table of laplace transforms
1
f t
( )
fs
( )
tn
n=1,2,...
s>0
†
n!
sn+1
†
1
s2
†
t
†
ta
†
a> -1
†
Ga+1
( )
sa+1
s>0
†
eat
†
te at
†
tneat
†
1
s-a
†
1
s-a
( )
2
†
n!
s-a
( )
n+1
†
s>a
†
s>a
†
s>a
†
t≥0
†
sinat
†
cosat
†
tcos at
†
tsin at
†
a
s2+a2
s>0
s>0
†
s
s2+a2
†
2as
s2+a2
( )
2
s>0
s>0
†
s2-a2
s2+a2
( )
2
†
eat sinbt
†
eat cos bt
†
b
s-a
( )
2+b2
†
s-a
s-a
( )
2+b2
†
s>a
†
s>a
†
sinhat
†
coshat
†
tcosh at
†
tsinh at
†
eat sinhbt
†
eat coshbt
†
a
s2-a2
†
s
s2-a2
†
s>a
†
s>a
†
s>a
†
s>a
†
2bs
s2-a2
( )
2
†
s2+b2
s2-a2
( )
2
†
b
s-a
( )
2-b2
†
s-a
s-a
( )
2-b2
†
s>a+b
†
s>a+b
maple
†
u t -a
( )
=0 t <a
1 t ≥a
Ï
Ì
Ó
†
a>0
s>0
†
e-as
s
f t
( )
†
t≥0
fs
( )
†
dt
( )
†
1
†
dt-a
( )
†
e-as
†
a≥0
†
s>0
s>0
s>0
†
J0at
( )
†
J0a t
( )
†
tpJpat
( )
†
p> - 1
2
†
s>0
†
s>0
†
s>0
†
1
s2+a2
†
2papGp+1
2
Ê
Ë
Á ˆ
¯
˜
ps2+a2
( )
p+1
2
†
e-a2
4s
s
†
p
Gk
( )
t
2a
Ê
Ë
Á ˆ
¯
˜
k-1
2
Jk-1
2
at
( )
†
p
Gk
( )
at
2a
Ê
Ë
Á ˆ
¯
˜
k-1
2
Jk-3
2
at
( )
†
k>0
†
k>1
2
†
1
s2+a2
( )
k
†
s
s2+a2
( )
k
†
s>0
†
s>0
†
erf at
( )
†
erf a t
( )
†
erfc a
2 t
Ê
Ë
Á
Á
ˆ
¯
˜
˜
†
a>0
†
a≥0
†
a≥0
†
a>0
†
e-a2t2
†
1
se
s2
4a2erfc s
2a
Ê
Ë
Á ˆ
¯
˜
†
s>0
†
a
s s+a2
†
s>0
†
1
se-a s
†
s>0
†
p
2a e
s2
4a2erfc s
2a
Ê
Ë
Á ˆ
¯
˜
†
s>0
Laplace transform is calculated
with the command laplace (f(t),t,s):
f(t) denotes the function to be transformed,
t is the independent variable of the function,
s is the variable of the transformed function
For calcualtaion of Laplace transform
or inverse Laplace transform
the package with integral transforms
has to be downloaded:
> with(inttrans);
†
[fourier,laplace,invlaplace,...]
Example 1:
> laplace(t^2,t,s);
†
2
s3
> f(t):=t^2*sin(5*t);
Example 2:
> laplace(f(t),t,s);
†
f(t) :=t2sin(5t)
†
1-at
( )
e-at
†
s
s+a
( )
2
†
Jnat
( )
†
s2+a2-s
Ê
Ë
Á ˆ
¯
˜
n
ans2+a2
†
n=0,1,2,..
†
10 3s2-25
( )
s2+25
( )
3
Inverse Laplace transform is calculated
with the command invlaplace ( (s),s,t):
(s) denotes the function to be transformed,
s is the independent variable of the function,
t is the variable of the transformed function
†
f
†
f
> phi(s):=exp(-4*s)/s;
Example 3:
> laplace( (s),s,t);
†
f
†
fs
( )
:=e-4s
( )
s
†
Heaviside(t -4)
> phi(s):=exp(-3*sqrt(s));
Example 4:
> laplace( (s),s,t);
†
f
†
fs
( )
:=e-3 s
( )
†
3e -9
4t
Ê
Ë
Á ˆ
¯
˜
2pt3 2
( )
†
1
s