Using infinite series and complex numbers to derive formulas involving Laplace transforms

Abstract

The formula for the Laplace transform of an exponential function, , can be derived with equal ease if the parameters a and s are complex numbers. This leads to formulas for the Laplace transforms of e^atsin (bt) and e^atcos (bt) (where a and b are complex) and to calculations of certain inverse Laplace transforms without the need to consider Laplace transforms of derivatives or convolutions. Simpler proofs also follow from coupling the formula , for which a simple proof is given, with the fact that the operator commutes with certain familiar infinite series.

Keywords:

• Laplace transform
• inverse Laplace transform
• infinite series
• complex integration