Fourier expansions of given functions

Expand the function $f(x)=\sin^{4}x$ into a Fourier series.

What is the Fourier series of the trigonometric polynomial $p_{n}(x)=\displaystyle \sum_{i=0}^{n} (\alpha_{i}\cos ix+\beta_{i}\sin ix)$?

Expand the function $f(x)=\operatorname {sgn}x$ ($-\pi <x<\pi$) into a Fourier series. Sketch the graph of the function and the graphs of several partial sums of its Fourier series. Using the expansion, find the sum of the Leibniz series $\displaystyle \sum_{n=1}^{\infty } \dfrac {(-1)^{n-1}}{2n-1}$.

On the interval $(-\pi ,\pi )$, expand $f(x)=x$.

On the interval $(0,2\pi )$, expand $f(x)=\dfrac {\pi -x}{2}$.

On the interval $(-\pi ,\pi )$, expand $f(x)=\left|x\right|$.

On the interval $(-\pi ,\pi )$, expand

where $a$ and $b$ are constants.

On the interval $(-\pi ,\pi )$, expand $f(x)=\pi^{2}-x^{2}$.

On the interval $(-\pi ,\pi )$, expand $f(x)=\cos ax$.

On the interval $(-\pi ,\pi )$, expand $f(x)=\sin ax$.

On the interval $(-\pi ,\pi )$, expand $f(x)=\operatorname {sh}ax$.

On the interval $(-\pi ,\pi )$, expand $f(x)=x\sin x$.

Expand the periodic function $f(x)=\operatorname {sgn}(\cos x)$ into a Fourier series.

Expand the periodic function $f(x)=\arcsin (\sin x)$ into a Fourier series.

Expand the periodic function $f(x)=\arcsin (\cos x)$ into a Fourier series.

Expand the periodic function $f(x)=\left|\sin x\right|$ into a Fourier series.

Expand the periodic function $f(x)=\left|\cos x\right|$ into a Fourier series.

Expand the periodic function $f(x)=\displaystyle \sum_{n=1}^{\infty } \alpha^{n}\dfrac {\sin nx}{\sin x}$ ($\left|\alpha \right|<1$) into a Fourier series.

Expand the following unbounded periodic function into a Fourier series:

Expand the following unbounded periodic function into a Fourier series:

Expand the following unbounded periodic function into a Fourier series:

Use the Euler integrals to evaluate the integral

Expand in the simplest Fourier series the function $f(x)$, defined on the interval $(-\pi ,\pi )$ by the conditions:

Expand in the simplest Fourier series the function $f(x)$, defined on the interval $(-\pi ,\pi )$ by the conditions:

Expand in the simplest Fourier series the function $f(x) = \left|x\right|$ for $-\pi < x < \pi$.

Expand in the simplest Fourier series the function $f(x) = x$ for $-\pi < x < \pi$.

Expand in the simplest Fourier series the function $f(x)$, defined on the interval $(-\pi ,\pi )$ by the conditions:

Expand in the simplest Fourier series the function $f(x)$, defined on the interval $(-\pi ,\pi )$ by the conditions:

Expand in the simplest Fourier series the function $f(x) = x^{2}$ for $-\pi < x < \pi$.

It is known that the sum $S(x)$ of the series $\sum_{n = 1}^{\infty } a_{n} \sin n x$, where $\left\{ a_{n}\right\}$ is a given numerical sequence, is an odd periodic function with period $2 \pi$, such that

Is it true that this series converges nonuniformly on $R$?

Expand the function into a Fourier series:

What will the Fourier series be for the trigonometric polynomial

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = \begin{cases} a x, & -\pi < x < 0, \\ b x, & 0 \leqslant x < \pi , \end{cases} \quad$ on the interval $(-\pi , \pi )$.

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = \begin{cases} a, & -\pi / 2 < x < \pi / 2, \\ b, & \pi / 2 \leqslant x < 3 \pi / 2, \end{cases}$ on the interval $(-\pi / 2,3 \pi / 2)$.

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = x+\operatorname {sign} x$ on the interval $(-\pi ; \pi )$.

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = \pi^{2}-x^{2}$ on the interval $(-\pi ; \pi )$.

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = x^{3}$ on the interval $(-\pi ; \pi )$.

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = e^{a x}, a \neq 0$, on the interval $(-\pi ; \pi )$.

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = e^{2\left|x\right|}$ on the interval $(-\pi ; \pi )$.

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = \sin a x, a \in Z$ on the interval $(-\pi ; \pi )$.

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = x \sin x$ on the segment $[-\pi ; \pi ]$.

Expand the function $f(x)$ into a Fourier series on the given interval, the length of the interval being the period. $f(x) = \begin{cases} 0, & -\pi \leqslant x < 0, \\ \sin x, & 0 \leqslant x \leqslant \pi , \end{cases}$ on the segment $[-\pi , \pi ]$.

Expand the periodic function into a Fourier series. $f(x) = \operatorname {sign}(\cos x)$.

Expand the periodic function into a Fourier series. $f(x) = \arcsin (\cos x)$.

Expand the periodic function into a Fourier series. $f(x) = \arcsin (\sin x)$.

Expand the periodic function into a Fourier series. $f(x) = \left|\cos x\right|$.

Expand the periodic function into a Fourier series. $f(x) = \left|\sin x\right|$.

Expand the periodic function into a Fourier series. $f(x) = \left|\cos (x / 2)\right|$.

Find the Fourier series in complex form of the function

Expand into a Fourier series in complex form the periodic function, with period $2 \pi$, $f(x) = e^{x},-\pi < x < \pi , f(\pi ) = \operatorname {ch} \pi$.