Laplace Transform Calculator

This calculator returns the Laplace transform F(s) for one of five common time-domain functions — exponential, sine, cosine, a power of t, or a delayed unit step — evaluated at whatever complex frequency value s you enter, alongside the function's pole location and residue. Switching Function moves the transform value more than either numeric input does, since it swaps the entire closed-form expression rather than shifting a number inside the same expression. For the default exponential function e^(at), the transform is the classic 1 over (s minus a), and evaluating it at a specific s makes that s value noticeably more influential on the numeric result than the rate parameter a is: at the calculator's default settings, nudging s by ten percent moves the transform value by about 1.6 times as much, proportionally, as nudging a by the same ten percent does, and the two pull the result in opposite directions — raising s shrinks the transform since it grows the denominator, while raising a grows the transform since it shrinks that same denominator by pushing s closer to it. That direction is specific to the exponential function, too: for the cosine function, F(s) = s/(s² + w²), raising the Parameter (there, the frequency w) grows the denominator and so lowers the transform instead — the sign of Parameter's pull depends on which function is active, not on Parameter alone.

The residue, by contrast, is not computed from a or s at all for the exponential case — it is fixed at exactly 1 regardless of what either field holds, since the residue of a simple pole at 1 over (s minus a) is always 1 by definition, independent of where that pole happens to sit. That is a property of the exponential branch specifically, though: for the polynomial function t^n, the residue is n factorial, so the Parameter (there, the degree n) directly determines it. What falls outside this calculator's scope: the classical Laplace transform of a growing exponential only converges for s values greater than the rate a, but the calculator does not check for or flag that condition — setting s below a still returns a numeric answer from the same formula, describing a result outside the transform's normal region of validity rather than refusing to compute one.