Section 7.5

Fourier Integral Formulas

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Problem 3881

Represent the function by its Fourier integral:

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

Represent the function by its Fourier integral:

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

Represent the function by its Fourier integral:

f(x)=\operatorname {sgn}(x-a)-\operatorname {sgn}(x-b) \quad (b>a).

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Problem 3884

Represent the function by its Fourier integral:

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

Represent the function by its Fourier integral:

f(x)=\dfrac {1}{a^{2}+x^{2}} \quad (a>0).

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Problem 3886

Represent the function by its Fourier integral:

f(x)=\dfrac {x}{a^{2}+x^{2}} \quad (a>0).

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Problem 3887

Represent the function by its Fourier integral:

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

Represent by a Fourier integral the function

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

Represent by a Fourier integral the function

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

Represent by a Fourier integral the function

f(x)=e^{-\alpha \left|x\right|} \quad (\alpha >0) .

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Problem 3891

Represent by a Fourier integral the function

f(x)=e^{-\alpha \left|x\right|}\cos \beta x \quad (\alpha >0) .

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Problem 3892

Represent by a Fourier integral the function

f(x)=e^{-\alpha \left|x\right|}\sin \beta x \quad (\alpha >0) .

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Problem 3893

Represent by a Fourier integral the function

f(x)=e^{-x^{2}} .

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Problem 3894

Represent by a Fourier integral the function

f(x)=xe^{-x^{2}} .

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Problem 3895

(a)(b)

Represent by a Fourier integral the function

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 3896

For the function $f(x)=e^{-\alpha \left|x\right|}$ ( $\alpha >0$ ), find its Fourier transform

F(x)=\dfrac {1}{\sqrt{2\pi }}\int _{-\infty }^{+\infty }f(t)e^{-itx}\, dt=\dfrac {1}{\sqrt{2\pi }}\lim _{l \to +\infty }\int _{-l}^{l}f(t)e^{-itx}\, dt .

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Problem 3897

For the function $f(x)=xe^{-\alpha \left|x\right|}$ ( $\alpha >0$ ), find its Fourier transform

F(x)=\dfrac {1}{\sqrt{2\pi }}\int _{-\infty }^{+\infty }f(t)e^{-itx}\, dt=\dfrac {1}{\sqrt{2\pi }}\lim _{l \to +\infty }\int _{-l}^{l}f(t)e^{-itx}\, dt .

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Problem 3898

For the function $f(x)=e^{-\frac{x^{2}}{2}}$ , find its Fourier transform

F(x)=\dfrac {1}{\sqrt{2\pi }}\int _{-\infty }^{+\infty }f(t)e^{-itx}\, dt=\dfrac {1}{\sqrt{2\pi }}\lim _{l \to +\infty }\int _{-l}^{l}f(t)e^{-itx}\, dt .

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Problem 3899

For the function $f(x)=e^{-\frac{x^{2}}{2}}\cos \alpha x$ , find its Fourier transform

F(x)=\dfrac {1}{\sqrt{2\pi }}\int _{-\infty }^{+\infty }f(t)e^{-itx}\, dt=\dfrac {1}{\sqrt{2\pi }}\lim _{l \to +\infty }\int _{-l}^{l}f(t)e^{-itx}\, dt .

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Problem 3900

(a)(b)

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

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(a)

$\displaystyle \int_{0}^{+\infty }\varphi (y)\cos xy\, dy=\dfrac {1}{1+x^{2}}$ ;

(b)

$\displaystyle \int_{0}^{+\infty }\psi (y)\sin xy\, dy=e^{-x}$ ( $x>0$ ).