# Tagged Questions

This covers all transformations of functions by integrals, including but not limited to the Radon, X-Ray, Hilbert, Mellin transforms. Use (wavelet-transform), (laplace-transform) for those respective transforms, and use (fourier-analysis) for questions about the Fourier and Fourier-sine/cosine ...

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### Connection between Fourier transform and Taylor series

Both Fourier transform and Taylor series are means to represent functions in a different form. My question: What is the connection between these two? Is there a way to get from one to the other (and ...
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### Fourier transform of $\text{sinc}^3 {\pi t}$

$$f(t)=\frac{\sin^3(\pi t)}{(\pi t)^3}$$ I want to calculate the Fourier transform. I can't calculate this integral: $$\int_0^\infty\frac{\sin^3(\pi t)}{(\pi t)^3}\cos(ut)\,\mathrm{d}t$$
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### Solving an initial value ODE problem using fourier transform

I am a physics undergrad and studying some transform methods. The question is as follows: $y^{\prime \prime} - 2 y^{\prime}+y=\cos{x}\,\,\,\,y(0)=y^{\prime}(0)=0\,\,\, x>0$ I am having some ...
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### Uniqueness for Integral Transform

What can be said about the uniqueness of the following integral transformation: $(Tf)(u) = \int_0^{\infty} f(t)G(tu)dt$ defined for all $u\geq 0$, where the kernel $G(z) \in [0,1]$ for all $z\geq0$,...
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### fancy about inverse discrete Fourier sine and cosine transform (i.e. Fourier sine and cosine series)

In order to find $f(x)$ so that $F(u)=\sum\limits_{x=0}^\infty f(x)\sin\dfrac{\pi ux}{L}$ and $F(u)=\sum\limits_{x=0}^\infty f(x)\cos\dfrac{\pi ux}{L}$ , we can borrow the idea from Fourier sine ...
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### Multiple Fourier Integrals involving Heaviside Theta Function

I want to evaluate the integral: $$I=\int_{-\infty}^{\infty}dx_1 \int_{-\infty}^{\infty}dx_2 \ \Theta(x_1-x_2) \ e^{i(ax_1+bx_2)}$$ where $\Theta(x)$ is the Heaviside function. What I was doing now ...
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### trouble with non-homogeneous ODE system… which method shall I use?

I am an undergrad statistics student and I am having troubles with non-homogeneous ODE systems. During my classes I went over just three methods for solving odes: Laplace transform, Fourier transform ...
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### Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z…: unified treatment of transforms?

I understand "transform methods" as recipes, but beyond this they are a big mystery to me. There are two aspects of them I find bewildering. One is the sheer number of them. Is there a unified ...
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### Laplace transformations for dummies

Is there a simple explanation of what the Laplace transformations do exactly and how they work? Reading my math book has left me in a foggy haze of proofs that I don't completely understand. I'm ...
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### Example for finite dimensional analog of integral transforms

I understand that integral transforms are generalisations of the dot product of functions that could be interpreted as infinite dimensional vectors. The most significant advantage then is that ...
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### What is the easiest known expression for inverse of Laplace transform?

There is a very simple expression for the inverse of Fourier transform. What is the easiest known expression for the inverse Laplace transform? Moreover, what is the easiest way to prove it?
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### Fourier transform convention: $\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x)e^{\pm ikx}dx$?

I've come across the Fourier transform being defined as: $$\tilde{f}(k)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x)e^{ikx}dx$$ But this convention is not present in the Wikipedia article. The ...
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### Series around $s=1$ for $F(s)=\int_{1}^{\infty}\text{Li}(x)\,x^{-s-1}\,dx$

Consider the function $$F(s)=\int_{1}^{\infty}\frac{\text{Li}(x)}{x^{s+1}}dx$$ where $\text{Li}(x)=\int_2^x \frac{1}{\log t}dt$ is the logarithmic integral. What is the series expansion around $s=1$? ...
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### An inequality involving arctan of complex argument

I have the following conjecture: $$\text{Re}\left[(1+\text{i}y)\arctan\left(\frac{t}{1+\text{i}y}\right)\right] \ge \arctan(t), \qquad \forall y,t\ge0.$$ Which seems to be ...
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### Prove that the Laplace transform of $I_0\left(2\sqrt t\right)$ is $\exp\left(1/s\right)/s$

Wolfram Alpha gave me the answer to this, but unfortunately Wolfram Alpha doesn't show its work, I can't find a proof anywhere else, and my feeble attempts to show it myself went nowhere. How can it ...
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### Decaying Fourier transform and smoothness

Suppose that $f\in L^1 (\mathbb{R})$ and that for any $n\in \mathbb{N}$ there is $C_n > 0$ such that its Fourier transform satisfies $$|\hat{f}(\xi )| \le C_n(1+|\xi |^2)^{-n}.$$ I want to show ...
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### Evaluating improper integrals using laplace transform

I want to calculate the following improper integral using Laplace and transforms (and laplace transforms only). $$\int_0^{\infty} x e^{-3x} \sin{x}\, dx$$ I propose the following method. I plan to ...
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### Consistency/range conditions for (integral) transform mapping into higher-dimensional space

I am interested in learning more about what the (formal) implications are when transforming a $n$-dimensional function space (e.g., the space of all $\mathbb{R}^n\to\mathbb{R}$) to a higherdimensional ...
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### Bilateral Laplace transform

My knowledge of Bilateral Laplace transform is less. Here are the few questions I need answer. What is the condition for existence of bilateral Laplace transform? How is the condition for existence ...
### Smoothness of Fourier transform of $\frac{1}{|x|^p}$
Consider the "function" (more precisely it is a tempered distribution) given by $f : \mathbb{R}^n \to \mathbb{R}$, $f(x) = \frac{1}{|x|^p}$, where $0 < p < n$. It can be calculated that the ...