The negative exponential transformation: a linear algebraic approach to the Laplace transform

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.

Keywords:

• Laplace transform
• pole
• Fourier transform
• commuting diagram
• basis
• Taylor inversion of real Laplace transform

Mathematics Subject Classification (2010):

• 97D80

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 (1(1) $\sum _{n=0}^{\mathrm{\infty }}a(n)x^n.$(1) ) is the finite sum 0. Furthermore, the basis functions $b_t$ 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 ${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{\mathrm{at}}\sin \mathrm{bt}\mathrm{d}t=\frac{b}{a^2+b^2}$ and ${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{\mathrm{at}}\cos \mathrm{bt}\mathrm{d}t=\frac{-a}{a^2+b^2}$.