CTFT of e^(−2jω₀t)

$x(t) = e^{-j2\omega_0 t}$ is a complex exponential of frequency $-2\omega_0$. It is not absolutely integrable ($|x(t)| = 1$ for all $t$), so its FT exists only with an impulse.

Step 1: FT of a constant. $1 \leftrightarrow 2\pi\delta(\omega)$, because

$$\frac{1}{2\pi}\int_{-\infty}^{\infty}2\pi\,\delta(\omega)\,e^{j\omega t}d\omega = 1$$

Step 2: frequency shifting property. $e^{j\beta t}x(t) \leftrightarrow X(j(\omega - \beta))$. Here $\beta = -2\omega_0$:

$$e^{-j2\omega_0t}\cdot 1 \;\longleftrightarrow\; 2\pi\,\delta(\omega + 2\omega_0)$$

Verification by inverse FT:

$$\frac{1}{2\pi}\int_{-\infty}^{\infty}2\pi\,\delta(\omega + 2\omega_0)\,e^{j\omega t}d\omega = e^{j(-2\omega_0)t} = e^{-j2\omega_0t}$$

Answer:

$$X(j\omega) = 2\pi\,\delta(\omega + 2\omega_0)$$

              X(jw)
   2*pi ^
        |
--------+--------+--------> w
     -2w0        0

Discussion: The spectrum is a single impulse at $\omega = -2\omega_0$ of strength $2\pi$. A complex exponential contains only one frequency, so the spectrum has no mirror image at $+2\omega_0$ (a real sinusoid would have impulses at both $\pm 2\omega_0$).