1/sqrt(2 π) integral_(-∞)^∞ (e^(-t^2) sin(t)) e^(i w t) dt = -1/2 i (e^(-1/4 (w + 1)^2)/sqrt(2) - e^(-1/4 (w - 1)^2)/sqrt(2))

(i e^(-1/4 (w + 1)^2) (e^w - 1))/(2 sqrt(2))

-(i e^(-1/4 (w - 1)^2 - 1/4 (w + 1)^2) (e^(1/4 (w - 1)^2) - e^(1/4 (w + 1)^2)))/(2 sqrt(2))

(i e^(-1/4 (w - 1)^2))/(2 sqrt(2)) - (i e^(-1/4 (w + 1)^2))/(2 sqrt(2))

(i (1/(e^((w - 1)^2))^(1/4) - 1/(e^((w + 1)^2))^(1/4)))/(2 sqrt(2))

ℱ_t[e^(-t^2) sin(t)](w) = i (ℱ_t^s[e^(-t^2) sin(t)](w))

sqrt(2/π) integral_0^∞ (e^(-t^2) sin(t)) sin(w t) dt = (e^(-1/4 (w + 1)^2) (e^w - 1))/(2 sqrt(2))