Mathquery

Existence And Property Of Laplace transforms : Existence Of Laplace Transforms :      In order to discuss the theorem on existence of transform we need the following definition . Definition :         A function f(t) is said to be piecewise continuous in a finite range a≤t≤b , if it is possible to divide the range into a finite number of sub-intervals such that f(t) is continuous inside each sub-interval and approaches finite values as t approaches either end of any interval from the interior. Theorem :        Let f(t) be a function which is piecewise continuous on every finite interval in the range t≥0 and satisfies          |f(t)|≤Me^αt , for all t≥0        .........(1)    and for some constants α and M . Then the Laplace transform of f(t) exists for all p>α . Proof :           Since f(t) is piecewise continuous ,      e⁻ᵖᵗf(t) is integrable over any finite interval on the t-axis ,            T i.e     ∫ e⁻ᵖᵗf(t) dt