Editing Laplace transform (section)

===Tauberian theory===
The Laplace transform of the measure <math>\mu</math> on <math>[0,\infty)</math> is given by
:<math>\mathcal L\mu(s) = \int_0^\infty e^{-st}d\mu(t).</math>
It is intuitively clear that, for small <math>s>0</math>, the exponentially decaying integrand will become more sensitive to the concentration of the measure <math>\mu</math> on larger subsets of the domain.  To make this more precise, introduce the distribution function:
:<math>M(t) = \mu([0,t)).</math>
Formally, we expect a limit of the following kind:
:<math>\lim_{s\to 0^+}\mathcal L\mu(s) = \lim_{t\to\infty} M(t).</math>
[[Tauberian theorem]]s are theorems relating the asymptotics of the Laplace transform, as <math>s\to 0^+</math>, to those of the distribution of <math>\mu</math> as <math>t\to\infty</math>.  They are thus of importance in asymptotic formulae of [[probability]] and [[statistics]], where often the spectral side has asymptotics that are simpler to infer.<ref>{{cite book|author=Feller|title=Introduction to Probability Theory, volume II,pp=479-483}}</ref>

Two Tauberian theorems of note are the [[Hardy–Littlewood Tauberian theorem]] and  [[Wiener's Tauberian theorem]].  The Wiener theorem generalizes the [[Ikehara Tauberian theorem]], which is the following statement:

Let ''A''(''x'') be a non-negative, [[monotonic function|monotonic]] nondecreasing function of ''x'', defined for 0&nbsp;≤&nbsp;''x''&nbsp;<&nbsp;∞. Suppose that

:<math>f(s)=\int_0^\infty A(x) e^{-xs}\,dx</math>

converges for ℜ(''s'')&nbsp;>&nbsp;1 to the function ''&fnof;''(''s'') and that, for some non-negative number ''c'',

:<math>f(s) - \frac{c}{s-1}</math>

has an extension as a [[continuous function]] for ℜ(''s'')&nbsp;≥&nbsp;1.
Then the [[Limit of a function|limit]] as ''x'' goes to infinity of ''e''<sup>&minus;''x''</sup>&thinsp;''A''(''x'') is equal to&nbsp;c.

This statement can be applied in particular to the [[logarithmic derivative]] of [[Riemann zeta function]], and thus provides an extremely short way to prove the [[prime number theorem]].<ref>{{citation| author=S. Ikehara | authorlink=Shikao Ikehara | title=An extension of Landau's theorem in the analytic theory of numbers | journal=Journal of Mathematics and Physics of the Massachusetts Institute of Technology | year=1931 | volume=10 | issue=1–4 | pages=1–12 | doi=10.1002/sapm19311011 |zbl=0001.12902}}</ref>