Projections of functions and bases

member 428835 · 2019-03-05T20:14:39+00:00

Homework Statement On ##L_2[0,2\pi]## where ##e = \{ 1/\sqrt{2 \pi},1/\sqrt{\pi}\sin x,1/\sqrt{2 \pi}\cos x \}##. Given ##f(x) = x##, find ##Pr_e f##. Homework Equations See solution. The Attempt at a Solution I take $$e \cdot \int_0^{2\pi} e f(x) \, dx = \pi - 2 \sin x.$$ Look correct?

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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Homework Statement

Homework Equations

The Attempt at a Solution

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

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