I never found a text book where Laplace and fourier transform is described rigorously. The domain of laplace transform was never clear to me. There are null functions whose laplace transform is 0. Then how is Laplace transform invertible? What are the theorems that cover the topic of region of convergence? I want a book material or course that covers laplace transform rigouruosly using riemann integrals,Compleax analysis, measure theory, where the definitions are concrete and clear. Also it would be very helpful if u can suggest me a book describing delta function and z-transform rigorously. Thank you.
2026-04-25 06:47:38.1777099658
Why does the mathematics of Laplace transform and fourier transform look so dodgy and non rigouruos and not well formulated?
238 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in COMPLEX-ANALYSIS
- Minkowski functional of balanced domain with smooth boundary
- limit points at infinity
- conformal mapping and rational function
- orientation of circle in complex plane
- If $u+v = \frac{2 \sin 2x}{e^{2y}+e^{-2y}-2 \cos 2x}$ then find corresponding analytical function $f(z)=u+iv$
- Is there a trigonometric identity that implies the Riemann Hypothesis?
- order of zero of modular form from it's expansion at infinity
- How to get to $\frac{1}{2\pi i} \oint_C \frac{f'(z)}{f(z)} \, dz =n_0-n_p$ from Cauchy's residue theorem?
- If $g(z)$ is analytic function, and $g(z)=O(|z|)$ and g(z) is never zero then show that g(z) is constant.
- Radius of convergence of Taylor series of a function of real variable
Related Questions in FOURIER-ANALYSIS
- An estimate in the introduction of the Hilbert transform in Grafakos's Classical Fourier Analysis
- Verifying that translation by $h$ in time is the same as modulating by $-h$ in frequency (Fourier Analysis)
- How is $\int_{-T_0/2}^{+T_0/2} \delta(t) \cos(n\omega_0 t)dt=1$ and $\int_{-T_0/2}^{+T_0/2} \delta(t) \sin(n\omega_0 t)=0$?
- Understanding Book Proof that $[-2 \pi i x f(x)]^{\wedge}(\xi) = {d \over d\xi} \widehat{f}(\xi)$
- Proving the sharper form of the Lebesgue Differentiation Theorem
- Exercise $10$ of Chapter $4$ in Fourier Analysis by Stein & Shakarchi
- Show that a periodic function $f(t)$ with period $T$ can be written as $ f(t) = f_T (t) \star \frac{1}{T} \text{comb}\bigg(\frac{t}{T}\bigg) $
- Taking the Discrete Inverse Fourier Transform of a Continuous Forward Transform
- Is $x(t) = \sin(3t) + \cos\left({2\over3}t\right) + \cos(\pi t)$ periodic?
- Translation of the work of Gauss where the fast Fourier transform algorithm first appeared
Related Questions in LAPLACE-TRANSFORM
- Solution to ODE with Dirac Delta satisfies ODE
- Calculating an inverse Laplace transform
- Laplace Transform working out
- How to solve the integral equation $f(x) = \int_0^x f(x-y)k(x,y)dy+g(x)$ for $f(x)$?
- Laplace Transform for an Initial Value Problem
- Laplace transform of a one-sided full-wave rectified...
- Laplace transform for the solution of a system of differential equations with no constant coefficients
- Question about Dirac comb
- Using Laplace transforms to solve a differential equation
- Prove $\int_0^{\infty} \frac{\cos xt}{1+t^2} dt = \frac{\pi}{2}e^{-x}$ by using Laplace Transform
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
The Fourier transform is a great tool for solving differential equations, which also means that it's often presented as a black box for that purpose without the underlying details. To some extent, that's fine; integration in general is another broadly useful operation that can be presented axiomatically or simply restricted to the continuous category, without going through all the infrastructure to develop it rigorously (and, at least for the Lebesgue integral, there's a substantial bit of infrastructure required). Still, there are quite a few technical issues in defining the Fourier transform, and getting (for example) the relationship between the smoothness of a function $f$ and the decay rate of $\hat f$, requires a bit of care. The sorts of questions in your post--- convergence issues, conditions for invertibility, etc.--- should have answers in any standard book that covers Fourier analysis at, say, the first-year grad student level.
As for specific references, take a look at Dudley's "Real Analysis and Probability," Rudin's "Real and Complex Analysis," or Adams and Guillemin's "Measure Theory and Probability." These books in particular discuss the characteristic function of a random variable, which is a bit different from the more elementary use of the transform to solve ODEs.