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Санкт-Петербургский политехнический университет Петра Великого (бывш. СПбГПУ)

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Таблица функции h(t0)					(продолжение)

t						x
t	0.0	0.05	0.10	0.15	0.20		0.25	0.30	0.35	0.40	0.45
	0.0	0.05	0.10	0.15	0.20		0.25	0.30	0.35	0.40	0.45
14.0	0.952	0.989	1.011	1.016	1.009		0.997	0.988	0.983	0.985	0.991
14.5	0.954	0.990	1.012	1.015	1.008		0.996	0.987	0.985	0.988	0.996

15.0	0.956	0.993	1.012	1.014	1.007		0.995	0.988	0.987	0.991	1.000

15.5	0.959	0.995	1.014	1.014	1.006		0.995	0.989	0.988	0.996	1.004

16.0	0.961	0.997	1.015	1.014	1.006		0.995	0.991	0.992	0.998	1.007
16.5	0.964	0.999	1.016	1.014	1.005		0.995	0.993	0.995	1.002	1.009

17.0	0.965	1.001	1.016	1.013	1.005		0.955	0.994	0.997	1.005	1.010

17.5	0.966	1.002	1.015	1.012	1.003		0.995	0.994	0.998	1.006	1.010
18.0.	0.966	1.002	1.015	1.011	1.002		0.995	0.995	1.001	1.008	1.010
18.5	0.966	1.001	1.015	1.009	1.001		0.994	0.995	1.001	1.007	1.009

19.0	0.967	1.000	1.015	1.008	0.998		0.992	0.995	1.001	1.006	1.006

19.5	0.967	1.000	1.014	1.006	0.996		0.991	0.995	1.001	1.005	1.004
20.0	0.967	1.000	1.013	1.005	0.995		0.991	0.995	1.001	1.005	1.002
20.5	0.968	1.002	1.012	1.004	0.994		0.991	0.996	1.002	1.004	1.001

21.0	0.968	1.002	1.011	1.003	0.994		0.992	0.997	1.003	1.004	1.001
21.5	0.969	1.002	1.011.	1.003	0.995		0.992	0.999	1.004	1.004	1.001
22.0	0.971	1.002	1.011	1.002	0.995		0.993	1.000	1.005	1.004	1.000

22.5	0.973	1.002	1.011	1.002	0.996		0.995	1.002	1.006	1.004	0.999

23.0	0.974	1.005	1.011	1.002	0.996		0.997	1.004	1.007	1.003.	0.999
23.5	0.975	1.005	1.010	1.002	0.996		0.998	1.004	1.008	1.003	0.998
24.0	0.975	1.005	1.010	1.001	0.996		0.999	1.005	1.007	1.002	0.997

24.5	0.975	1.005	1.009	1.000	0.996		0.999	1.005	1.006	1.001	0.997

25.0	0.975	1.005	1.008	1.000	0.995		0.999	1.005	1.004	1.000	0.996
25.5	0.975	1.005	1.008	0.999	0.995		0.999	1.004	1.003	0.998	0.996
26.0	0.975	1.005	1.007	0.999	0.995		0.999	1.004	1.002	0.997	0.996

Таблица функции h(t0)					(продолжение)

t						x
t	0.50	0.55	0.60	0.65	0.70	0.75	0.80	0.85	0.90	0.95	1.00
	0.50	0.55	0.60	0.65	0.70	0.75	0.80	0.85	0.90	0.95	1.00
14.0	1.000	1.006	1008	0.999	0.987	0.974	0.965	0.961	0.965	0.976	0.990
14.5	1.002	1.006	1.005	0.994	0.983	0.970	0.969	0.971	0.981	0.997	1.010

15.0	1.005	1.007	1.002	0.993	0.983	0.976	0.978	0.987	1.001	1.017	1.030

15.5	1.008	1.007	1.001	0.993	0.985	0.984	0.991	1.003	1.019	1.032	1.040

16.0	1.011	1.008	1.001	0.994	0.990	0.993	1.003	1.018	1.031	1.039	1.039
16.5	1.011	1.008	1.001	0.996	0.995	1.001	1.014	1.027	1.036	1.038	1.028

17.0	1.012	1.007	0.999	0.997	0.999	1.008	1.020	1.030	1.032	1.027	1.012

17.5	1.009	1.005	0.997	0.998	1.002	1.012	1.023	1.027	1.023	1.013	0.988
18.0.	1.008	1.002	0.997	0.998	1.004	1.014	1.020	1.018	1.008	0.993	0.979
18.5	1.006	0.999	0.995	0.998	1.003	1.012	1.014	1.007	0.993	0.978	0.969

19.0	1.001	0.995	0.993	0.997	1.004	1.009	1.006	0.995	0.981	0.969	0.967

19.5	0.998	0.992	0.992	0.996	1.003	1.005	0.998	0.985	0.973	0.967	0.973
20.0	0.996	0.991	0.992	0.995	1.003	1.001	0.991	0.979	0.972	0.974	0.985
20.5	0.995	0.991	0.994	0.996	1.001	0.996	0.986	0.976	0.974	0.990	1.001

21.0	0.995	0.993	0.997	0.996	0.999	0.993	0.983	0.975	0.981	1.002	1.016
21.5	0.996	0.995	1.000	0.995	0.998	0.992	0.986	0.988	0.997	1.013	1.024
22.0	0.996	0.996	1.000	0.997	0.997	0.991	0.991	0.997	1.012	1.024	1.029

22.5	0.997	1.000	1.004	1.000	0.996	0.992	0.998	1.008	1.022	1.028	1.026

23.0	0.998	1.001	1.006	1.001	0.997	0.994	1.002	1.015	1.025	1.027	1.016
23.5	0.999	1.002	1.007	1.002	0.998	0.997	1.007	1.017	1.023	1.023	1.002
24.0	1.000	1.002	1.008	1.003	0.999	1.000	1.008	1.017	1.015	1.012	0.988

24.5	1.000	1.002	1.006	1.003	1.000	1.002	1.008	1.014	1.005	0.995	0.979

25.0	1.000	1.002	1.004	1.003	1.001	1.003	1.005	1.008	0.991	0.985	0.975
25.5	1.000	1.002	1.002	1.002	1.002	1.004	1.004	1.001	0.986	0.978	0.977
26.0	1.000	1.002	1.000	1.001	1.002	1.004	1.002	0.987	0.984	0.977	0.983

Таблица преобразования Лапласа для рациональных изображений

F (s) = D1 (s) D(s)

№

F(s)

F(t) (t 0)

1.1

1/s

1.2

eat

s − a

1.3

A=1/a , K=-1/a

s(s − a)

Aeat+K

1.4

s + d

A=(1+d/a) , K=-d/a

s(s − a)

1.5

A=1/(a-b) ,

B=1/(b-a)

(s − a)(s − b)

Aeat+Bebt

1.6

s + d

A =

a + d

;

B =

b + d

;

(s − a)(s − b)

a − b

b − a

1.7

A =

;

B =

; K =

;

s(s − a)(s − b)

a(a − b)

b(b − a)

1.8

s + d

Aeat+Bebt+K

A =

a + d

;

B =

b + d

; K =

;

s(s − a)(s − b)

a(a − b)

b(b − a)

s 2 + gs + d

A =

a 2

+ ga + d

B =

b2 + gb + d

K =

1.9

;

s(s − a)(s − b)

a(a −b)

b(b − a)

A =

;

1.10

(a − b)(a − c)

− a)(s − b)(s − c)

B =

; C =

;

(b − a)(b − c)

(c − a)(c − d )

A =

a + d

;

1.11

s + d

Aeat+Bebt+Cect

(a − b)(a − c)

− a)(s − b)(s − c)

B =

b + d

; C =

c + d

;

(b − a)(b − c)

(c − a)(c − d )

A =

a 2

+ ag + d

;

1.12

s 2 + gs + d

(a − b)(a − c)

− a)(s − b)(s − c)

B =

b 2 + bg + d

; C =

c2 cg + d

;

(b − a)(b − c)

(c − a)(c

− d )

2.1

A =

;

α = 0;

(s − a) 2 +ω2

Aeatsin( ωt +α )

2.2

s + d

A =

[(a + d )2

+ω2 ]1/ 2

.α = arctg

(s − a) 2 +ω2

a + b

2.3

A =

. K

; α = −arctg

(s − a)2

+ω2

(a 2

+ω2 )1/ 2

a 2

+ω2

(a + b) 2 + ω2

1 / 2

s + d

Ae sin( ωt +α )+K

A =

[

] . K =

;

2.4

+ω

(s − a)2

+ω2

α = arctg

− arctg ω

a + d

(продолжение)

№

F(s)

F(t) (t 0)

(a 2

− ω 2 + ag + d ) 2 + ω 2 (2a + g) 2

1/ 2

A =

[

]

s 2 + gs + d

a 2 + ω 2

2.5

Aeatsin( ωt + α )+K

. K =

;

(s − a) 2 + ω

2 + ω 2

α = arctg

ω(2a + g)

− arctg ω

a 2 − ω 2

+ ag + d

A =

ω [(a − b)2 + ω 2 ]1 / 2

2.6

(s − b)

(s − a)2

+ ω 2

. B =

α = −arctg

;

(a − b) 2

+ ω 2

a − b

s + d

A =

[

(a + d ) 2 + ω 2

]1/ 2

(a − b) 2

+ ω 2

2.7

(s − b)

(s − a) 2

+ ω 2

Aeatsin( ωt + α )+Bebt

. B =

b + d

; α = arctg

− arctg

(a − b) 2

+ ω 2

a + d

a − b

(a 2

− ω 2 + ag + d ) 2 + ω 2 (2a + g 2 )

1/ 2

A =

[

]

s 2 + gs + d

(a − b) 2 + ω 2

. B =

b2 + bg + d

2.8

(s − b)

(s − a) 2

+ ω 2

;

(a − b) 2

+ ω 2

α = arctg

− arctg

a 2

−

ω 2 + ag + d

a − b

A =

(a 2 + ω 2 )1 / 2 [(a − b)2 + ω 2 ]1/ 2

. B =

K = −

2.9

s(s − b)

(s − a) 2

+ ω 2

;

b[( b

− a )2 + ω2 ]

b(a 2

+ ω 2 )

α = −arctg

− arctg ω

− b

(d + a)2 + ω 2

1/ 2

A =

[

]

s + d

ω(a 2 + ω 2 )1 / 2

(a − b)2 + ω 2

Aeatsin( ω t + α )+

b + d

s(s − b)

(s − a) 2

+ ω 2

2.10

+Bebt+K

. B = b[( b − a )2 + ω2 ] ;

K = − b(a 2

+ ω 2 )

;

α = arctg

− arctg

− arctg ω

+ d

a − b

A =

1 ( a2

− ω2 + ag + d )2 + ω2 ( 2a + g )2

1 / 2

{

}

s 2 + gs + d

( a2 + ω2 )[( a − b )2 + ω2 ]

B =

+ bg + d

K = −

2.11

s(s − b)

(s − a) 2

+ ω 2

;

b[( b − a )2 + ω2 ]

b(a 2 + ω 2 )

α = arctg

ω( 2a + g )

− arctg

− arctg ω

a2 − ω2 + ag + d

a − b

(продолжение)

A =

ω (a 2 + ω 2 )1 / 2 [(a − b)2 + ω 2 ]1/ 2

2.12

. B =

[(s-a 2 )+щ12 ](s2+щ22 )

ω 2

[(a2 + ω 2

− ω 2 )2

+ 4a2ω

2 ]1/ 2

2aω1

2aω2

α = −arctg

β = arctg

− ω

2 + ω 2

+ ω 2

− ω

Aeatsin( ω

t + α )+

(a + d )2ω 2

1/ 2

A =

[

]

+Bsin( ω2t + β )

ω1

(a2 + ω12 − ω22 )2 + 4a2ω22

B =

d 2 + ω22

1/ 2

s + d

[

]

2.13

ω 2

(a2 + ω12 − ω22 )2 + 4a2ω22

[( s − a2 ) + ω12 ]( s2 + ω22 )

α = arctg

− arctg

2aω

a2 −

ω 2 + ω 2

+ d

β = arctg ω2

+ arctg

2aω2

a2 + ω 2 − ω 2

A =

1 (a2 − ω 2 + ag + d )2 + ω

(2a + g)2

1/ 2

[

]

ω1

(a2 + ω12 − ω22 )2 + 4a2ω22

B =

(d − ω22 )2 g 2ω22

1/ 2

s2 + gs + d

Aeatsin( ω 1t + α )+

[

]

ω 2

(a2 + ω 2

− ω

2 )2

+ 4a2ω 2

2.14

+Bsin( ω2t + β )

[(s − a2 ) +ω12 ](s2 +ω22 )

α = arctg

ω (2a + g)

− arctg

2aω

a2 − ω 2

+ ag + d

− ω 2 + ω

β = arctg

gω2

+ arctg

2aω2

d − ω 2

a2 + ω 2

− ω 2

3.1

3.2

A = 0

A1 = 1

(s −a)2

( A + A t)eat

s + d

A = 1

A1 = a + d

3.3

(s −a)2

3.4

A =

K = − A

K1 = −

s2 (s − a)2

3.5

s + d

Aeat + K + K t

A =

a + d

K = − A

= −

2 2

s (s − a)

3.6

s2 + gs + d

a2 ag + d

s2 (s − a)2

A =

K = 1− A K1 = − a

3.7

A = −

K = − A

A1 =

s(s −a)2

3.8

s + d

( A + A t)eat + K

A = −

K = − A

A1 =

a + d

s(s −a)

3.9

s2 + gs + d

A =

a2 − d

K = 1− A A1 =

a2 + ag + d

s(s −a)2

(продолжение)

№

F(s)

F(t)

(t 0)

3.10

A = −

A 1 = −

B = − A

(s −a)2 (s −b)

− b)2

a − b

3.11

s + d

A = −

b − d

B = − A A =

a + d

(s −a)2 (s −b)

(a − b)2

a − b

( A + A t)eat + Bebt

a2 − 2ab − bg − d

a2 + ag + d

A =

s2 + gs + d

(a − b)2

a − b

3.12

(s −a) (s −b)

B =

+ bg + d

(a − b)2

A = −

B =

3.13

a2 (a − b)

b2 (b − a)

s2 (s − a)2 (s −b)

K = −

a + b

a2b2

s + d

A = −

a + d

B =

b + d

3.14

Aeat + Bebt K + K1t

a2 (a − b)

b2 (b − a)

ab + d (a + b)

(s − a) (s −b)

K = −

a2b2

s2 + gs + d

A = −

a2 + ag + d

B =

b2 + bg + d

3.15

a2 (a − b)

b2 (b − a)

s2 (s − a)2 (s −b)

K = −

abg + d (a

+ b)

a2b2

A = −

b − 2a

B =

3.16

a2 (a − b)2

b2 (a − b)2

s(s − a)2 (s −b)

K = −

a2b

a(a − b)

s + d

A = −

a2 + 2ad − bd

B =

b + d

b(a − b)2

3.17

( A + A1t)e

+ Be

+ K

a2 (a − b)2

s(s − a)2 (s −b)

K = −

a + d

a2b

a(a − b)

A = −

a2 (b + g) + d (2a − b)

B =

b2 + bg + d

s2 + gs + d

3.18

a2 (a − b)2

b(a − b)2

s(s − a) (s −b)

K = −

a + ag + d

a2b

a(a − b)

A =

(b + c) − 2a

B =

3.19

(b − a)2 (a − c)

− a)2 (b − c)

(s −a)2 (s −b)(s −c)

C =

A 1 =

(c − a)2 (c

− b)

− b)(a − c)

( A + A t)eat + Bebt + Cect

A = −

a2 + 2ad − d (b + c) − bc

(a − b)2 (a − c)2

3.20

s + d

A1 =

a + d

(s −a)2 (s −b)(s −c)

(a − b)(a − c)

B =

b + d

C =

c + d

(b − a)2 (b

− c)

(c − a)2 (c − b)

(продолжение)

№

F(s)

F(t)

(t 0)

3.21

A = −

A 1 = −

K = − A K1 =

s2 (s − a)2

s + d

A = −

a + 2d

K = − A

3.22

A1 =

a + d

s2 (s − a)2

( A + A t)eat + K + K t

+ gs + d

A = −

ag + 2d

A =

a2 + ag + d

3.23

s2 (s − a)2

K = − A K =

A = −

B = − A

3.24

(a − b)3

(s −a)2 (s −b)2

A1 =

− b)2

s + d

A = −

a + b + 2d

B = − A

3.25

( A + A1t)eat

+ (B + B1t)ebt

(a − b)3

+ d

b + d

(s −a) (s

−b)

A =

(a − b)2

− b)2

A = −

(a + b)g + 2(ab + d )

B = − A

s2 + gs + d

3.26

(a − b)3

a2 + ag + d

b2 + bg + d

(s −a)2 (s −b)2

A =

(a − b)2

A =

B = − A

3.27

(a − b)3

(s −a)3 (s −b)

A 1 = −

( A + A t + A t2 )eat + Bebt

2(a

− b)

(a − b)2

b + d

A =

B = − A

3.28

s + d

(a − b)3

(s −a)3 (s −b)

A =

a + d

= −

b + d

2(a

− b)

(a − b)2

+ gs + d

A =

ag + 3d

A =

a2 + ag + d

3.29

( A + A t + A t 2 )eat + K + K t

s2 (s −a)3

1 2

K = − A K = −

= −

ag + 2d

(продолжение)

№

F(s)

F(t)

(t 0)

A =

4.1

ω (a2 + ω 2 )

(a2 + ω 2 )2

s2 ((s −a)2 +ω2 )

K =

α = −2arctg ω

+ ω 2

s + d

A =

((a + d )2 + ω 2 )1/ 2

K =

a2 + ω 2 + 2ad

(a2 + ω 2 )2

4.2

ω (a2 + ω 2 )

s2 ((s −a)2 +ω2 )

Aeat sin(ωt + α ) +

α = arctg

− 2arctg ω

a2 + ω 2

a + d

+K + K t

A =

((a2 − ω 2

+ ag + d )2

+ ω 2 (2a + g)2 )1/ 2

ω (a2 + ω 2 )

s2 + gs + d

K =

(a2 + ω 2 )g + 2ad

4.3

(a2 + ω 2 )2

s2 ((s −a)2 +ω2 )

K1 =

a2 + ω 2

α = arctg

ω (2a + g)

− 2arctg ω

a2 − ω 2 + ag + d

A =

2(a − b)

4.4

ω ((a − b)2 + ω 2 )

((a − b)2 + ω 2 )2

(s −b)2 ((s −a)2 +ω2 )

α = −2arctg

− b)

2 + ω

a − b

A =

((a + d )2 + ω 2 )1/ 2

ω ((a − b)2 + ω 2 )

s + d

B = (a − b)2 + ω 2 + 2(a − b)(b + d )

4.5

((a − b)2 + ω 2 )2

(s −b)2 ((s −a)2 +ω2 )

B1 =

b + d

Aeat sin(ωt + α ) +

(a − b)2 + ω 2

+(B + B1t)ebt

α = arctg

− 2arctg

a + d

− b

A =

((a2 − ω 2 + ag + d )2 + ω 2 (2a + g)2 )1/ 2

ω ((a − b)2 + ω 2 )

s2 + gs + d

B =

((a − b)2 + ω 2 )(2b + g) + 2(a − b)(b2 + bg + d )

4.6

((a − b)2 + ω 2 )2

(s −b)2 ((s −a)2 +ω2 )

B =

b 2 + bg + d

(a − b)2 + ω 2

= arctg

ω (2a + g)

− 2arctg

a − b

a2 − ω 2 + ag + d

eat

(sinωt −

4.7

2ω3

((s −a)2 +ω2 )2

−ωt cosωt)

4.8

s −a

teat sinωt

((s −a)2 +ω2 )2

2ω

(s −a)2 −ω2

4.9

((s −a)2 +ω2 )2

te cosωt

Таблица преобразований Лапласа

(а, b, с – различные переменные)

F(s)

F(t) (t>0)

1/ s

1/ s 2

t n−1

1/ s n (n = 1,2,3...)

(n -1)!

πt

s −3 / 2

t / π

s −( n+1/ 2) (n = 1,2,3...)

2n t n−1 / 2

1× 3 × 5...(2n -1)

Г(k )

(k > 0)

t k −1

s k

(s - a)

teat

(s - a) 2

(n = 1,2,3...)

t n−1eat

(s - a) n

Г(k)

(n -

1)!

(k > 0)

t k −1eat

(s - a)k

(s - a)(s - b)

(eat - ebt )

(a - b)

(s - a)(s - b)

(eat - bebt )

(a - b)

(b - c)eat + (c - a)ebt + (a - b)ect

(s - a)(s - b)(s - c)

(a - b)(b - c)(c - a)

sin at

s 2

+ a 2

cos at

s 2

+ a 2

shat

s 2 - a 2

chat

s 2 - a 2

(1 - cos at)

a 2

s(s 2

+ a 2 )

(1 - sin at)

a 2

s(s 2

+ a 2 )

(s 2 + a 2 )2

(sin at - at cos at)

2a 2

(продолжение)

F(s)	F(t) (t>0)

sin at

(s 2

+ a 2 )2

s 2

(sin at + at cos at)

(s 2

+ a 2 )2

s 2 - a 2

t × cos× at

(s 2

+ a 2 )2

(a 2 = b 2 )

cos at - cos bt

+ a

)(s

+ b

)

b 2 - a 2

eat

sin bt

(s - a) 2

+ b 2

s - a

eat

cos bt

(s 2 - a 2 ) 2 + b 2

3a 2

e−at - eat / 2 (cos

sin

at 3

)

+ a 3

4a3

sin at × chat - cos at × shat

s 4

+ 4a 4

sin at × shat

2a 2

s 4

+ 4a 4

(shat - sin at)

2a 2

s 4 - a 4

(chat - cos at)

2a 2

8a 3 s 2

(1 + a 2t 2 ) sin at - at cos at

(s 2

+ a 2 )3

(

s -1

(t n e−t ) =

Ln (t)

n! dt

(1 + 2at)

πt

(s - a)

3 / 2

(ebt

- eat )

πt 3

s - a - s - b

- aea2t erfc(a

+ a

t )

πt

+ aea2t erf (a

s - a 2

t )

πt

s + a 2

e -a2t ∫ eλ2 dλ

πt

(s - a 2 )

ea2t erf (a

t )

a t

s (s + a

)

∫ eλ

dλ

-a

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