Laplace transform of tf(t) and f(t)/t

[@blackpenredpen]

Here’s a question that I have in mind for a while now but haven’t seen any proof or page about it. Q: does differentiation under the integral sign work for indefinite integral? To me, it seems like it should be. But I haven’t seen anyone talks or does it.

[@MuPrimeMath]

Based on this proof, I think the Leibniz rule does work for indefinite integrals: en.m.wikipedia.org/wiki/Leibniz_integral_rule#Proof_of_basic_form If you follow the process that I used in my integral of ln(x) with Feynman's trick video for the indefinite integral, you still get the right answer. I think it's just that +C makes it harder to deal with.

[@blackpenredpen]

@@MuPrimeMath Ah, thanks!

[@user-wu8yq1rb9t]

Great as always. I hope you'll decide to continue this playlist (Laplace). Thank you so much 💕

[@isaswa1602]

4:50 is Fubini's theorem even valid under implicit integral ? I'd like to see a formal prove if so, thanks !

[@anegativecoconut4940]

3:58 Why when I hear this passage I hear some french music and see Fubini, some weird Lebesgue, and the Wikipedia page for the Dominated Convergence theorem?

[@marcschmidtpujol550]

Great explanation!

[@shreyasujagar643]

crystal clear, thank you!

[@minicachoro]

You saved me! Thank you so much!!!

[@l.JAI.SHREE.RAM.l]

Is there not any theorem as - ℒ⁻¹{∫[s ∞] f(u) du} = F(t)/t

[@benjaminbrady2385]

Interchange of partial derivative and integral may not be valid as you've pointed out yourself my boi: kzbin.info/www/bejne/sHKqgn5_fL-Hm9E

[@MuPrimeMath]

The rule applies for any function f(t) such that f(t)e^(-st) has a partial derivative that exists and can be integrated over (0, inf). The Dirac delta function is a special case!

[@Raizensen9415]

Let f(t)=1 When t>=0 Find L{f(t)}

[@ayoubachak01]

This seems useless 😧 But nice work ❤ I'm a fan 😍

[@MuPrimeMath]

You can see a video where I used this identity to solve a differential equation here: kzbin.info/www/bejne/Y6WpgWiFqbymd6M