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?
They are an alternate way of describing a function just like Taylor series. Every function is imagined in terms of sums of harmonics with each having a particular frequency, and the fourier transform tells us the magnitude and phase that that function has corresponding to every frequency. Laplace is just an extension of fourier transform Coming to inverse transforms, they just use the fourier components (from the information gathered by fourier transforms) to generate the time domain signal. This is the real motivation of Fourier transforms to come to this step. The same works for laplace transforms Thus as every signal (as quoted by Dr Fourier himself) can be expressed as sum of harmonics, Frequency has became a domain in itself!
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)
You don't need a PhD, but you do need prior knowledge of Laplace transforms. For example, you need to know that the *s* parameter is itself a complex number (a + bi), so that you re not confused when he talks about the Real part of *s*
@@dozog I mean when at 4:56 he said Re(α)>-1 and then Re(s-a)>0,why only real part matter here not the imaginary part?can imaginary part can take any value here?
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
@@dozog I feel like it's a sort of philosophy. ∫f(x)dx is like parentheses around f(x), whereas ∫dx f(x) is like an operator acting on f(x). Also in this form if you have lots of integrals it's easier to see what is integrated over like ∫ dx dz dy dz dw G(x-y)G(y-w)G(w-z)G(z-x), which happens in Quantum Field Theory all the time when calculating Feynman diagrams. When writing down those integrals for the diagrams it's also nice to first write down the variables that are integrated over and then think of what the integrand looks like and writing it down.
Aadityakiran S. thats kinda true.. but sometimes what is written is not shown at all, and having upped the playback speed i got a little dizzy on this one. Too much unnecessary movement. That being said i regret writing such a mean comment, now im just thankful they actually recorded this genius. Also the camera work is still better than most lectures on youtube. that being said i still think there is too much zoom.
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"
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.
Look at the basic mechanical system comprising of a spring, a mass and a dampener and the analogous electrical system of a inductance, capacitance and resistance.
Prof balki is not that type of someone.. besides he's a physicsist, this is the course, mathematical methods in physics just tp show the mathematics that's required to..
gentle singularity.., professor enriched with deep n beautiful mathematical code
Interesting mathematical formulation of laplace transform.it help to have a fathom view of problem. Thanks Sir
now i understood the initial and final value theorem
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?
They are an alternate way of describing a function just like Taylor series. Every function is imagined in terms of sums of harmonics with each having a particular frequency, and the fourier transform tells us the magnitude and phase that that function has corresponding to every frequency. Laplace is just an extension of fourier transform
Coming to inverse transforms, they just use the fourier components (from the information gathered by fourier transforms) to generate the time domain signal. This is the real motivation of Fourier transforms to come to this step. The same works for laplace transforms
Thus as every signal (as quoted by Dr Fourier himself) can be expressed as sum of harmonics, Frequency has became a domain in itself!
Interesting lecture. So good
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)
Superb lectures
I think u probably need to have a PHD already to be eligible to follow this guy!
You don't need a PhD, but you do need prior knowledge of Laplace transforms.
For example, you need to know that the *s* parameter is itself a complex number (a + bi), so that you re not confused when he talks about the Real part of *s*
@@dozog but why only concerned with the real part of s not its imaginary part
@@rajeshsalvi8249 ... So that you are not confused when he talks about either the real or the imaginary part of *s*
@@dozog I mean when at 4:56 he said Re(α)>-1 and then Re(s-a)>0,why only real part matter here not the imaginary part?can imaginary part can take any value here?
Ya
EXCELLENT SIR
Sir, what is the Laplace transform of u(sin2t) and in general f(g(t))
Please reply sir
Hehe
Good evening Sir,
The amount of CF need to be added dictated by the initial conditions
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
Where can I find the next parts?
why the professor is mentioning "dt" before the function ?
Because he s old.
It's just his preferred notation. There is literally no difference.
@@dozog I feel like it's a sort of philosophy. ∫f(x)dx is like parentheses around f(x), whereas ∫dx f(x) is like an operator acting on f(x). Also in this form if you have lots of integrals it's easier to see what is integrated over like ∫ dx dz dy dz dw G(x-y)G(y-w)G(w-z)G(z-x), which happens in Quantum Field Theory all the time when calculating Feynman diagrams. When writing down those integrals for the diagrams it's also nice to first write down the variables that are integrated over and then think of what the integrand looks like and writing it down.
The camera was moving too fast!
Ty very much sir
I need Laplace's equation in spheical polar cordinate. Plz send me solutions on my email id. Plz sir
show the board, not the professor
Khalil Idiab Yeah man but during a lecture, we tend to look at the professor a lot, the body language comes into play and not much was written anyway.
Aadityakiran S.
thats kinda true.. but sometimes what is written is not shown at all, and having upped the playback speed i got a little dizzy on this one. Too much unnecessary movement.
That being said i regret writing such a mean comment, now im just thankful they actually recorded this genius. Also the camera work is still better than most lectures on youtube.
that being said i still think there is too much zoom.
K I fuck off idiot 😂😂😂
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"
Shut the fuck up. Take your Jesus elsewhere. Otherwise Laplace will kick his ass.
Hi, it’s me Jesus. Sorry I am late. How can I help?
Show the board not the professor
you can also say,"why even this video lacture is here,we have got books of our own. we dont need this".
look at board not the teacher
My brain too dumb to understand
35:22 They're not independent... independent and mutually exclusive events must have 0% or 100% probability.
They are.
Dhyv
wtf...he just directly writes down the formulas and blabber something..no proofs for anything
This is a Physics lecture, not a Mathematics lecture, you want proof, you see a Mathematician's treatment.
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.
Look at the basic mechanical system comprising of a spring, a mass and a dampener and the analogous electrical system of a inductance, capacitance and resistance.
Prof balki is not that type of someone.. besides he's a physicsist, this is the course, mathematical methods in physics just tp show the mathematics that's required to..
not so good... just blabbering
It sucked big time