Laplace Transform of Real Power

Theorem

Let $n$ be a constant real number such that $n > -1$

Let $f: \R \to \R$ be the real function defined as:

$\map f t = t^n$

Then $f$ has a Laplace transform given by:

$\ds \laptrans {\map f t}$

$=$

$\ds \int_0^\infty e^{-s t} t^n \rd t$

$\ds $

$=$

$\ds \frac {\map \Gamma {n + 1} } {s^{n + 1} }$

where $\Gamma$ denotes the gamma function.

$\ds \laptrans {t^n}$

$=$

$\ds \int_0^\infty e^{-s t} t^n \rd t$

Definition of Laplace Transform

$\ds $

$=$

$\ds \int_0^\infty e^{-u} \paren {\dfrac u s}^n \rd \paren {\dfrac u s}$

Integration by Substitution: $u := s t$ where $s > 0$ is assumed

$\ds $

$=$

$\ds \dfrac 1 {s^{n + 1} } \int_0^\infty u^n e^{-u} \rd u$

$\ds $

$=$

$\ds \dfrac {\map \Gamma {n + 1} } {s^{n + 1} }$

Definition of Gamma Function

$\blacksquare$

can also be seen presented in the form:

$\laptrans {\dfrac {t^n} {\map \Gamma n} } = \dfrac 1 {s^n}$

1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Laplace Transforms of Special Functions: $1$
1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: The Gamma Function: $31$
1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Appendix $\text B$: Table of Special Laplace Transforms: $4.$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Exponential Functions: $15.76$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of Special Laplace Transforms: $32.28$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 18$: Definite Integrals: Definite Integrals involving Exponential Functions: $18.76.$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 33$: Laplace Transforms: Table of Special Laplace Transforms: $33.28.$