Given the possibilities of errors through several transcription, please do not rely on the fornulas given below without confirmation check or proof. Corrections and additions welcome.
Laplace transforms are related to Fourier transforms and useful in solving various linear differential equations, and useful in their own right in the study of automorphisms of various linear spaces and algebras.
A Table of Some Laplace Transforms and Relationships
By definition, when it exists, the Laplace transform is
usually defined as a Riemann integral:
oo
f(s) := INT exp( -st ) F(t) dt := L{F(t)}
0
L{} is a obviously a linear transform.
NB: L{.} maps Real valued functions to Real valued functions
within the same linear space of functions, defined on the
positive t-axis.
Columnarly, the table follows, with LHS being the Laplace
transform of the RHS:
f(s) := L{F(t)} F(t)
------------------------------------------------------------
Derivatives of F(t):
s f(s) - F(0) F'(t)
s² f(s) - s F(0) - F'(0) F''(t)
s^3 f(s)
- s² F(0) - s F'(t) - F(0) F'''(t)
. . .
and generally, for any n = 0, 1, 2, ...
s^n f(s)
n-1
- Sum s^(n-1-k) F^(n)(0) F^(n)(t)
k=0
--------------------------------------------------------------
t
f(s)/s INT F(x) dx
0
oo
INT f(r) dr F(t)/t
s=0
(d/ds)^k f(s) (-t)^k F(t)
Convolution Theorem
t
f(s) g(s) INT F(t-x) G(x) dx
0
INT f(s-x) g(s) F(t) G(t) IS THERE A FORMULA?
Translation Theorems
f(s-a) exp(at) F(t)
exp( -as ) f(s) F(t-a) NOT TRUE, instead:
exp( -as ) (f(s) + S(s,c)) F(t-a)
where
c
S(s,c) := INT exp(sx) F(-x) dx (see proof below).
0
OR
exp( -as ) f(s) F(t-a) H(t-a)
H(): the Heaviside function = 0 for x < 0, = 1, x > 0
f(as + b) (1/a) exp( -(b/a)t ) F(t/a)
Powers
For s > 0,
s^-(n+1) t^n / n!
For s > k
n! / (s - k)^(n+1) t^n exp( kt )
Periodic functions of period T
T
INT exp(-st) F(t) dt
0
--------------------- F(t + T) = + F(t)
(1 - exp(-sT))
T
INT exp(-st) F(t) dt
0
--------------------- F(t + T) = - F(t)
(1 + exp(-sT))
------------------------------------------------------------
More Specifically:
a/s a
n!/s^(n+1) t^n
1/s^x, x > 0 t^(x-1) / Gamma(x)
1/(s + a) exp(-at)
1/(s + a)² t exp(-at)
exp(-at) / t is Divergent
s/(s + a)² (1 - at) exp(-at)
1/(s + a)^n, n > 0 exp(-at) t^(n-1)/(n-1)!
a / s(s+a) 1 - exp( -at )
exp(-at) - exp(-bt)
1 / (s+a)(s+b), a ≠ b --------------------
(b-a)
exp(-at) exp(-bt)
1 / s(s+a)(s+b), a ≠ b 1/(ab) + -------- + --------
a(a-b) b(b-a)
a exp(-at) - b exp(-bt)
s / (s+a)(s+b), a ≠ b ------------------------
(a-b)
Trigonometric Functions
a /(s²+a²) sin(at)
s sin w + a cos w
----------------- sin(at + w)
(s²+a²)
a cos ac - s sin ac
------------------- sin(a(t-c))
(s²+a²)
s / (s²+a²) cos(at)
a cos w + s sin w
------------------- cos(at - w))
(s²+a²)
s cos ac + a sin ac
------------------- cos(a(t-c))
(s²+a²)
(s²-a²)
-------- t cos(at)
(s²+a²)²
1 / s(s²+a²) (1 - cos(at))/a²
sin(at) - at cos(at)
1 / (s²+a²)² --------------------
2 a^3
t sin(at)
s / (s²+a²)² ---------
2 a
(1/a)sin(at) + (1/b) sin(bt)
1 / (s²+a²)(s²+b²) ----------------------------
(b² - a²)
a / s²(s+a) t - (1 - exp(-at))/a
b
------------- exp(-at) sin(bt)
(s + a)² + b²
b cos w - (s+a) sin w
--------------------- exp(-at) sin(bt-w)
(s + a)² + b²
s + a
------------- exp(-at) cos(bt)
(s + a)² + b²
b sin w + (s+a) cos w
--------------------- exp(-at) cos(bt-w)
(s + a)² + b²
s + c
------------- exp(-at) [cos(bt) + (c-a)/b sin(bt)]
(s + a)² + b²
sqrt(a²+b²) sin(bt + w)
(s+a) / (s²+b²) ------------------------
b
with w = arctan(b/a)
Hyperbolic Functions
a /(s²-a²) sinh(at)
(1 - exp(2ac)) s + (1 + exp(2ac)) a
----------------------------------- sinh(a(t-c))
2 exp(ac) (s² - a²)
s /(s²-a²) cosh(at)
(1 - exp(2ac)) a + (1 + exp(2ac)) s
----------------------------------- cosh(a(t-c))
2 exp(ac) (s² - a²)
Miscellaneous and Mixed Functions
arctan(a/s) (1/t) sin(at)
1/sqrt(s) 1/sqrt(πt)
1/sqrt(s+a) exp(-at)/sqrt(πt)
1/s^(3/2) 2 sqrt(t/π)
1/sqrt(s²+a²) J_0(at) Bessel function
1/(s²+a²)^(3/2) (t/a) J_1(at) Bessel function
1/sqrt(s²-a²) I_0(at) modified Bessel function
1/(s²-a²)^(3/2) (t/a) I_1(at) modified Bessel function
(1/s)^(n+1) exp( -a/s) (t/a)^(n/2) J_n( 2 sqrt(at) )
1 - s/sqrt(s²+a²) a J_1(at)
exp(bt) - exp(at)
sqrt(s-a) - sqrt(s-b) ------------------
2t sqrt( πt )
exp(-as) δ(t-a), Dirac delta function
exp(-as)/s H_a(t), Heaviside function
For a > 0
(1/s) exp( -a sqrt(s)) erfc( a / (2 sqrt(t)) )
1 / s sqrt(s+1) erf( sqrt(t) )
ln( 1 + 1/s ) ( 1 - exp(-t) ) / t
ln (s+k)/(s-k) (2/t) sinh(kt)
oo 2 x
INT -------- dx (2/t) cosh(kt),
s x² - k² k > 0, s-k < 0, s+k > 0
ln( 1 - s²/a²) (2/t) (1 - cosh(at))
ln( 1 + s²/a²) (2/t) (1 - cos(at))
-------------------
where,
x
erf(x) := (2/sqrt(π)) INT exp(-y²) dy
0
erfc(x) := 1 - erf(x)
--------------------------------------------------------------------------
d(uv) = u dv + v du
oo
L{F(t) G(t)} = INT exp(-st) F(t) G(t) dt
0
oo
= INT ( exp(-st) F(t) dt ) G(t)
0
oo oo
= | exp(-st) F(t) G(t) - INT exp(-st) F(t) G'(t) dt
0=t 0
oo
= F(0) G(0) - INT exp(-st) F(t) G'(t) dt
0
-------------------------------------------------------------------------
Derivation for L{F(t-c)}:
oo
f(s) := INT exp( -st ) F(t) dt := L{F(t)}
0
oo
INT exp( -st ) F(t-c) dt := L{F(t-c)}
0
Let x := t - c, with c < oo
t=0 => x=-c
t=oo => x=oo
oo
INT exp( -s(x+c) ) F(x) dx := L{F(t-c)}
-c
oo
exp( -sc ) INT exp( -sx ) F(x) dx := L{F(t-c)}
-c
0 oo
exp( -sc ) (INT + INT) exp( -sx ) F(x) dx = L{F(t-c)}
-c 0
0
exp( -sc ) (INT exp( -sx ) F(x) dx + L{F(t}} = L{F(t-c)}
-c
0
L{F(t-c)} = exp( -sc ) L{F(t)} - exp( -sc ) INT exp( sx ) F(-x) dx
c
c
L{F(t-c)} = exp( -sc ) L{F(t)} + exp( -sc ) INT exp( sx ) F(-x) dx
0
c
L{F(t-c)} = exp( -sc ) ( f(s) + INT exp( sx ) F(-x) dx)
0
L{F(t-c)} = exp( -sc ) ( f(s) + S(s,c) )
--------------------------------------------------------------------------
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