Question

how to prove L{t^n} = \(\large \frac{n!}{s^{n+1}}\) using the definite for laplace transform?

Answer 1

ok
are you familiar with Gamma function? because it will use its definition?

Answer 2

@lgbasallote

Answer 3

gamma function?

Answer 4

yes .

Answer 5

i dont think i am...not sure though...im not really familiar with fancy terms

Answer 6

ok let me solve this first.

Answer 7

first check gamma function here .
http://en.wikipedia.org/wiki/Gamma_function

Answer 8

@lgbasallote

Answer 9

\[\text{ \let } I_n=\int\limits_{0}^{+\infty}t^{n}e^{-st}dt\]
let \[n \in \mathbb{N}^{*}\]
using an integration by parts
\[I_n=\int\limits_{0}^{+\infty}(-\frac{e^{-st}}{s}) \prime t^n dt =\left[ (-\frac{e^{-st}}{s}) \prime t^n \right]"=0"+\frac{n}{s}\int\limits_{0}^{+\infty}t^{n-1}e^{-st}dt\]
then \[I_n=\frac{n}{s}I_{n-1}\] (**)
now there is two methods to do this :
1)\[I_n=\frac{n}{s}I_{n-1}=\frac{n}{s}\frac{(n-1)}{s}I_{n-2}=\frac{n}{s}\frac{(n-1)}{s}\frac{(n-2)}{s}I_{n-3}\]...........................
\[I_n=\frac{n}{s}\frac{(n-1)}{s}\frac{(n-2)}{s}.....................\frac{1}{s}I_0=\frac{n!}{s^n}I_0\]
if you calculate I_0 you will find 1/s
2) using induction for n=0 \[I_0=\frac{0!}{s^{0+1}}\]
so the "statement" is true for n=0
let n be a positive integer
let's assume that \[I_n=\frac{n!}{s^{n+1}}\]
let's prove that \[I_{n+1}=\frac{(n+1)!}{s^{n+1+1=n+2}}\]
using(**)
\[I_{n+1}=\frac{(n+1)}{s}I_n=\frac{(n+1)}{s}\frac{n!}{s^{n+1}}=\frac{(n+1)!}{s^{n+2}}\]
so we proved that \[\forall n \in \mathbb{N} I_n=L(t^n)=\frac{n!}{s^{n+2}}\]
**"I don't like the first one" !

Answer 10

@Neemo i think in the denominator it should be
\[\Large S^{n+1}\] not n+2 !

Answer 11

yes ! you're right !I made a mistake in the last line (n+1) How do I correct it .?

Answer 12

@sami-21

Answer 13

i am not Quite good with the Latex codes .
you have to write it again:(
or
take snap till where the answer is correct. then write the remainingg:P

Answer 14

I hope that @lgbasallote will read your comment !
or this "I made a typo in the last line"
the correction " \[I_n=L(t^n)=\frac{n!}{s^{n+1}}\]" not n+2 :D

Answer 15

there was typo in response as well :P
corrected below.

Answer 16

i have no idea what these mean =_=

Answer 17

all of these !!!!! :p :p :p :p ???

Answer 18

then take it :P
before understanding gamma function you will not be able to do this :)
so remember for time being just like !
\[\Large (a+b)^2=a^2+b^2+2ab\]
so is
\[\Large L(t^n)=\frac{n!}{s^{n+1}}\]
:)