restart; Fourier transforms using Maple with(inttrans); NiM3L0kpYWRkdGFibGVHNiJJKGZvdXJpZXJHRiVJK2ZvdXJpZXJjb3NHRiVJK2ZvdXJpZXJzaW5HRiVJJ2hhbmtlbEdGJUkoaGlsYmVydEdGJUkraW52Zm91cmllckdGJUkraW52aGlsYmVydEdGJUkraW52bGFwbGFjZUdGJUkqaW52bWVsbGluR0YlSShsYXBsYWNlR0YlSSdtZWxsaW5HRiVJKnNhdmV0YWJsZUdGJQ== Let us lok at the definitions convert(fourier(f(t),t,w),int); NiMtSSRpbnRHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkKiYtSSJmR0YoNiNJInRHRigiIiItSSRleHBHRiU2IyooXiMhIiJGL0YuRi9JIndHRihGL0YvL0YuOywkSSlpbmZpbml0eUdGJkY1Rjo= convert(invfourier(g(w),w,t),int); NiMsJComSSNQaUdJKnByb3RlY3RlZEdGJiEiIi1JJGludEc2JEYmSShfc3lzbGliRzYiNiQqJi1JImdHRiw2I0kid0dGLCIiIi1JJGV4cEdGKjYjKiheI0YzRjNGMkYzSSJ0R0YsRjNGMy9GMjssJEkpaW5maW5pdHlHRiZGJ0Y9RjMjRjMiIiM= fourier(exp(-abs(t)),t,w); NiMsJCokLCYiIiJGJiokSSJ3RzYiIiIjRiYhIiJGKg== fourier(exp(-a*abs(t)),t,w); NiMtSShmb3VyaWVyRzYiNiUtSSRleHBHNiRJKnByb3RlY3RlZEdGKkkoX3N5c2xpYkdGJTYjLCQqJkkiYUdGJSIiIi1JJGFic0dGKjYjSSJ0R0YlRjAhIiJGNEkid0dGJQ== assume(a>0); fourier(exp(-a*abs(t)),t,w); NiMsJComSSNhfGlyRzYiIiIiLCYqJEYlIiIjRicqJEkid0dGJkYqRichIiJGKg== invfourier(2*a/(a^2+w^2),w,t); NiMsJiomLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRihJKF9zeXNsaWJHNiI2IywkKiZJI2F8aXJHRioiIiJJInRHRipGLyEiIkYvLUkqSGVhdmlzaWRlR0YnNiNGMEYvRi8qJi1GJjYjRi1GLy1GMzYjLCRGMEYxRi9GLw== fourier(sin(t)*exp(-abs(t)),t,w); NiMqKl4jIiIlIiIiSSJ3RzYiRiYsKCEiI0YmKiRGJyIiIyEiIkYnRixGLSwoRixGJkYrRiZGJ0YsRi0= invfourier(4*I*w/((2+w^2+2*w)*(-2-w^2+2*w)),w,t); NiMqJi1JJHNpbkc2JEkqcHJvdGVjdGVkR0YnSShfc3lzbGliRzYiNiNJInRHRikiIiIsJiomLUkqSGVhdmlzaWRlR0YmRipGLC1JJGV4cEdGJjYjLCRGKyEiIkYsRiwqJi1GMEYzRiwtRjJGKkYsRixGLA== fourier(Heaviside(t)*exp(-t),t,w); NiMqJCwmIiIiRiUqJl4jRiVGJUkid0c2IkYlRiUhIiI= assume(b,real); assume(b<0); fourier(Heaviside(t-b)*Heaviside(a-t),t,w); NiMqKl4jIiIiRiUtSSRleHBHNiRJKnByb3RlY3RlZEdGKUkoX3N5c2xpYkc2IjYjLCYqKF4jISIiRiVJI2J8aXJHRitGJUkid0dGK0YlRiUqKEYvRiVJI2F8aXJHRitGJUYyRiVGJUYlRjJGMCwmLUYnNiMqKEYkRiVGMkYlRjRGJUYwLUYnNiMqKEYkRiVGMkYlRjFGJUYlRiU= fourier(Heaviside(t+1)*Heaviside(1-t),t,w); NiMsJComSSJ3RzYiISIiLUkkc2luRzYkSSpwcm90ZWN0ZWRHRitJKF9zeXNsaWJHRiY2I0YlIiIiIiIj fourier(Heaviside(t+a)*Heaviside(a-t),t,w); NiMsJComSSJ3RzYiISIiLUkkc2luRzYkSSpwcm90ZWN0ZWRHRitJKF9zeXNsaWJHRiY2IyomRiUiIiJJI2F8aXJHRiZGL0YvIiIj fourier(Heaviside(t+a)*Heaviside(a-t)*t,t,w); NiMqKF4jIiIjIiIiLCYtSSRzaW5HNiRJKnByb3RlY3RlZEdGK0koX3N5c2xpYkc2IjYjKiZJIndHRi1GJkkjYXxpckdGLUYmISIiKihGMUYmLUkkY29zR0YqRi5GJkYwRiZGJkYmRjAhIiM= fourier(Heaviside(t+a)*Heaviside(a-t)*t^2,t,w); NiMsJComLCgqKEkjYXxpckc2IiIiIy1JJHNpbkc2JEkqcHJvdGVjdGVkR0YtSShfc3lzbGliR0YoNiMqJkkid0dGKCIiIkYnRjJGMkYxRilGMkYqISIjKihGJ0YyLUkkY29zR0YsRi9GMkYxRjJGKUYyRjEhIiRGKQ==