Property Function exp

1. Aug 25, 2010

juaninf

How can prove this

$$\exp(At)\exp(-At_0)=\exp(A(t-t_0))$$?

using $$\displaystyle\sum_{i=0}^n{(1/k!)A^kt^k}$$

and this properties
in t=0
$$[\exp(At)]_{t} = I$$

$$exp(At)exp(-At)=I$$
$$\frac{dexp(At)}{dt}=Aexp(At)=exp(At)A$$

Last edited: Aug 26, 2010
2. Aug 25, 2010

ross_tang

For your first equation, please refer to this question in http://www.voofie.com/concept/Mathematics/" [Broken]:

http://www.voofie.com/content/152/how-to-prove-eat-e-at_0-eat-t_0/" [Broken]

I think you typed wrong in this formula:

$$exp(At)_t=0 = I$$

0 is not equal to I. And your what's your meaning of $$exp(At)_t$$?

For this one $$exp(At)exp(-At)=I$$, you can use my result to prove easily. For the last one, you should try to use the power series expansion and differentiate term by term. You will get the answer easily too.

Last edited by a moderator: May 4, 2017
3. Aug 26, 2010

Petr Mugver

To conclude, i suppose you mean

$$[\exp(At)]_{t=0} = I$$

Well, it's pretty simple:

$$[\exp(At)]_{t=0} = \left[\sum_{k=0}^\infty\frac{A^kt^k}{k!}\right]_{t=0}=I+0+0+\cdots=I$$

4. Aug 26, 2010

juaninf

fix question my question is
How prove this,
$$\exp(At)\exp(-At_0)=\exp(A(t-t_0))$$
using as above properties

5. Aug 26, 2010

juaninf

http://www.voofie.com/content/152/how-to-prove-eat-e-at_0-eat-t_0/ [Broken]

but i dont understand how change sumatoria infinite to finite, Where i can read this?

Last edited by a moderator: May 4, 2017
6. Aug 26, 2010