Волков Интеграл и преобразование Фуре Учебно-методическое 2012

2

1

2

sin tx

1

F (x) =

t costx dt =

t

−

π ∫

π

x

C

0

0

1

1

sin tx

2

sin tx

costx

−∫

x

dt

=

t

x

+

x

2

=

π

0

0

=

2 sin x

+

cos x −1

;

π

x

x

2

2

1

2

costx

1

F (x) =

∫

t sin tx dt =

−t

x

+

π

π

S

0

0

1 costx

2

costx

sin tx

1

+∫

x

dt

=

−t

x

+

x

2

=

0

π

0

2

sin x

cos x

=

−

.

π

x

2

x

Пример 6. Найти функцию f (x) , если

+∞∫ f (t)sin tx dt = e−x (x > 0).

(17)

0

2

+∞

=

Im ∫et (ix−1)

dt =

π

0

2

t (ix−1)

+∞

2

1

=

e

=

=

Im

Im

π

π

ix −1

0

1

−ix

=

2

1+ix

=

2

x

Im

,

x ≥ 0 .

π

π

1+ x2

1+ x2

1

+∞∫ f (t)eitx dt = e−ax (x > 0, a > 0).

(18)

2π

−∞

0

= +∞∫e−t (a+ix) dt =

e−t (a+ix)

+∞ =

−(a +ix)

0

0

=

1

=

a −ix

.

a +ix

a2 + x2

Пример 8. Найти функцию f (x) , если

1

+∞

sin

x

,

x

≤π;

∫

itx

π

f (t)e

dt =

x

> π.

−∞

0,

Решение.

f (x) = 1

∫π sin

t

e−itx dt = ∫π sin t costx dt =

2

−π

0

=

1

cost(x −1)

−

cost(x +1)

π

=

2

x −1

x +1

0

=

1

−cos πx −1

−

−cos πx −1

=

2

x −1

x +1

=

1+cos πx 1

−

1

=

1+cos πx

.

2

1

− x

2

x +1

x −1

1,

x

<1;

1.

f (x) =

x

>1.

0,

x

h

1−

,

x

≤ a;

2.

f (x) =

a

x

> a.

0,

3.

f (x) =

1

,

a > 0 .

a2

+ x2

4.

f (x) = e−α

x

cosβx , α > 0 .

5.

f (x) = xe−x2 .

−1 ≤ x ≤ −

1

−1,

;

2

1

1

−

< x <

;

f (x) = 0,

2

2

1

≤ x ≤1.

1,

2

+∞∫ f ( y)cos xy dy =

1

,

x ≥ 0 .

1+ x

2

0

0

3.

f (x) =

1 +∞∫e−at costx dt .

a

0

α +∞

1

1

4.

f (x) =

+

costx dt .

π

(t −β)

2

+α

2

(t

+β)

2

+α

2

∫0

5.

f (x) =

1

+∞∫te−t2 /4 sin tx dt .

2

π

0

8.

F (x) =

2

sin x −sin(x / 2) ,

F (x) =

2

cos(x / 2) −cos x .

C

π

x

S

π

x

9.

f (x) =e−x ,

x ≥ 0 .