What is Transform: Definition and 1000 Discussions

In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable


{\displaystyle t}
(often time) to a function of a complex variable


{\displaystyle s}
(complex frequency). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms linear differential equations into algebraic equations and convolution into multiplication.For suitable functions f, the Laplace transform is the integral








{\displaystyle {\mathcal {L}}\{f\}(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt.}

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  1. durandal

    Engineering Fourier transform of a shifted sine wave

    This is my attempt at a solution. I have used Eulers formula to rewrite the sine function and then used the Fourier transform of complex exponentials. My solution is not correct and I don't understand if I have approached this problem correctly. Please help. $$ \mathcal{F}\{\sin (4t-4) \} =...
  2. C

    Use the Shifting Theorem to find the Laplace transform

    For (b), I'm confused on the highlighted step. Does someone please explain to me how they got from the left to the right? Thanks!
  3. C

    Laplace transform of cosine squared function

    For part (b), I have tried finding the Laplace transform of via the convolution property of Laplace transform. My working is, ##L[\cos^2 (2t)] = L[\cos 2t] * L[\cos 2t]## ##L[\cos^2 (2t)] = \frac{s}{s^2 + 4} * \frac{s}{s^2 + 4}## ##\int_0^t \frac{s^2}{(s^2 + 4)^2} dt = \frac{ts^2}{(s^2 +...
  4. LCSphysicist

    Fourier transform of triangle diagram

    OBS: Ignore factors of ## (2 \pi) ##, interpret any differential ##dx,dp## as ##d^4x,d^4p##, ##\int = \int \int = \int ... \int##. I am using ##x,y,z## instead of ##x_i##. Honestly, i am a little confused how to show this "triangle-star duality". Look, the propagators in positions space gives...
  5. flyusx

    Fourier Transform - Solutions Error?

    This is technically a Fourier transform of a quantum function, but the problem I'm having is solely mathematical. Conducting this integral is relatively straightforward. We can pull the square roots out since they are constants, rewrite the bounds of the integral to be from ##-a## to ##a##...
  6. T

    I Fourier Transform of Photon Emission Hamiltonian

    Hey all, I just wanted to double check my logic behind getting the Fourier Transform of the following Hamiltonian: $$H(x) = \frac{ie\hbar}{mc}A(x)\cdot\nabla_{x}$$ where $$A(x) = \sqrt{\frac{2\pi\hbar c^2}{\omega L^3}}\left(a_{p}\epsilon_{p} e^{i(p\cdot x)} + a_{p}^{\dagger}\epsilon_{p}...
  7. Sciencemaster

    I Gravitational Field Transformations Under Boosted Velocity

    Let's say we have some observer in some curved spacetime, and we have another observer moving relative to them with some velocity ##v## that is a significant fraction of ##c##. How would coordinates in this curved spacetime change between the two reference frames? For example, imagine a...
  8. T

    Can Left Handed Up Quarks Transform Into Right Handed Ones?

    Isospin must be transformable if it is the case. In which case there must be additional Bosons for doing the transformation.
  9. M

    I Laplace Transform of Sign() or sgn() functions

    Trying to model friction of a linear motor in the process of creating a state space model of my system. I've found it easy to model friction solely as viscous friction in the form b * x_dot, where b is the coefficient of viscous friction (N/m/s) and x_dot represents the motor linear velocity...
  10. redtree

    I Properties of the Fourier transform

    I was wondering if the following is true and if not, why? $$ \begin{split} \hat{f}_1(\vec{k}) \hat{f}_2(\vec{k}) &= \hat{f}_1(\vec{k}) \int_{\mathbb{R}^n} f_2(\vec{x}) e^{-2 \pi i \vec{k} \cdot \vec{x}} d\vec{x} \\ &= \int_{\mathbb{R}^n} \hat{f}_1(\vec{k}) f_2(\vec{x})...
  11. S

    Fourier transform of ##e^{-a |t|}\cos{(bt)}##

    First, ##\tilde{f}(\omega)=\int_{-\infty}^{\infty}e^{a|t|}\cos(bt)e^{-i\omega t} \mathrm{d}t## We can get rid of the absolute value by splitting the integral up ##\int_{-\infty}^{0}e^{at}\cos(bt)e^{-i\omega t} \mathrm{d}t+ \int_{0}^{\infty}e^{-at}\cos(bt)e^{-i\omega t} \mathrm{d}t## Using...
  12. J

    Does the Galilean transform rely on 2 events?

    From my limited understanding the Galilean transform has 2 frames but 4 four perspectives. For example x is the stationary frame when using ## ∆x = ∆x′ + v ∆t ## and x' is moving. When using ## ∆x' = ∆x - v ∆t ## and x' is stationary and x is moving. Now lets use the example of ## ∆x = ∆x′ + v...
  13. H

    Solving ##y'' - 5 y' - 6y = e^{3x}## using Laplace Transform

    We have to solve $$ \begin{align*} y'' - 5y' - 6y = e^{3x} \\ y(0) = 2,~~ y'(0) = 1 \\ \end{align*} $$ Applying Laplace Transform the equation $$ \begin{align*} L [ y''] - 5 L[y'] - 6 L[y] = L [ e^{3x} ] \\ s^2 Y(s) - \left( s y(0) + y'(0) \right) - 5s Y(s) + y(0) - 6 Y(s) = \frac{1}{s-3} \\...
  14. L

    A Applying the Laplace transform to solve Differential equations

    Is it possible to apply Laplace transform to some equation of finite order, second for instance, and get the differential equation of infinite order?
  15. S

    Fourier transform of wave packet

    I am unsure if ##h(x,t)## really is a wave packet, but it looks like one, hence the title. Anyway, so I'd like to determine ##\hat{h}(k,t=0)##. My attempt so far is recognizing that, without the real part in the integral, i.e. ##g(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} a(k)e^{i(kx-\omega...
  16. dRic2

    I Transform a 2x2 matrix into an anti-symmetric matrix

    Hi, I have a 2x2 hermitian matrix like: $$ A = \begin{bmatrix} a && b \\ -b && -a \end{bmatrix} $$ (b is imaginary to ensure that it is hermitian). I would like to find an orthogonal transformation M that makes A skew-symmetric: $$ \hat A = \begin{bmatrix} 0 && c \\ -c && 0 \end{bmatrix} $$ Is...
  17. greg_rack

    Check on proof for property of the Laplace transform

    Could someone check whether my proof for this simple theorem is correct? I get to the result, but with the feeling of having done something very wrong :) $$\mathcal{L} \{f(ct)\}=\int_{0}^{\infty}e^{-st}f(ct)dt \ \rightarrow ct=u, \ dt=\frac{1}{c}du, \ \mathcal{L}...
  18. H

    A Polar Fourier transform of derivatives

    The 2D Fourier transform is given by: \hat{f}(k,l)=\int_{\mathbb{R}^{2}}f(x,y)e^{-ikx-ily}dxdy In terms of polar co-ordinates: \hat{f}(\rho,\phi)=\int_{0}^{\infty}\int_{-\pi}^{\pi}rf(r,\theta)e^{-ir\rho\cos(\theta-\phi)}drd\theta For Fourier transforms in cartesian co-ordinates, relating the...
  19. S

    "Simple" Fourier transform problem

    I am unsure about what is being asked for in the question. At first I thought the question asks one to calculate the inverse Fourier transform and then to analyze its depends on ##t##, however, the "estimate" makes me think otherwise.
  20. Salmone

    I Doubts about Fourier transform of IR spectroscopy

    I was studying a Michelson interferometer for infrared absorption in Fourier transform and I've found these two images (taken from https://pages.mtu.edu/~scarn/teaching/GE4250/ftir_lecture_slides.pdf ) representing an infrared monochromatic beam of light going into the interferometer and the...
  21. DuckAmuck

    A Non-unitary gauge transformation

    You see in the literature that the vector potentials in a gauge covariant derivative transform like: A_\mu \rightarrow T A_\mu T^{-1} + i(\partial_\mu T) T^{-1} Where T is not necessarily unitary. (In the case that it is ##T^{-1} = T^\dagger##) My question is then if T is not unitary, how is...
  22. K

    A Fourier Transform MW spectroscopy in a FB cavity

    Hello! I am reading about Fourier Transform MW spectroscopy in a FB cavity, which seems to be quite an old technique and I want to make sure I got it right. As far as I understand, this is very similar to normal NRM, i.e. one applies a MW ##\pi/2## pulse which puts the molecules in a linear...
  23. A

    Hilbert Transform, Causality, PI Controller

    I was told that PI controller is a causal filter, and has frequency response represented by H(w) = Ki/(iw)+ Kp. I was also told that causal filter should satisfy this relationship H(w) = G(w) -i G_hat(w) where G_hat(w) is the Hilbert transform of G(w). Does this mean that we cannot freely...
  24. Salmone

    I How Does Fourier Transform Analyze Beats in Signals?

    What is the Fourier transform of a beat? For example, I want to calculate the Fourier transform of the function ##f(t)=\cos((\omega_p+\omega_v) t)+\cos((\omega_p-\omega_v)t),## where ##$\omega_p+\omega_v=\Omega,\space\omega_p-\omega_v=\omega## and ##\Omega\simeq\omega.## I think it is equal to...
  25. F

    Fourier transform: duality property and convolution

    Hello, First of all, I checked several other threads mentioning duality, but could not find a satisfying answer, and I don't want to revive years old posts on the subject; if this is bad practice, please notify me (my apologies if that is the case). For the past few days, I have had a lot of...
  26. D

    Fourier transform ##f(t) = te^{-at}##

    Doing the Fourier transform for the function above I'm getting a result, but since I can't get the function f(t) with the inverse Fourier transform, I'm wondering where I made a mistake. ##F(w) = \frac{1}{\sqrt{2 \pi}} \int_{0}^{\infty} te^{-t(a + iw)} dt## By integrating by part, where G = -a...
  27. R

    Galilean transform and Lorentz transform questions

    I have a quick question about the Galilean transform. If I have Alice running and Bob stationary. The Galilean transform will tell me the position of Alice from Bob's stationary position. Also if I have Alice's position which is moving it will tell me Bob's stationary position. If I want Bob...
  28. H

    Mellin transform of Dirac delta function ##\delta(t-a)##

    Hi, I found Laplace transform of this Dirac delta function which is ##F(s) = e^{-st}## since ##\int_{\infty}^{-\infty} f(t) \delta (t-a) dt = f(a)## and that ##\delta(x) = 0## if ##x \neq 0## Then the Mellin transform ##f(t) = \frac{1}{2 \pi i} \int_{\gamma - i \omega}^{\gamma +i \omega}...
  29. B

    Python Sklearn LabelEncoder inverse transform

    Hi everyone! I need to inverse an label transform with sklearn. I found this example on web: from sklearn.preprocessing import LabelEncoder np.random.seed(1) y = np.random.randint(0, 2, (10, 7)) y = y[np.where(y.sum(axis=1) != 0)[0]] array([[1, 1, 0, 0, 1, 1, 1], [1, 1, 0, 0, 1, 0...
  30. A

    Engineering I/Q of Signals and Hilbert Transform

    Hello, would anyone be willing to provide help to the following problem? I can find the Fourier Transform of the complex envelope of s(t) and the I/Q can be found by taking the Real and imaginary parts of that complex envelope, but how can I approach the actual question of finding the carrier...
  31. L

    I Laplace transform of a simple equation (Simple question)

    Lets consider very simple equation ##x''(t)=0## for ##x(0)=0##, ##x'(0)=0##. By employing Laplace transform I will get s^2X(s)=0 where ##X(s)## is Laplace transform of ##x(t)##. Why then this is equivalent to X(s)=0 why we do not consider ##s=0##?
  32. L

    Inverse Laplace transform

    \mathcal{L}^{-1}[\frac{e^{-5s}}{s^2-4}]=Res[e^{-5s}\frac{1}{s^2-4}e^{st},s=2]+Res[e^{-5s}\frac{1}{s^2-4}e^{st},s=-2] From that I am getting f(t)=\frac{1}{4}e^{2(t-5)}-\frac{1}{4}e^{-2(t-5)}. And this is not correct. Result should be f(t)=\theta(t-5)(\frac{1}{4}e^{2(t-5)}-\frac{1}{4}e^{-2(t-5)})...
  33. L

    A Laplace transform of derivatives

    I have a question regarding Laplace transforms of derivatives \mathcal{L}[f'(t)]=p\mathcal{L}[f(t)]−f(0^−) Can anyone explain me why ##0^-##?
  34. Swordwhale

    Fourier transform radial component of magnetic field

    Hello everybody! I have a question concerning the Fourier transformation: So far I have experimentially measured the magnetic field of a quadrupole but as the hall effect sensor had a fixed orientation I did two series, one for the x, one for y component of the magnetic field, I have 50 values...
  35. F

    Engineering Transform question about this circuit

    I Normally can solve these but when Vo is embeded into the circuit, is throwing me for a loop (no pun is intended). My question is, if I apply Kirchoff's voltage law: VR+VC+VL-Vi=0, I'm not quite sure how this would be set up when Vo is between R1R2 & C1C2? I believe this ia 4 Loop circuit: I1 =...
  36. M

    I Transform from polar to cartesian

    Probability distribution - uniform on unit circle. In polar coordinates ##dg(r,a)=\frac{1}{2\pi}\delta(r-1)rdrda##. This transforms in ##df(x,y)=\frac{1}{2\pi}\delta(\sqrt{x^2+y^2}-1)dxdy##. The problem I ran into was the second integral was 1/2 instead of 1.
  37. L

    Is the Fourier Transform Correctly Applied in Solving This Laplace Equation?

    I have tried to Fourier transform in ##x## and get the result in the transformed coordinates, please check my result: $$ \tilde{u}(k, y) = \frac{1-e^{-ik}}{ik}e^{-ky} $$ However, I'm having some problems with the inverse transform: $$ \frac{1}{2\pi}\int_{-\infty}^\infty...
  38. LCSphysicist

    How do I know "what Fourier transform" to use?

    Homework Statement:: . Relevant Equations:: . I am having a hard time thinking about Fourier transform, because there are so many conventions that i think i got more confused each time i think about it. See an example, "Find the Fourier transform of $$V(t) = Ve^{iwt} \text{ if } nT \leq t...
  39. S

    MHB Fourier Transform Help: Issues Solving for a & b

    Hello again. Having some issues on Fourier transform. Can someone please tell me how to proceed? Need to solve this then use some software to check my answer but how to solve for a and b. Plzz help
  40. redtree

    I Fourier transform of a function in spherical coordinates

    I am trying to understand the relationship between Fourier conjugates in the spherical basis. Thus for two functions ##f(\vec{x}_3)## and ##\hat{f}(\vec{k}_3)##, where \begin{equation} \begin{split} \hat{f}(\vec{k}_3) &= \int_{\mathbb{R}^3} e^{-2 \pi i \vec{k}_3 \cdot \vec{x}_3} f(\vec{x}_3...
  41. Joan Fernandez

    A Why is the MGF the Laplace transform?

    The Laplace transform gives information about the exponential components in a function, as well as oscillatory components. To do so there is a need for the complex plane (complex exponentials). I get why the MGF of a distribution is very useful (moment extraction and classification of the...
  42. tworitdash

    A Fourier Transform of an exponential function with sine modulation

    I want to know the frequency domain spectrum of an exponential which is modulated with a sine function that is changing with time. The time-domain form is, s(t) = e^{j \frac{4\pi}{\lambda} \mu \frac{\sin(\Omega t)}{\Omega}} Here, \mu , \Omega and \lambda are constants. A quick...
  43. A

    Help with Haar wavelet transform problems?

    Attempt: This is where I am having difficulties understanding and getting a start at a solution. I get a lot of definitions and yet no worked examples. The plot will be constant piece-wise functions, as I understand, but I am having trouble visualizing. Any guidance is much appreciated.
  44. A

    I Understanding the Laplace Transform of cos(t)/t

    So, I know the direct definition of the Laplace Transform: $$ \mathcal{L}\{f(t) \} = \int_0^\infty e^{-st}f(t)dt$$ So when I plug in: $$\frac{\cos(t)}{t}$$ I get a divergent integral. however:https://www.wolframalpha.com/input/?i=+Laplace+transform+cos%28t%29%2F%28t%29 is supposed to be the...
  45. H

    Fourier transform to solve PDE (2nd order)

    I just want to make sure I am on the right track here (hence have not given the other information in the question). In taking the Fourier transform of the PDE above, I get: F{uxx} = iω^2*F{u}, F{uxt} = d/dt F{ux} = iω d/dt F{u} F{utt} = d^2/dt^2 F{u} Together the transformed PDE gives a second...
  46. H

    What is the Inverse Laplace Transform of e^(-sx^2/2)?

    My attempt at finding this was via convolution theorem, where we take F(s) = 1/s^2 and G(s) = e^(-sx^2/2). Then to use convolution we need to find the inverses of those transforms. From a table of Laplace transforms we know that f(t) = t. But I am sort of struggling with e^(-sx^2/2). My 'guess'...
  47. F

    A Fourier transform and Cosmic variance - a few precisions

    I cite an original report of a colleague : 1) I can't manage to proove that the statistical error is formulated like : ##\dfrac{\sigma (P (k))}{P(k)} = \sqrt{\dfrac {2}{N_{k} -1}}_{\text{with}} N_{k} \approx 4\pi \left(\dfrac{k}{dk}\right)^{2}## and why it is considered like a relative error ...
  48. M

    MHB Inverse laplace transform pf infinite product

    I have to do inverse laplace transform of infinite product that is shown below. Can somebody help me with that?
  49. C

    I Proof check for Plancherel's theorem (Fourier Transform version)

    I'm trying to prove Plancherel's theorem for functions $$f\in L^1\cap L^2(\mathbb{R})$$. I've included below my attempt and I would really appreciate it if someone could check this for me please, and give me any feedback they might have. **Note:** I am working with a slightly different...