Math Problems Vault

Problem 2936

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

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Problem 2937

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

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Problem 2938

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 $\displaystyle \sum_{n=1}^{\infty } \dfrac {(-1)^{n-1}}{2n-1}$ .

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Problem 2939

On the interval $(0,2l)$ , expand

f(x)=\begin{cases} A,& \text{if } 0<x<l, \\ 0,& \text{if } l<x<2l, \end{cases}

where $A$ is a constant.

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Problem 2940

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

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Problem 2941

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

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Problem 2942

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

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Problem 2943

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

f(x)=\begin{cases} ax,& \text{if } -\pi <x<0, \\ bx,& \text{if } 0<x<\pi , \end{cases}

where $a$ and $b$ are constants.

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Problem 2944

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

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Problem 2945

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

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Problem 2946

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

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Problem 2947

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

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Problem 2948

On the interval $(-h,h)$ , expand $f(x)=e^{ax}$ .

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Problem 2949

On the interval $(a,a+2l)$ , expand $f(x)=x$ .

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Problem 2950

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

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Problem 2951

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

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Problem 2952

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

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Problem 2953

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

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Problem 2954

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

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Problem 2955

Expand the periodic function $f(x)=x-\left[x\right]$ into a Fourier series.

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Problem 2956

Expand the periodic function $f(x)=(x)$ into a Fourier series, where $(x)$ denotes the distance from $x$ to the nearest integer.

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Problem 2957

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

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Problem 2958

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

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Problem 2959

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.

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Problem 2960

Expand the function $f(x)=\sec x$ ( $-\dfrac {\pi }{4}<x<\dfrac {\pi }{4}$ ) into a Fourier series.

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Problem 2961

(a)(b)(c)

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

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(a)

as a cosine expansion;

(b)

as a sine expansion;

(c)

as an expansion on the interval $(0,2\pi )$ .

Plot the graph of the function and the graphs of the sums of the Fourier series in cases (1), (2), and (3). Using these expansions, find the sums of the series:

\sum _{n=1}^{\infty } \frac{1}{n^{2}}, \quad \sum _{n=1}^{\infty } \frac{(-1)^{n+1}}{n^{2}} \quad \text{and} \quad \sum _{n=1}^{\infty } \frac{1}{(2n-1)^{2}}.

Problem 2962

From the expansion

x=2\sum _{n=1}^{\infty } (-1)^{n+1}\frac{\sin nx}{n} \quad (-\pi <x<\pi ),

using term-by-term integration, find the Fourier series of the functions $x^{2}$ , $x^{3}$ , and $x^{4}$ on the interval $(-\pi ,\pi )$ .

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Problem 2963

Write out the Lyapunov equation for the function

f(x)= \begin{cases} 1,& \text{when} \left|x\right|<a,\\ 0,& \text{when} a<\left|x\right|<\pi \end{cases}

Using the Lyapunov equation, find the sums of the following series:

\sum _{n=1}^{\infty } \frac{\sin ^{2}na}{n^{2}} \quad \text{and} \quad \sum _{n=1}^{\infty } \frac{\cos ^{2}na}{n^{2}}.

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Problem 2964

Expand the function

f(x)= \begin{cases} x,& \text{if} 0 \leqslant x \leqslant 1,\\ 1,& \text{if} 1<x<2,\\ 3-x,& \text{if} 2 \leqslant x \leqslant 3 \end{cases}

into a Fourier series.

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Problem 2965

Using the formulas

\cos x=\frac{1}{2}(t+\bar{t}), \quad \sin x=\frac{1}{2i}(t-\bar{t}),

where $t=e^{ix}$ and $\bar{t}=e^{-ix}$ , expand the following function into a Fourier series: $\cos^{2m}x$ ( $m$ a positive integer).

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Problem 2966

Using the formulas

\cos x=\frac{1}{2}(t+\bar{t}), \quad \sin x=\frac{1}{2i}(t-\bar{t}),

where $t=e^{ix}$ and $\bar{t}=e^{-ix}$ , expand the following function into a Fourier series:

\frac{q\sin x}{1-2q\cos x+q^{2}} \quad (\left|q\right|<1).

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Problem 2967

Using the formulas

\cos x=\frac{1}{2}(t+\bar{t}), \quad \sin x=\frac{1}{2i}(t-\bar{t}),

where $t=e^{ix}$ and $\bar{t}=e^{-ix}$ , expand the following function into a Fourier series:

\frac{1-q^{2}}{1-2q\cos x+q^{2}} \quad (\left|q\right|<1).

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Problem 2968

Using the formulas

\cos x=\frac{1}{2}(t+\bar{t}), \quad \sin x=\frac{1}{2i}(t-\bar{t}),

where $t=e^{ix}$ and $\bar{t}=e^{-ix}$ , expand the following function into a Fourier series:

\frac{1-q\cos x}{1-2q\cos x+q^{2}} \quad (\left|q\right|<1).

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Problem 2969

Using the formulas

\cos x=\frac{1}{2}(t+\bar{t}), \quad \sin x=\frac{1}{2i}(t-\bar{t}),

where $t=e^{ix}$ and $\bar{t}=e^{-ix}$ , expand the following function into a Fourier series:

\ln (1-2q\cos x+q^{2}) \quad (\left|q\right|<1).

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Problem 2970

Expand the following unbounded periodic function into a Fourier series:

f(x)=\ln \left|\sin \frac{x}{2}\right|.

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Problem 2971

Expand the following unbounded periodic function into a Fourier series:

f(x)=\ln \left|\cos \frac{x}{2}\right|.

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Problem 2972

Expand the following unbounded periodic function into a Fourier series:

f(x)=\ln \left|\operatorname {tg}\frac{x}{2}\right|.

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Problem 2973

Expand the function

f(x)=\int _{0}^{x} \ln \sqrt{\left|\operatorname {ctg}\frac{t}{2}\right|}\, dt \quad (-\pi \leqslant x \leqslant \pi )

into a Fourier series.

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Problem 2974

The functions

x=x(s), \quad y=y(s) \quad (0 \leqslant s \leqslant 4a)

are the parametric equations of the boundary of the square: $0<x<a$ , $0<y<a$ , where $s$ is the arc length measured counterclockwise starting from the point $O(0,0)$ . Expand this function into a Fourier series.

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Problem 2975

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

f(x)=\sum _{n=1}^{\infty } a_{n}\cos (2n-1)x \quad (-\pi <x<\pi )?

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Problem 2976

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

f(x)=\sum _{n=1}^{\infty } b_{n}\sin (2n-1)x \quad (-\pi <x<\pi )?

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Problem 2977

(a)(b)

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

f(x)=x\left(\frac{\pi }{2}-x\right)

as follows:

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(a)

in cosines of odd multiples of the angle;

(b)

in sines of odd multiples of the angle.

Sketch the graphs of the sums of the Fourier series in cases (1) and (2).

Problem 2978

Let $f(x)$ be an anti-periodic function with period $\pi$ , that is,

f(x+\pi )=-f(x).

What special property does the Fourier series of this function on the interval $(-\pi ,\pi )$ have?

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Problem 2979

Let $f(x+\pi )=f(x)$ . What special property does the Fourier series of the function $f(x)$ on the interval $(-\pi ,\pi )$ have?

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Problem 2980

(a)(b)

A function $y=f(x)$ with period $2\pi$ has a graph that is:

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(a)

symmetric about the points $(0,0)$ , $\left(\pm \dfrac {\pi }{2},0\right)$ as centers;

(b)

symmetric about the origin as a center and about $x=\pm \dfrac {\pi }{2}$ as axes of symmetry,

What special properties do its Fourier coefficients $a_{n}$ , $b_{n}$ ( $n=1,2,3,\cdots$ ) have in each case?

Problem 2981

If the functions satisfy $\varphi (-x)=\psi (x)$ , what relation holds between the Fourier coefficients $a_{n},b_{n}$ of $\varphi (x)$ and $\alpha_{n},\beta_{n}$ of $\psi (x)$ ( $n=0,1,2,\cdots$ )?

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Problem 2982

If the functions satisfy $\varphi (-x)=-\psi (x)$ , what relation holds between the Fourier coefficients $a_{n},b_{n}$ of $\varphi (x)$ and $\alpha_{n},\beta_{n}$ of $\psi (x)$ ( $n=0,1,2,\cdots$ )?

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Problem 2983

Given that the integrable function $f(x)$ with period $2\pi$ has Fourier coefficients $a_{n},b_{n}$ ( $n=0,1,2,\cdots$ ), compute the Fourier coefficients $\overline{a_{n}},\overline{b_{n}}$ ( $n=0,1,2,\cdots$ ) of the "shifted" function $f(x+h)$ ( $h=$ constant).

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Problem 2984

Given that the integrable function $f(x)$ with period $2\pi$ has Fourier coefficients $a_{n},b_{n}$ ( $n=0,1,2,\cdots$ ), compute the Fourier coefficients $A_{n},B_{n}$ ( $n=0,1,2,\cdots$ ) of the Steklov function

f_{h}(x)=\frac{1}{2h}\int _{x-h}^{x+h} f(\xi )\, d\xi

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Problem 2985

Let $f(x)$ be a continuous function with period $2\pi$ , and let $a_{n},b_{n}$ ( $n=0,1,2,\cdots$ ) be its Fourier coefficients. Find the Fourier coefficients $A_{n},B_{n}$ ( $n=0,1,2,\cdots$ ) of the convolution function

F(x)=\frac{1}{\pi }\int _{-\pi }^{\pi } f(t)f(x+t)\, dt

Using the result obtained, derive Lyapunov's equality.

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