Half-range Fourier expansions

Expand the function $f(x)=x^{2}$ into a Fourier series:

How should the integrable function $f(x)$, given on the interval $\left(0,\dfrac {\pi }{2}\right)$, be extended to the interval $(-\pi ,\pi )$ so that its expansion into a Fourier series has the form

How should the integrable function $f(x)$, given on the interval $\left(0,\dfrac {\pi }{2}\right)$, be extended to the interval $(-\pi ,\pi )$ so that its expansion into a Fourier series has the form

On the interval $\left(0,\dfrac {\pi }{2}\right)$, expand the function

as follows:

On the interval $(0,\pi )$, expand $x^{3}$ in cosines.

On the interval $(0,\pi )$, expand $x^{3}$ in sines.

On the interval $(0,\pi )$, expand $\sin x$ in cosines.

The function $f(x) = x^{2}(a-x)$ is given on the interval $(0,a)$. Expand it in a Fourier series as an even function.

The function $f(x) = x^{2}(a-x)$ is given on the interval $(0,a)$. Expand it in a Fourier series as an odd function.

The function $f(x) = x$ is given on the interval $(0,a)$. Expand it in a Fourier series as an even function.

The function $f(x) = x$ is given on the interval $(0,a)$. Expand it in a Fourier series as an odd function.

The function $f(x) = x$ is given on the interval $(0,a)$. Expand it in a Fourier series as an odd function, in sines of odd multiples of the angle.

The function $f(x)$, given on the interval $\left(0,\dfrac {\pi }{2}\right)$, is continued onto the interval $\left(\dfrac {\pi }{2},\pi \right)$, so that $f\left(\dfrac {\pi }{2}+x\right) = -f\left(\dfrac {\pi }{2}-x\right)$; prove that for $0 < x < \pi$ the equality holds

where

How should the function $f(x)$, given on the interval $\left(0,\dfrac {\pi }{2}\right)$, be continued onto the interval $\left(\dfrac {\pi }{2},\pi \right)$ so that for $0 < x < \pi$

and how can the coefficients be computed?

Expand into a Fourier series the function $f(x) = \pi -2 x, 0 < x \leqslant \pi$, extending it to the segment $[-\pi ; 0]$:

Expand into a Fourier series the function $f(x)$, given on the segment $[0 ; \pi ]$ and extended to the segment $[-\pi ; 0]$ evenly. Draw the graph of the sum of the series:

Expand the function $f(x) = x, 0 \leqslant x \leqslant \pi$, into a Fourier series in cosines.

Expand the function $f(x) = \cos 2 x, 0 \leqslant x \leqslant \pi$, into a Fourier series in sines.

Expand into a Fourier series on $(0 ; \pi )$ in cosines the function

Expand into a Fourier series on the interval $(0 ; \pi )$ in sines the function

Expand the function

into a Fourier series in cosines on the segment $[0 ; 2]$.

Expand the function $f(x) = x^{2}$ into a Fourier series:

Expand into a Fourier series the function $f(x) = x-x^{2} / 2,0 \leqslant x \leqslant 1$:

Expand into a Fourier series in sines the function

Prove that $\frac{3 x^{2}-6 \pi x+2 \pi^{2}}{12} = \sum_{n = 1}^{\infty } \frac{\cos n x}{n^{2}}, 0 \leqslant x \leqslant \pi$.

Hint. Use the results of problems 40 and 45.

Expand into a Fourier series in cosines the function $f(x) = e^{x}$ on the interval $(0 ; \ln 2)$.

Expand into a Fourier series in sines the function

Expand into a Fourier series in sines the function

Expand the function $f(x) = e^{a x}, 0 < x < \pi$:

Expand the function $f(x) = \sin a x, 0 \leqslant x \leqslant \pi$, into a Fourier series in cosines.

Expand into a Fourier series in cosines the function

Expand into a Fourier series in cosines the function

Prove that

Prove that if a function $f$, absolutely integrable on the segment $[0 ; \pi ]$, satisfies the condition $f(\pi -x) = f(x)$, then its Fourier coefficients have the following properties:

How should a function, absolutely integrable on the segment $[0 ; \pi / 2]$, be extended to the segment $[-\pi ; \pi ]$ so that its Fourier series has the form

How should a function, absolutely integrable on the segment $[0 ; \pi / 2]$, be extended to the segment $[-\pi ; \pi ]$ so that its Fourier series has the form

Expand the function $f(x) = x(\pi / 2-x)$ into a Fourier series on the segment $[0 ; \pi / 2]$:

Prove that a piecewise smooth function on the segment $[0 ; l]$ can be expanded into a series of the form

Prove that a piecewise smooth function on the segment $[0 ; l]$ can be expanded into a series of the form

Prove that if the function $f$ is continuously differentiable on the segment $[x ; x+\pi ]$, then

(Steklov).