Скачать или смотреть What are Dirichlet's conditions?

Welcome back MechanicaLEi, did you know that Peter Gustav Lejeune Dirichlet who was one of the first mathematicians to define a function was the one on whose name the Dirichlet's conditions are named? This makes us wonder, what are Dirichlet's conditions? Before we jump in check out the previous part of this series to learn about what Fourier series are? Now, The Dirichlet conditions are sufficient conditions for a real-valued, periodic function f to be equal to the sum of its Fourier series at each point where f is continuous. The conditions are: first f must be absolutely integrable over a period. Second, f must be of bounded variation in any given bounded interval and finally f must have a finite number of discontinuities in any given bounded interval, and the discontinuities cannot be infinite. This leads us to the Dirichlet’s theorem. We state the Dirichlet's theorem assuming that f is a periodic function of period 2 pi with Fourier series expansion whose a n term is given by the following equation. Dirichlet’s theorem states that: If f satisfies Dirichlet's conditions, then for all x, we have that the series obtained by plugging x into the Fourier series is convergent, and is given by summation of a n into e power i n x from n equals minus infinity to infinity is equal to half the sum of f of x plus and f of x minus. Here, x plus and x minus denote the right and left limits of f respectively. Hence, we first saw what Dirichlet's conditions are and then went on to see what Dirichlet's theorem is?
In the next episode of MechanicaLEi find out what half sine and cosine series are?

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Doh De Oh by Kevin MacLeod is licensed under a Creative Commons Attribution licence (https://creativecommons.org/licenses/...)
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