Fourier integral representation

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Problem 3823 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.3 "Change of variable in improper integrals. Differentiation and integration under the integral sign for improper integrals"

For different values of xx, find the Dirichlet discontinuous multiplier

D(x)=2π∫0+∞sin⁡λcos⁡λx dλλ, D(x)=\dfrac {2}{\pi }\int _{0}^{+\infty }\sin \lambda \cos \lambda x\, \dfrac {d\lambda }{\lambda } ,

and sketch the graph of the function y=D(x)y=D(x).

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Problem 3881 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.5 "Fourier Integral Formulas"

Represent the function by its Fourier integral:

f(x)={1,if ∣x∣<1;0,if ∣x∣>1. f(x)= \begin{cases} 1, & \text{if } \left|x\right|<1; \\ 0, & \text{if } \left|x\right|>1. \end{cases}
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Problem 3882 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.5 "Fourier Integral Formulas"

Represent the function by its Fourier integral:

f(x)={sgn⁡x,if ∣x∣<1;0,if ∣x∣>1. f(x)= \begin{cases} \operatorname {sgn}x, & \text{if } \left|x\right|<1; \\ 0, & \text{if } \left|x\right|>1. \end{cases}
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Problem 3883 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.5 "Fourier Integral Formulas"

Represent the function by its Fourier integral:

f(x)=sgn⁡(x−a)−sgn⁡(x−b)(b>a). f(x)=\operatorname {sgn}(x-a)-\operatorname {sgn}(x-b) \quad (b>a).
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Problem 3884 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.5 "Fourier Integral Formulas"

Represent the function by its Fourier integral:

f(x)={h(1−∣x∣a),if ∣x∣⩽a;0,if ∣x∣>a. f(x)= \begin{cases} h\left(1-\dfrac {\left|x\right|}{a}\right), & \text{if } \left|x\right| \leqslant a; \\ 0, & \text{if } \left|x\right|>a. \end{cases}
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Problem 3885 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.5 "Fourier Integral Formulas"

Represent the function by its Fourier integral:

f(x)=1a2+x2(a>0). f(x)=\dfrac {1}{a^{2}+x^{2}} \quad (a>0).
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Problem 3886 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.5 "Fourier Integral Formulas"

Represent the function by its Fourier integral:

f(x)=xa2+x2(a>0). f(x)=\dfrac {x}{a^{2}+x^{2}} \quad (a>0).
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Problem 3887 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.5 "Fourier Integral Formulas"

Represent the function by its Fourier integral:

f(x)={sin⁡x,if ∣x∣⩽π;0,if ∣x∣>π. f(x)= \begin{cases} \sin x, & \text{if } \left|x\right| \leqslant \pi ; \\ 0, & \text{if } \left|x\right|>\pi . \end{cases}
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Problem 3888 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.5 "Fourier Integral Formulas"

Represent by a Fourier integral the function

f(x)={cos⁡x,if ∣x∣⩽π2;0,if ∣x∣>π2. f(x)= \begin{cases} \cos x , & \text{if } \left|x\right| \leqslant \dfrac {\pi }{2}; \\ 0 , & \text{if } \left|x\right|>\dfrac {\pi }{2}. \end{cases}
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Problem 3889 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.5 "Fourier Integral Formulas"

Represent by a Fourier integral the function

f(t)={Asin⁡ωt,if ∣t∣⩽2πnω;0,if ∣t∣>2πnω(n a natural number). f(t)= \begin{cases} A\sin \omega t , & \text{if } \left|t\right| \leqslant \dfrac {2\pi n}{\omega }; \\ 0 , & \text{if } \left|t\right|>\dfrac {2\pi n}{\omega } \end{cases} \quad (n \text{ a natural number}) .
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Problem 3890 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.5 "Fourier Integral Formulas"

Represent by a Fourier integral the function

f(x)=e−α∣x∣(α>0). f(x)=e^{-\alpha \left|x\right|} \quad (\alpha >0) .
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Problem 3891 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.5 "Fourier Integral Formulas"

Represent by a Fourier integral the function

f(x)=e−α∣x∣cos⁡βx(α>0). f(x)=e^{-\alpha \left|x\right|}\cos \beta x \quad (\alpha >0) .
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Problem 3892 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.5 "Fourier Integral Formulas"

Represent by a Fourier integral the function

f(x)=e−α∣x∣sin⁡βx(α>0). f(x)=e^{-\alpha \left|x\right|}\sin \beta x \quad (\alpha >0) .
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Problem 3893 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.5 "Fourier Integral Formulas"

Represent by a Fourier integral the function

f(x)=e−x2. f(x)=e^{-x^{2}} .
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Problem 3894 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.5 "Fourier Integral Formulas"

Represent by a Fourier integral the function

f(x)=xe−x2. f(x)=xe^{-x^{2}} .
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Problem 3895 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.5 "Fourier Integral Formulas"

Represent by a Fourier integral the function

f(x)=e−x(0<x<+∞). f(x)=e^{-x} \quad (0<x<+\infty ) .
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(a)

Extend it as an even function;

(b)

Extend it as an odd function.

Problem 3900 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.5 "Fourier Integral Formulas"

Find the functions φ(x)\varphi (x) and ψ(x)\psi (x), if

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(a)

∫0+∞φ(y)cos⁡xy dy=11+x2\displaystyle \int_{0}^{+\infty }\varphi (y)\cos xy\, dy=\dfrac {1}{1+x^{2}};

(b)

∫0+∞ψ(y)sin⁡xy dy=e−x\displaystyle \int_{0}^{+\infty }\psi (y)\sin xy\, dy=e^{-x} (x>0x>0).

Problem 4790 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.8 "Miscellaneous Problems"

By splitting the interval (−∞,∞)(-\infty ,\infty ) into the parts (−∞,x−h)(-\infty ,x-h), (x−h,x)(x-h,x), (x,x+h)(x,x+h), (x+h,∞)(x+h,\infty ), where hh is an arbitrarily chosen number, prove the equality:

lim⁡n→∞1π∫−∞∞f(u)sin⁡n(u−x)u−x du=f(x). \lim _{n \rightarrow \infty } \frac{1}{\pi } \int _{-\infty }^{\infty } f(u) \frac{\sin n(u-x)}{u-x}\, du = f(x) .

Here, for simplicity, it is assumed that f(x)f(x) is continuous everywhere, f′(x)f'(x) exists everywhere and is bounded on every given interval, and ∫−∞∞f(u) du\int_{-\infty }^{\infty } f(u)\, du is absolutely convergent.

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Problem 4791 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.8 "Miscellaneous Problems"

Under the same conditions on f(x)f(x) as in the preceding problem, prove Fourier's formula:

f(x)=12π∫−∞∞du∫−∞∞f(v)cos⁡u(v−x) dv. f(x) = \frac{1}{2\pi } \int _{-\infty }^{\infty } du \int _{-\infty }^{\infty } f(v) \cos u(v-x)\, dv .
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Problem 4792 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.8 "Miscellaneous Problems"

Show that the Fourier integral (see the preceding problem) remains valid also in the case when f(x)f(x) has finite discontinuities at isolated points, provided the earlier conditions are preserved at the remaining points. In that case, at the points of discontinuity the value of the integral equals 12(f(x+0)+f(x−0))\dfrac {1}{2}\left(f(x+0)+f(x-0)\right), where f(x+0)f(x+0) and f(x−0)f(x-0) are the limits from the right and from the left of f(ξ)f(\xi ) as ξ→x\xi \rightarrow x.

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Problem 4947 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.9 "Fourier Series and Related Questions"

Represent as a Fourier integral the function

f(x)={e−mx,x>0;emx,x<0;m>0. f(x) = \begin{cases} e^{-mx}, & x > 0; \\ e^{mx}, & x < 0; \quad m > 0 . \end{cases}
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Problem 4948 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.9 "Fourier Series and Related Questions"

Represent as a Fourier integral the function

f(x)={e−mx,x>0;−emx,x<0;m>0. f(x) = \begin{cases} e^{-mx}, & x > 0; \\ -e^{mx}, & x < 0; \quad m > 0 . \end{cases}
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Problem 4949 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.9 "Fourier Series and Related Questions"

Represent as a Fourier integral the function

f(x)={e−mx,x>0;0,x<0;m>0. f(x) = \begin{cases} e^{-mx}, & x > 0; \\ 0, & x < 0; \quad m > 0 . \end{cases}
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Problem 4950 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.9 "Fourier Series and Related Questions"

Represent as a Fourier integral the function

f(x)={1,∣x∣<τ;0,∣x∣>τ. f(x) = \begin{cases} 1, & \left|x\right| < \tau ; \\ 0, & \left|x\right| > \tau . \end{cases}
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Problem 4951 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.9 "Fourier Series and Related Questions"

Represent as a Fourier integral the function

f(x)={1,0<x<τ;−1,−τ<x<0;0,∣x∣>τ. f(x) = \begin{cases} 1, & 0 < x < \tau ; \\ -1, & -\tau < x < 0; \\ 0, & \left|x\right| > \tau . \end{cases}
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Problem 4952 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.9 "Fourier Series and Related Questions"

Represent as a Fourier integral the function

f(x)={A(1−∣x−x0∣a),∣x−x0∣<a;0,∣x−x0∣>a. f(x) = \begin{cases} A\left(1-\dfrac {\left|x-x_{0}\right|}{a}\right), & \left|x-x_{0}\right| < a; \\ 0, & \left|x-x_{0}\right| > a . \end{cases}
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Problem 4953 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.9 "Fourier Series and Related Questions"

Represent as a Fourier integral the function

f(x)={e−kxsin⁡ωx,x>0;0,x<0;k>0. f(x) = \begin{cases} e^{-kx}\sin \omega x, & x > 0; \\ 0, & x < 0; \quad k > 0 . \end{cases}
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Problem 4954 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.9 "Fourier Series and Related Questions"

Represent as a Fourier integral the function

f(x)={sin⁡x,0<x<nπ;n an integer;0,x<0; x>nπ. f(x) = \begin{cases} \sin x, & 0 < x < n\pi ; \quad n \text{ an integer}; \\ 0, & x < 0; \, x > n\pi . \end{cases}
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Problem 3.17.1 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.17.2 "Problems"

Represent the function f(x)f(x) by a Fourier integral.

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(1)

f(x)={1, if ∣x∣<τ0, if ∣x∣>τf(x) = \begin{cases} 1, & \text{ if }\left|x\right| < \tau \\ 0, & \text{ if }\left|x\right| > \tau \end{cases}

(2)

f(x)={1−∣x∣/a, if ∣x∣⩽a,0, if ∣x∣>a;f(x) = \begin{cases} 1-\left|x\right| / a, & \text{ if } & \left|x\right| \leqslant a, \\ 0, & \text{ if } & \left|x\right| > a ; \end{cases}

(3)

f(x)=sign⁡(x−a)−sign⁡(x−b),b>af(x) = \operatorname {sign}(x-a)-\operatorname {sign}(x-b), b > a;

(4)

f(x)=1/(x2+a2),a≠0f(x) = 1 /\left(x^{2}+a^{2}\right), a \neq 0.

Problem 3.17.2 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.17.2 "Problems"

Represent the function f(x)f(x) by a Fourier integral.

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(1)

f(x)=xx2+a2,a≠0f(x) = \frac{x}{x^{2}+a^{2}}, \quad a \neq 0;

(2)

f(x)={sin⁡x, if ∣x∣⩽π,0, if ∣x∣>π;f(x) = \begin{cases} \sin x, & \text{ if } & \left|x\right| \leqslant \pi , \\ 0, & \text{ if } & \left|x\right| > \pi ; \end{cases}

(3)

f(x)={cos⁡x, if ∣x∣⩽π/2,0, if ∣x∣>π/2;f(x) = \begin{cases} \cos x, & \text{ if } & \left|x\right| \leqslant \pi / 2, \\ 0, & \text{ if } & \left|x\right| > \pi / 2 ; \end{cases}

(4)

f(x)={sin⁡ωx, if ∣x∣⩽2πn/ω,n∈N,ω>0.0, if ∣x∣>2πn/ω,f(x) = \begin{cases} \sin \omega x, & \text{ if } & \left|x\right| \leqslant 2 \pi n / \omega , \quad n \in N, \quad \omega > 0. \\ 0, & \text{ if } & \left|x\right| > 2 \pi n / \omega , \end{cases}

Problem 3.17.3 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.17.2 "Problems"

Represent the function f(x)f(x) by a Fourier integral.

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(1)

f(x)=e−α∣x∣sin⁡βx,α>0f(x) = e^{-\alpha \left|x\right|} \sin \beta x, \alpha > 0

(2)

f(x)=e−α∣x∣cos⁡βx,α>0f(x) = e^{-\alpha \left|x\right|} \cos \beta x, \alpha > 0;

(3)

f(x)=e−x2f(x) = e^{-x^{2}};

(4)

f(x)=xe−x2f(x) = x e^{-x^{2}}.

Problem 3.17.4 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.17.2 "Problems"

Represent the function f(x)f(x) by a Fourier integral.

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(1)

f(x)={e−αx, if x>0,α>0;0, if x<0;f(x) = \begin{cases} e^{-\alpha x}, & \text{ if } x > 0, \quad \alpha > 0 ; \\ 0, & \text{ if } x < 0 ; \end{cases}

(2)

f(x)={e−αxsin⁡ωx, if x>0,α>0;0, if x<0;f(x) = \begin{cases} e^{-\alpha x} \sin \omega x, & \text{ if } x > 0, \quad \alpha > 0 ; \\ 0, & \text{ if } x < 0 ; \end{cases}

(3)

f(x)={sin⁡x, if 0<x<πn,0, if x<0 or x>πn,n∈N.f(x) = \begin{cases} \sin x, & \text{ if } 0 < x < \pi n, \\ 0, & \text{ if } x < 0 \text{ or } x > \pi n, n \in N. \end{cases}

Problem 3.17.5 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.17.2 "Problems"

Represent the function f(x)f(x) by a Fourier integral, extending it as an odd function to the interval (−∞;0)(-\infty ; 0), if:

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(1)

f(x)={sin⁡x, if 0⩽x⩽π0, if x>π;f(x) = \begin{cases} \sin x, & \text{ if } 0 \leqslant x \leqslant \pi \\ 0, & \text{ if } \quad x > \pi ; \end{cases}

(2)

f(x)={2−3x, if 0⩽x⩽2/3,0, if x>2/3.f(x) = \begin{cases} 2-3 x, & \text{ if } 0 \leqslant x \leqslant 2 / 3, \\ 0, & \text{ if } \quad x > 2 / 3. \end{cases}

Problem 3.17.6 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.17.2 "Problems"

Represent the function f(x)f(x) by a Fourier integral, extending it as an even function to the interval (−∞;0)(-\infty ; 0), if:

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(1)

f(x)=e−αx,x⩾0,α>0f(x) = e^{-\alpha x}, x \geqslant 0, \alpha > 0;

(2)

f(x)={1, if 0⩽x⩽10, if x>1.f(x) = \begin{cases} 1, & \text{ if } 0 \leqslant x \leqslant 1 \\ 0, & \text{ if } \quad x > 1. \end{cases}

Problem 3.17.17 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.17.2 "Problems"

Find φ(y)\varphi (y), if:

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(1)

∫0+∞φ(y)cos⁡xydy=11+x2\int_{0}^{+\infty } \varphi (y) \cos x y d y = \frac{1}{1+x^{2}};

(2)

∫0+∞φ(y)sin⁡xydy=e−x,x>0\int_{0}^{+\infty } \varphi (y) \sin x y d y = e^{-x}, x > 0.

References

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.
[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.
[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Функции многих переменных. 2-е изд., ФИЗМАТЛИТ, 2003.