An infinite series transformed from Laplace transform

[SAT999]

hello. I have transformed the Laplace transform into the infinite series by repeatedly using integration by parts.
What is this infinite series? may be Laplace transform series, or only an infinite series without name?

$$L(t)= \int_{t}^{\infty}\frac{f(t)}{e^{st}} dt =-0 + <br /> <br /> \frac{1}{e^{st}}\sum_{n=0}^{\infty}\frac{f^{(n)}(t)}{s^{(n+1)}}$$

$$<br /> IL(L(t))= \frac{1}{2\Pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}e^{st}L(t) ds=\frac{1}{2\Pi <br /> <br /> i}\oint_{c}e^{st}L(t) ds = \frac{1}{2\Pi i}\sum_{n=0}^{\infty}\oint_{c}\frac{f^{(n)}(t)}{s^{(n+1)}} <br /> <br /> ds$$

$$= \frac{1}{2\Pi i}f(t)\oint_{c}\frac{1}{s} ds = \frac{1}{2\Pi i}f(t)\int_{0}^{2\Pi}\frac{is}{s} <br /> <br /> d\Theta = f(t)$$

 

[fresh_42]

I'm not quite sure what you have done there, but the answer is: just another infinite series without name.