# Post's inversion formula

Post's inversion formula

Post's inversion formula for Laplace transforms, named after Emil Post, is a simple-looking but usually impractical formula for evaluating an inverse Laplace transform.

The statement of the formula is as follows: Let "f"("t") be a continuous function on the interval [0, &infin;) of exponential order, i.e.

: $sup_\left\{t>0\right\} frac\left\{f\left(t\right)\right\}\left\{e^\left\{bt < infty$

for some real number "b". Then for all "s" > "b", the Laplace transform for "f"("t") exists and is infinitely differentiable with respect to "s". Furthermore, if "F"("s") is the Laplace transform of "f"("t"), then the inverse Laplace transform of "F"("s") is given by

: $f\left(t\right) = mathcal\left\{L\right\}^\left\{-1\right\} \left\{F\left(s\right)\right\} = lim_\left\{k o infty\right\} frac\left\{\left(-1\right)^k\right\}\left\{k!\right\} left\left( frac\left\{k\right\}\left\{t\right\} ight\right) ^\left\{k+1\right\} F^\left\{\left(k\right)\right\} left\left( frac\left\{k\right\}\left\{t\right\} ight\right)$

for "t" > 0, where "F"("k") is the "k"-th derivative of "F".

As can be seen from the formula, the need to evaluate derivatives of arbitrarily high orders renders this formula impractical for most purposes. With the advent of powerful home computers, the main efforts to use this formula have come from dealing with approximations or asymptotic analysis of the Inverse Laplace transform, using the Grunwald-Letnikov differintegral to evaluate the derivatives. Post inversion has attracted interest due to the improvement in computational science and the fact that you don't need to know where the poles of F(s) lie, which make it interesting to calculate the asymptotic behaviour for big 'x' using inverse Mellin transforms for several arithmetical functions related to the Riemann Hypothesis.

References

* [http://www.rose-hulman.edu/~bryan/invlap.pdf Elementary inversion of the Laplace transform] . Bryan, Kurt. Accessed June 14 2006.

Wikimedia Foundation. 2010.

### См. также в других словарях:

• Emil Leon Post — Infobox Scientist name = Emil Leon Post image width = birth date = February 11, 1897 birth place = Augustów, then Russian Empire death date = April 21 1954, death place = New York City, flagicon|USA U.S. residence = nationality = field =… …   Wikipedia

• Inverse Laplace transform — Contents 1 Mellin s inverse formula 2 Post s inversion formula 3 See also 4 References 5 Ext …   Wikipedia

• List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… …   Wikipedia

• Laplace transform — In mathematics, the Laplace transform is one of the best known and most widely used integral transforms. It is commonly used to produce an easily soluble algebraic equation from an ordinary differential equation. It has many important… …   Wikipedia

• Switzerland — /swit seuhr leuhnd/, n. a republic in central Europe. 7,248,984; 15,944 sq. mi. (41,294 sq. km). Cap.: Bern. French, Suisse. German, Schweiz. Italian, Svizzera. Latin, Helvetia. * * * Switzerland Introduction Switzerland Background: Switzerland s …   Universalium

• Continental philosophy — Collective term for the many distinct philospohical traditions, methods, and styles that predominated on the European continent (particularly in France and Germany) from the time of Immanuel Kant. It is usually understood in contrast with… …   Universalium

• Barack Obama — «Barack» y «Obama» redirigen aquí. Para otras acepciones véase Barack (desambiguación) y Obama (desambiguación). Barack Obama …   Wikipedia Español

• climate — /kluy mit/, n. 1. the composite or generally prevailing weather conditions of a region, as temperature, air pressure, humidity, precipitation, sunshine, cloudiness, and winds, throughout the year, averaged over a series of years. 2. a region or… …   Universalium

• Europe, history of — Introduction       history of European peoples and cultures from prehistoric times to the present. Europe is a more ambiguous term than most geographic expressions. Its etymology is doubtful, as is the physical extent of the area it designates.… …   Universalium

• heredity — /heuh red i tee/, n., pl. heredities. Biol. 1. the transmission of genetic characters from parents to offspring: it is dependent upon the segregation and recombination of genes during meiosis and fertilization and results in the genesis of a new… …   Universalium

### Поделиться ссылкой на выделенное

##### Прямая ссылка:
Нажмите правой клавишей мыши и выберите «Копировать ссылку»