Definition and Basic Transforms

Differential Equations · Axiom Academy

LESSON Definition and Basic Transforms The Laplace transform turns a function of time into a function of s — one integral, one kernel, and a whole table of results falls out. Take a function f(t) defined for . Multiply it by the exponential kernel e^ -st — this damps the function so the product settles down as — then integrate from 0 to . Whatever area accumulates under that damped product is F(s) . Here s is a real (or complex) parameter and is the transform — a new function, of s instead of t . The integral only converges if e^ -st can damp f(t) fast enough. We need f(t) to not outrun the exponential — precisely, f(t) should be exponentially bounded . If f(t) is exponentially bounded by a , then exists for every — that half-line is the region of convergence (ROC) . 3. Basic Transforms by Direct Integration Every entry below follows the same three moves : substitute f(t) into the definition, evaluate the (improper) integral as a limit , and simplify. Watch the boundary term vanish at t=T because of the kernel's decay — that vanishing is exactly what makes the transform converge. The bracket term at vanishes exactly because when — that's the kernel damping at work again. For and , write them via Euler's formula as combinations of complex exponentials — then reuse the exponential-transform result from Step 3 (with a complex ) instead of integrating by parts twice. Both and are built from and — so their transforms fall straight out of .

This is the written version of the interactive lesson above. See the full Differential Equations course.