# 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?

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

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

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

#### fresh_42

Mentor
2018 Award
I'm not quite sure what you have done there, but the answer is: just another infinite series without name.

"An infinite series transformed from Laplace transform"

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