Tables of Integral Transformations - Heat Conduction: Mathematical Models and Analytical Solutions - page 475

Tabl e C. 1 continued

Inverse Image Functions Image Functions

e i bx

− π
α

e | ω − b |

x 2 ,

Re ( α ) < 0 , b is a real number

α

2

+

e α | ω − b | +

e α | ω + b |

cos bx
α

− 2

x 2 ,

Re ( α ) < 0 , b is a real number

2

+ α

− 2α i e α | ω − b | − e α | ω + b |

sin bx
α

+ x 2 ,

Re ( α ) < 0 , b is a real number

2

sin b √ x 2 a 2

0

+

, | ω | > b

π J 0 a √ b 2

2

√ x 2

, a , b > 0

− ω , | ω | < b

+ a 2

sin b √ x 2 a 2

0

−

, | ω | > b

I 0 a √ b 2

2

√ x 2 , | x | > a

π − ω , | ω | < b

− a 2

cos b √ a 2

x 2

J 0 a √ ω b 2

− , | x | < α

π

2

−

, | x | > α

0

ch b √ a 2

− x 2 π J 0 a √ ω − b 2

2

, | x | < α , | ω | > b

0 , | x | > α 0 , | ω | < b

Tabl e C. 2 Laplace transfomations

Inverse Image Functions Image Functions

β + i∞ + ∞

1

f f

e st ds e − st d t f ( t )= ( s ) ( s )= f ( t )

2 π i

β − i ∞ 0

1
s

1

n !
s n + 1

t n

, n = 1 , 2 , 3 , ···

1

e α t

s − α

1

t e α t

2

( s − α )