Tables of Integral Transformations - Heat Conduction: Mathematical Models and Analytical Solutions - page 473

Appendix C
Tables of Integral Transformations

Tabl e C. 1 Fourier transformations

Inverse Image Functions Image Functions

+ ∞ + ∞

1
2

f ( ω ) e iω x dω f ( ω )=

f ( x ) e − iω x d x f ( x )=

π

− ∞ − ∞

π , | ω | < α
0

sin α x

, α > 0

, | ω | > α

x

e i kx

e i ( k − ω ) a

e i ( k − ω ) b

i

a < x < b

−

0 , x < a or x > b k − ω

e − cx + i kx

, x > 0 i

, c > 0

0 , x < 0

k + ω + i c

2
4η

e − ω

π
η

e − η x 2 , η > 0

cos ω

π
η

2

η − 4

x 2 cos η , η > 0

4

cos ω

π
η

2
4η + 4

x 2

sin η , η > 0

2
| ω |

sin 1 | − s | x , 0 < Re ( s ) < 1 Γ ( 1 − s )

2 π

s

1 − s

2 √ α

1
| x |

2 π

e − α | x | ,

2 2

α > 0 + ω + α

2

+ ω

α