dictionnnaire images laplace

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dictionnaire des images A nnée 2007/2008. Fonction : f (t) Transformée : L (f )(p) Fonction : f (t) Transformée : L (f )(p) 1 1 1 p e at 1 p - a 2 t n n! p n+1 t α , α> -1 Γ(α + 1) p α+1 3 e kt sin at a (p - k) 2 + a 2 e kt cos at p - k (p - k) 2 + a 2 4 e kt sh at a (p - k) 2 - a 2 e kt ch at p - k (p - k) 2 - a 2 5 t sin at 2ap (p 2 + a 2 ) 2 t cos at p 2 - a 2 (p 2 + a 2 ) 2 6 t e αt 1 (p + α) 2 t n e αt n! (p + α) n+1 7 Log t Γ (1) - Log p p e at - e bt t Log p + b p + a 8 t n f (t) (-1) n d n dp n [L (f )] (p) tf (t) -L (f )(p) - p d dp [L (f )(p)] 9 f (t) pL (f )(p) - f (0) f ′′ (t) p 2 L (f )(p) - pf (0) - f (0) 10 f (t - α); α> 0 e αp L (f )(p) f (n) (t) p n L (f )(p) - n1 k=0 p n1k f (k) (0) 11 e kt f (t) L (f )(p - k) f (t) t p L (f )(τ ) 12 (f⋆g )(t) L (f )(p)L (g )(p) t 0 f (τ ) L (f )(p) p 13 f (kt); k> 0 1 k L (f ) p k f (t)= f (t + ω); ω> 0 1 1 - e ωp ω 0 e pt f (t)dt lim p−→∞ L (f )(p)=0 lim p−→∞ pL (f )(p)= f (0) (f⋆g )(t)= t 0 f (t - x)g (x)dx M r 1

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Page 1: Dictionnnaire Images Laplace

dictionnaire des images A nnée 2007/2008.

Fonction : f(t) Transformée : L (f)(p) Fonction : f(t) Transformée : L (f)(p)

1 11

peat 1

p − a

2 tnn!

pn+1tα, α > −1

Γ(α + 1)

pα+1

3 ekt sin ata

(p − k)2 + a2ekt cos at

p − k

(p − k)2 + a2

4 ekt sh ata

(p − k)2− a2

ekt ch atp − k

(p − k)2− a2

5 t sin at2ap

(p2 + a2)2t cos at

p2− a2

(p2 + a2)2

6 t e−αt 1

(p + α)2tn e−αt n!

(p + α)n+1

7 Log tΓ′(1) − Log p

p

e−at− e−bt

tLog

p + b

p + a

8 tnf(t) (−1)ndn

dpn[L (f)] (p) tf ′(t) −L (f)(p) − p

d

dp[L (f)(p)]

9 f ′(t) pL (f)(p) − f(0) f ′′(t) p2L (f)(p) − pf(0) − f ′(0)

10 f(t − α); α > 0 e−αpL (f)(p) f (n)(t) pnL (f)(p) −

n−1∑

k=0

pn−1−kf (k)(0)

11 ekt f(t) L (f)(p − k)f(t)

t

p

L (f)(τ) dτ

12 (f ⋆ g)(t) L (f)(p)L (g)(p)

∫ t

0

f(τ) dτL (f)(p)

p

13 f(kt); k > 01

kL (f)

(p

k

)

f(t) = f(t + ω); ω > 01

1 − eωp

∫ ω

0

e−pt f(t)dt

• limp−→∞

L (f)(p) = 0 • limp−→∞

pL (f)(p) = f(0) • (f ⋆ g)(t) =

∫ t

0

f(t − x)g(x)dx

Mr Amroun 1