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Niles
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Homework Statement
Hi all.
I have to find the half-range sine expansion of a function f(x) = 1 for 0<x<2. My question is: This function is not piecewise smooth, so why does the book ask me to do this?
Niles said:I have to find the half-range sine expansion of a function f(x) = 1 for 0<x<2. My question is: This function is not piecewise smooth, so why does the book ask me to do this?
Niles said:That f and f' must be piecewise continuous, i.e. that there are a finite number of discontinuities, where the limits exists.
f is "piecewise" continuous, so that is OK. But f' is defined on the interval 0 < x < 2 with no discontinuities?
A half-range sine expansion is a mathematical technique used to approximate a periodic function using a series of sine functions. It is often used in signal processing and other areas of science and engineering.
A half-range sine expansion allows for a more efficient representation of a periodic function, as it only requires half the number of sine terms compared to a full-range expansion. This can save computational resources and make calculations easier.
To calculate a half-range sine expansion, the function is first expressed as a Fourier series with only odd terms. Then, the coefficients of the sine terms are modified to account for the half-range. The final expression is a sum of sine functions with different frequencies and amplitudes.
A half-range sine expansion can only approximate periodic functions that are symmetric about the origin. It also assumes that the function has a finite range, and may not accurately represent functions with sharp changes or discontinuities.
A half-range sine expansion can be used in various applications, such as in the analysis of electrical signals, sound waves, and vibrations. It can also be used in image processing and data compression to efficiently represent periodic patterns or signals.