Lerch's theorem

This article deals with Lerch's uniqueness theorem. For the theorem of elementary number theory see Lerch's theorem (number theory) .

The uniqueness theorem of Lerch (after Mathias Lerch ) has in the context of functional analysis in the Laplace transform a meaning.

statement

Lerch's theorem of uniqueness states: If of and the Laplace transforms exist, and if ${\displaystyle f\,}$ ${\displaystyle g\,}$

${\displaystyle {\mathcal {L}}\{f(t)\}(s)={\mathcal {L}}\{g(t)\}(s)}$

holds for all with a sufficiently large real part, then is ${\displaystyle s\,}$

${\displaystyle f(t)=g(t)\,}$

in all points where both functions are continuous. ${\displaystyle t\,}$

literature

• Christian Blatter: Complex analysis, Fourier and Laplace transform fo ̈ r electrical engineers . Lecture notes from the ETH Zurich, Department of Electrical Engineering and Information Technology, 2006 ( online (PDF; 1.7 MB)).