# Why are functions for the frequency domain expressed in terms of a whole complex number rather than just omega?

I don't understand why some formulas used in electrical engineering, especially why using fourier analysis, include static numbers in function inputs, rather than just the changing variable.

For example, the formula for the Fourier Transform of $$x(t)=\cos\omega_0t$$ is $$X(j\omega)=\pi\delta(\omega-\omega_0)+\pi\delta(\omega+\omega_0)$$ Why is this in terms of $j\omega$ rather than just $\omega$, because $j$ never changes. I've seen other formulas use complex numbers as the function input, such as $$X(e^{j\Omega})=1$$ Again, if $e$ and $j$ are not changing, why include them as a function input?

## 1 Answer

Why is this in terms of jω rather than just ω,

In general, we have the complex frequency $s = \sigma + j\omega$.

For the Fourier transform, we set $\sigma = 0$ such that $s = j \omega$.

Thus, $X(s) \rightarrow X(j\omega)$

In the case of the Z transform, we have in general $z = Ae^{j\Omega}$.

For the Discrete Time Fourier Transform (DTFT), $A = 1$.

Thus, $X(z) \rightarrow X(e^{j\Omega})$

The notation varies among disciplines and authors. As long as the context is clear, one can vary the notation.