How can I compute Laplace transforms when the time shift is a variable, in Symbolic Math Toolbox 5.7 (R2011b)?

I am working with Laplace transforms of functions involving sine waves and Heaviside functions. When I execute the following code:

% Numeric time shift

syms t w

x = sin(w*(t-10))*heaviside(t-10);

X = laplace(x);

pretty(simplify(X))

I receive the answer I expect:

w  /  ( exp(10 s) (s^2  +  w^2 ) )

However, when there is one more symbolic variable, T instead of 10:

% Symbolic time shift

syms t w T

x = sin(w*(t-T))*heaviside(t-T);

X = laplace(x);

pretty(simplify(X))

I receive the following answer:

-laplace(sin(w (T - t)) heaviside(t - T), t, s)

How can I make the answer resemble what I received in the first case, with 'T' instead of 10?

Answer 1

[MathWorks Support Team]

You can impose the assumption that T is positive:

syms T positive

syms t w

   x = sin(w*(t-T))*heaviside(t-T);

X = laplace(x);

pretty(simplify(X))

Then you can receive the following answer:

w  /  ( exp(T s) (s^2  +  w^2) )