory to extend the concept of a harmonic function, by allowing the definition of a function which is 'harmonic at infinity'. This technique is also used in the study of subharmonic and superharmonic functions. In order to define the Kelvin transform f* of a function f, it is necessary to first consider the concept of inversion in a sphere in Rn as follows. It is possible to use inversion in any sphere, but the ideas are clearest when considering a sphere with centre at the origin. Given a fixed sphere S(0,R) with centre 0 and radius R, the inversion of a point x in Rn is defined to be A useful effect of this inversion is that the origin 0 is the image of , and is the image of 0. Under this inversion, spheres are transformed into spheres, and the exterior of a sphere is transformed to the interior, and vice versa. The Kelvin transform of a function is then defined by: If D is an open subset of Rn which does not contain 0, then for any function f defined on D, the Kelvin transform f* of f with respect to the sphere S(0,R) is One of the important properties of the Kelvin transform, and the main reason behind its creation, is the following result: Let D be an open subset in Rn which does not contain the origin 0. Then a function u is harmonic, subharmonic or superharmonic in D if and only if the Kelvin transform u* with respect to the sphere S(0,R) is harmonic, subharmonic or superharmonic in D*. This follows from the formula (Wikipedia).
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Visit http://ilectureonline.com for more math and science lectures! In this video I will explain the mathematical definition and equation of a Fourier transform. Next video in this series can be seen at: https://youtu.be/yl6RtWp7y4k
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Visit http://ilectureonline.com for more math and science lectures! In this video I will explain what is a Fourier transform and how is it different from the Fourier series. Next video in this series can be seen at: https://youtu.be/fMHk6_1ZYEA
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Visit http://ilectureonline.com for more math and science lectures! In this video I will solve F(w)=? of a simple example of a Fourier transform. Next video in this series can be seen at:
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http://AllSignalProcessing.com for more great signal processing content, including concept/screenshot files, quizzes, MATLAB and data files. Introduces the definition of the z-transform, the complex plane, and the relationship between the z-transform and the discrete-time Fourier transfor
From playlist The z-Transform
This video will discuss the Fourier Transform, which is one of the most important coordinate transformations in all of science and engineering. Book Website: http://databookuw.com Book PDF: http://databookuw.com/databook.pdf These lectures follow Chapter 2 from: "Data-Driven Science an
From playlist Fourier
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Visit http://ilectureonline.com for more math and science lectures! In this video I will explain the concept of the Fourier transform delta function in the time domain to the Fourier transform to the frequency domain. Next video in this series can be seen at: https://youtu.be/GKKv9T-noO0
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