Mod-07 Lec-18 Laplace transforms (Part I)

[TheSpringRain-yivx]

gentle singularity.., professor enriched with deep n beautiful mathematical code

[tulanimtunzi]

Interesting mathematical formulation of laplace transform.it help to have a fathom view of problem. Thanks Sir

[ritanandy]

Ur videos r very helpful. Please make videos on geometrical or physical meanings of Laplace and Fourier transforms. What are the geometrical or physical meanings of Inverse Laplace and Fourier transforms. What do Laplace and Fourier transforms really tell us?

[MyChannel-vwcy]

now i understood the initial and final value theorem

[fadiadaghestani]

1 first is it an ballistic function
2 how do we continue analyrically?
Secret is exp t =
F hat (s)= gamma[a+1]/(s- a)^(a-1)
Convergence test
For what values you'd expect this integral to converge?
Re (a) > -1
Now can analytically continue this
Note Re [s-a] >0
Expect some problem if there's a singularity
When s =-a
I.e for complex a, it's a branch point
Takeaway the explicit form tells you what the singularity is
1Just do integral
2 analytic continuation
The large t behavior is governed by small s behavior
[Inverese relationship t & s
There's a theory says how find asymptomatic t
Tending to inf (behavior of f t )
Based on s tend 0 behavior of Laplace transform
Similar t tend 0 behavior based on the s tending to infinity
Abelian theorems
S-a shifting laplace transform by a
Inverse laplace:
Integral=Ftilsa(s+a)
Convulutikn property of g f between times t t'
Equals to ftilda (s)* gtilda (s)
Convultiontion transform to multiplicat
General convulsions of many functions
Integration by parts :
Laplate f prime t =
S ftilda(s) - f(0)
2nd moment
L[f doublePrime(t)=
S² ftilda(s)- s f'(0) - s(0)
Transformed to operation of multiplication
Inversion of transform
Inverse L T
L f[(t)]
See contour integrals in region of analyticity
Om right region s
As long s is rational
Note branch cuts of s plane
Fold original contour over to ther side
Provided line exists into line integral
Note ensure initial conditions are satisfied for any inhomogenous case
Trigonometric functions go over hyperbolic functions
Sqrt[ w²0 gama²/4]
Is pure imaginary
Probability theory random processes bessil function [the the first kknd] of order n

1. Define it as fun of complex variable
Order my (complex as well
Jv(z)= sum[n =0, inf]
(Z/2 )^(v+2n)/n
n! gamma v
( n +1)

[byxfan]

I think u probably need to have a PHD already to be eligible to follow this guy!

[alute]

If noise ft converges to e^t^2 then noise larger than conergent signal thus unable to converge (such) functions
(they don't have a Laplace transform
F is analytic function (to a right of some (scalar) number )
S is real ecp(-t) gives oscillatory part
L(t^alpha)
Takeaway
It's large quantity is governed by the small s' transform behavior
If t inf and alpha positive real part
This thing diverges
Divergence in Laplace transform apprears near s=0
Theory exists in finding asymptomatic t at inf based on s=0
Similarly how to tell t=0 from s tending to infinity
I. E. Abelian turberian theorems
In general any f multiplied by exp(-at)
Is shifting Laplace transform by a (shift function)
Ftilda(s+a)
Cobvilution
What happens to the product of 2 functions

[ajaysinghrajawat]

Sir, what is the Laplace transform of u(sin2t) and in general f(g(t))

Please reply sir

[sam]

The amount of CF need to be added dictated by the initial conditions

[kantasuresh]

why the professor is mentioning "dt" before the function ?

[YPmath-dot-com]

This is a great lecture, so many valid points, so detailed, but prof gives zero fucks about making it simple and understandable for non iitian public :D

[Scaletheuniverse]

I need Laplace's equation in spheical polar cordinate. Plz send me solutions on my email id. Plz sir

[marcielalves]

accept the Lord Jesus Christ who has not accepted yet because He is coming back ... sanctify more and more inside and outside ... doing works worthy of repentance and leaving worldliness ... leaving the vanities the tinctures, earrings, makeup, enamels, the fashions of hair and clothes, the short and tight clothes because the Lord is Holy and we must be holy in all our way of living "1 Peter 1: 15, 16"

[aarongoldsmith]

35:22 They're not independent... independent and mutually exclusive events must have 0% or 100% probability.

[babunandha]

These mathematicians just play with formulae without explaining with application examples, 98% of them don't know where these r used. Just a reproducing machine.

[vishwasgajawada]

wtf...he just directly writes down the formulas and blabber something..no proofs for anything