Abstract
We view the (real) Laplace transform through the lens of linear algebra as a continuous analogue of the power series by a negative exponential transformation that switches the basis of power functions to the basis of exponential functions. This approach immediately points to how the complex Laplace transform is a generalisation of the Fourier transform where the pole of the transform realises the linear algebraic intuition. The exponential transformation also motivates the Taylor inversion of the real Laplace transform.
Mathematics Subject Classification (2010):
Disclosure statement
No potential conflict of interest was reported by the author.
Data availability statement
The datasets generated during and/or analysed during the current study are available in the following bibliography (please see References).
Notes
1 As far as evaluating the series is concerned, the case of x = 0 is pathological since the power series (Equation1(1)
(1) ) is the finite sum 0. Furthermore, the basis functions
all reduce to the 0 function. For the purposes of the inversion of the real Laplace transform, the case of x = 0 plays a critical role (see Section 6, Remark 6.1 and Proposition 6.2).
2 This follows from and
.