laplace

Compute the Laplace transform of 1/sqrt(x). By default, the transform is in terms of s.

syms x y
f = 1/sqrt(x);
F = laplace(f)

F = 
$$\frac{\sqrt{\pi }}{\sqrt{s}}$$

Compute the Laplace transform of exp(-a*t). By default, the independent variable is t, and the transformation variable is s.

syms a t y
f = exp(-a*t);
F = laplace(f)

F = 
$$\frac{1}{a+s}$$

Specify the transformation variable as y. If you specify only one variable, that variable is the transformation variable. The independent variable is still t.

F = laplace(f,y)

F = 
$$\frac{1}{a+y}$$

Specify both the independent and transformation variables as a and y in the second and third arguments, respectively.

F = laplace(f,a,y)

F = 
$$\frac{1}{t+y}$$

Compute the Laplace transforms of the Dirac and Heaviside functions.

syms t s
syms a positive
F = laplace(dirac(t-a),t,s)

F = $${\mathrm{e}}^{-a\hspace{0.17em}s}$$

F = laplace(heaviside(t-a),t,s)

F = 
$$\frac{{\mathrm{e}}^{-a\hspace{0.17em}s}}{s}$$

Show that the Laplace transform of the derivative of a function is expressed in terms of the Laplace transform of the function itself.

syms f(t) s
Df = diff(f(t),t);
F = laplace(Df,t,s)

F = $$s\hspace{0.17em}\mathrm{laplace}\left(f\left(t\right),t,s\right)-f\left(0\right)$$

Find the Laplace transform of the matrix M. Specify the independent and transformation variables for each matrix entry by using matrices of the same size. When the arguments are nonscalars, laplace acts on them element-wise.

syms a b c d w x y z
M = [exp(x) 1; sin(y) 1i*z];
vars = [w x; y z];
transVars = [a b; c d];
F = laplace(M,vars,transVars)

F = 
$$\left(\begin{array}{cc}\frac{{\mathrm{e}}^x}{a}& \frac{1}{b}\\ \frac{1}{c^2+1}& \frac{\mathrm{i}}{d^2}\end{array}\right)$$

If laplace is called with both scalar and nonscalar arguments, then it expands the scalars to match the nonscalars by using scalar expansion. Nonscalar arguments must be the same size.

F = laplace(x,vars,transVars)

F = 
$$\left(\begin{array}{cc}\frac{x}{a}& \frac{1}{b^2}\\ \frac{x}{c}& \frac{x}{d}\end{array}\right)$$

Compute the Laplace transform of symbolic functions. When the first argument contains symbolic functions, then the second argument must be a scalar.

syms f1(x) f2(x) a b
f1(x) = exp(x);
f2(x) = x;
F = laplace([f1 f2],x,[a b])

F = 
$$\left(\begin{array}{cc}\frac{1}{a-1}& \frac{1}{b^2}\end{array}\right)$$

If laplace cannot transform the input then it returns an unevaluated call.

syms f(t) s
f(t) = 1/t;
F(s) = laplace(f,t,s)

F(s) = 
$$\mathrm{laplace}\left(\frac{1}{t},t,s\right)$$

Return the original expression by using ilaplace.

f(t) = ilaplace(F,s,t)

f(t) = 
$$\frac{1}{t}$$