# using Fourier Transforms to solve the question.

I am given a question of Fourier Transform:

$$e^{2(t-1)}u(t-1)$$ My teacher solved it by using the formula which I couldn't understand so I tried to apply the properties on it. Now I have solved it by the following method:

$$e^{2(t)}u(t) \rightarrow \frac{1}{2+j\omega}$$

Now we know that: $$\delta(t-t_0) \rightarrow e^{-j\omega t_0}$$ So I used the above property on $$u(t-1)$$ and got the following answer which is same as my teacher got, which is:

$$\frac{e^{-j\omega}}{2+j\omega}$$

Is my method correct?

Assuming that your definition of the Fourier transform is $$\hat f(\omega) = \int_{\mathbb R} f(t)e^{j\omega t}\ \mathsf dt,$$ then yes, your answer is correct. We can use a change of variables $$s=t+1$$ to compute \begin{align} \hat f(\omega) &= \int_1^\infty e^{-2(t+1)}e^{j\omega t}\ \mathsf dt\\ &= e^{-j\omega}\int_0^\infty e^{-2s}e^{j\omega s}\ \mathsf ds\\ &= e^{-j\omega}\cdot\frac{1}{2+j\omega}. \end{align}