Fourier integral

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Problem: Find the Fourier image of function

f (x)

f (x) \equiv 0

x \notin [x_{1}, x_{4}]

x \in [x_{1}, x_{4}]

the graph of this function consists of links of a polygonal chain, passing through points

A (x_{1}, y_{1}), B (x_{2}, y_{2})

C (x_{3}, y_{3}), D (x_{4}, y_{4})

. The coordinates of the points are presented in the table:

Unknown environment 'tabular'

2.6.1.1 Fourier integral

5.09 $

Problem: Find the Fourier image of function

f (x)

f (x) \equiv 0

x \notin [x_{1}, x_{4}]

x \in [x_{1}, x_{4}]

the graph of this function consists of links of a polygonal chain, passing through points

A (x_{1}, y_{1}), B (x_{2}, y_{2})

C (x_{3}, y_{3}), D (x_{4}, y_{4})

. The coordinates of the points are presented in the table:

Unknown environment 'tabular'

2.6.1.2 Fourier integral

5.09 $

Problem: Find the Fourier image of function

f (x)

f (x) \equiv 0

x \notin [x_{1}, x_{4}]

x \in [x_{1}, x_{4}]

the graph of this function consists of links of a polygonal chain, passing through points

A (x_{1}, y_{1}), B (x_{2}, y_{2})

C (x_{3}, y_{3}), D (x_{4}, y_{4})

. The coordinates of the points are presented in the table:

Unknown environment 'tabular'

2.6.1.3 Fourier integral

5.09 $

Problem: Find the Fourier image

F [f (x)] (v)

= \int_{- \infty}^{+ \infty} f (t) e^{- i v t} d t

f (t)

, using equalities

F [

t] (v) = sinc \frac{v}{2}

F [Λ (t)] (v) = {sinc}^{2} \frac{v}{2}

. The graph of function

f (t)

consists a polygonal chain, connecting points

A, B, C

D

, as well as from parts of the

x

-axis (to the left of

A

and to the right of

D

Unknown environment 'tabular'

2.6.1.4 Fourier integral

5.09 $

Problem: Find the Fourier image of function

f (x)

f (x) \equiv 0

x \notin [x_{1}, x_{4}]

x \in [x_{1}, x_{4}]

the graph of this function consists of links of a polygonal chain, passing through points

A (x_{1}, y_{1}), B (x_{2}, y_{2})

C (x_{3}, y_{3}), D (x_{4}, y_{4})

. The coordinates of the points are presented in the table:

Unknown environment 'tabular'

2.6.1.5 Fourier integral

5.09 $

f (x) = e^{- | x |} \cos ω x

. Calculate the integral:

F (y) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{+ \infty} f (x) e^{i y x} d x .

2.6.1.6 Fourier integral

3.05 $

Problem: Represent the following function as a Fourier integral, extending it evenly to the interval

(- \infty, 0)

f (x) = {\begin{cases} 1, & if 0 \leq x \leq 1 \\ 0, & if x > 1 \end{cases} .

2.6.1.7 Fourier integral

2.54 $