Que es laplace transform

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L = LAPLACE(F) is the Laplace transform of the scalar sym F with
default independent variable t. The default return is a function
of s. If F = F(s), then LAPLACE returns a function of t: L = L(t).
By definition L(s) = int(F(t)*exp(-s*t),0,inf), where integration
occurs with respect to t.

L = LAPLACE(F,t) makes L a function of t instead of the default s:LAPLACE(F,t) L(t) = int(F(x)*exp(-t*x),0,inf).

L = LAPLACE(F,w,z) makes L a function of z instead of the
default s (integration with respect to w).
LAPLACE(F,w,z) L(z) = int(F(w)*exp(-z*w),0,inf).

Examples:
syms a s t w x
laplace(t^5) returns 120/s^6
laplace(exp(a*s)) returns 1/(t-a)
laplace(sin(w*x),t) returns w/(t^2+w^2)laplace(cos(x*w),w,t) returns t/(t^2+x^2)
laplace(x^sym(3/2),t) returns 3/4*pi^(1/2)/t^(5/2)
laplace(diff(sym('F(t)'))) returns laplace(F(t),t,s)*s-F(0)

See also sym/ilaplace, sym/fourier, sym/ztrans.

Reference page in Help browser
doc sym/laplace

>> help ilaplace
--- help for sym/ilaplace ---

ILAPLACE Inverse Laplace transform.F = ILAPLACE(L) is the inverse Laplace transform of the scalar sym L
with default independent variable s. The default return is a
function of t. If L = L(t), then ILAPLACE returns a function of x:
F = F(x).
By definition, F(t) = int(L(s)*exp(s*t),s,c-i*inf,c+i*inf)
where c is a real number selected so that all singularities
of L(s) are to the left of the lines = c, i = sqrt(-1), and
the integration is taken with respect to s.

F = ILAPLACE(L,y) makes F a function of y instead of the default t:
ILAPLACE(L,y) F(y) = int(L(y)*exp(s*y),s,c-i*inf,c+i*inf).
Here y is a scalar sym.

F = ILAPLACE(L,y,x) makes F a function of x instead of the default t:
ILAPLACE(L,y,x) F(y) = int(L(y)*exp(x*y),y,c-i*inf,c+i*inf),integration is taken with respect to y.

Examples:
syms s t w x y
ilaplace(1/(s-1)) returns exp(t)
ilaplace(1/(t^2+1)) returns sin(x)
ilaplace(t^(-sym(5/2)),x) returns 4/3/pi^(1/2)*x^(3/2)
ilaplace(y/(y^2 + w^2),y,x) returns cos(w*x)
ilaplace(sym('laplace(F(x),x,s)'),s,x) returns F(x)

See alsosym/laplace, sym/ifourier, sym/iztrans.

Reference page in Help browser
doc sym/ilaplace

>> help residue
RESIDUE Partial-fraction expansion (residues).
[R,P,K] = RESIDUE(B,A) finds the residues, poles and direct term of
a partial fraction expansion of the ratio of two polynomials B(s)/A(s).
If there are no multiple roots,
B(s) R(1) R(2)R(n)
---- = -------- + -------- + ... + -------- + K(s)
A(s) s - P(1) s - P(2) s - P(n)
Vectors B and A specify the coefficients of the numerator and
denominator polynomials in descending powers of s. The residues
are returned in the column vector R, the pole locations in column
vector P, and the direct terms in row vector K. The numberof
poles is n = length(A)-1 = length(R) = length(P). The direct term
coefficient vector is empty if length(B) < length(A), otherwise
length(K) = length(B)-length(A)+1.

If P(j) = ... = P(j+m-1) is a pole of multplicity m, then the
expansion includes terms of the form
R(j) R(j+1) R(j+m-1)
-------- + ------------ +... + ------------
s - P(j) (s - P(j))^2 (s - P(j))^m

[B,A] = RESIDUE(R,P,K), with 3 input arguments and 2 output arguments,
converts the partial fraction expansion back to the polynomials with
coefficients in B and A.

Warning: Numerically, the partial fraction expansion of a ratio of
polynomials represents an ill-posed problem. If the...
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