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Laplace Transforms are used in circuit analysis. https://web.stanford.edu/~boyd/ee102/laplace_ckts.pdf

What do you do with it; check for stability, like a unit test?

Laplace and z-transform are widely used in control systems and signal processing. Both are more general forms of the Fourier transform in continuous and discrete time, respectively.

Is it used in neural networks as approximators? From my understanding they're quite good but they suffer from overfitting.

For a brief period i did work on audio, video, and image processing along with signal processing.