Jump discontinuity with fourier series

yoq_bise
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Hi all,
I couldn't find any proof of the following statement: The Fourier series expansion of f(x), which has a discontinuity at y, takes on the mean of the left and right limits
i.e. f(y)= (1/2)(f(y+)+f(y-))

is there anyone who can help me?
Thanks
 
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Have you tried looking in a book?
 
Yes I looked in Hassani, Arfken and Diprima but I couldn't find
Can anyone suggest other books or web sites?
 
Last edited:

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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