A Collection of Identities and Formulas
Involving Laplace Transforms

Given the possibilities of errors through several transcriptions, please do not rely on the formulas 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.

This table is good for finding the "easy" inverse Laplace transforms. For the harder ones, try using the maxima system that runs over many LISP interpreters, including CMUCL and GNU Common LISP (gcl) and clisp, all of which have packages for any modern GNU/Linux system. The Debian distribution is probably the best for scientific and mathematical work.

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A Table of Some Laplace Transforms and Relationships

By definition, when it exists, the Laplace transform is
usually defined as a Riemann integral on a space of functions
defined on a real semiaxis [0, oo]:

oo
f(s)  :=   INT exp( -st ) F(t) dt  :=  L{F(t)}
0

L{} is 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'(0) - 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) (nth derivative of F(t)
k=0				w.r.t. t)

--------------------------------------------------------------
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
t
INT f(s-x) g(s)			F(t) G(t)  IS THERE A FORMULA?
x=0

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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