[ \int_0^\infty e^-\alpha t e^-i\omega t dt = \int_0^\infty e^-(\alpha + i\omega) t dt = \frac1\alpha + i\omega ]
The unit step function, often denoted ( u(t) ), is one of the most fundamental, yet mathematically troublesome, signals in engineering and physics. Defined as: fourier transform step function
The Fourier transform of ( \textsgn(t) ) is ( 2/(i\omega) ) (without a delta, since its average is zero). Thus: [ \int_0^\infty e^-\alpha t e^-i\omega t dt =
Here, ( e^-\alpha t ) ensures convergence for ( \alpha > 0 ). Then: Then: Now, take the limit as ( \alpha
Now, take the limit as ( \alpha \to 0^+ ):
confirming the result. | Function | Fourier Transform | |----------|------------------| | ( u(t) ) (unit step) | ( \pi\delta(\omega) + \frac1i\omega ) | | ( \textsgn(t) ) (sign) | ( \frac2i\omega ) | | Constant ( 1 ) | ( 2\pi\delta(\omega) ) |