# Tagged Questions

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### Fourier Inversion transform

Is what i have highlighted in green a typo? Should it be $\pi e^{-|\xi|}$?
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### Poles of Fourier transform

Let $f\in L_2(\mathbb R_+)$ and consider its Fourier transform $$F(\zeta)=\int_0^\infty f(x)e^{ix\zeta}dx$$ Is it true that analytic continuation of $F(\zeta)$ has at most finitely many poles in a ...
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### Paley-Wiener theorem for a sector $\{\zeta:-\epsilon<\arg(\zeta)<\pi+\epsilon\}$

One of the variations of the Paley-Wiener theorem yields: If $f\in L^2(\mathbb R_+)$, then the Fourier transform $F$, defined by $$F(\zeta)=\int_0^\infty f(x)e^{ix\zeta}dx$$ is a holomorphic function ...
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### Fourier transform and inverse transorm

I have to prove that for $f\in L^1(\mathbb{R})$ $$\check{\hat{f}}=\hat{\check{f}},$$ where $$\hat{f}(\xi):=\int\limits_{\mathbb{R}}e^{-i\xi x}f(x)\mathrm{d}x$$ and ...
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### Find an harmonic function in $\mathbb R^n$ which is a polynomial of degree 4 and is 1 at the origin

Find an harmonic function in $R^n$ which It is a polynomial of degree 4 and is =1 at the origin. It is a polynomial of degree 5 and its partial derivatives are both =0 at the origin. Important ...
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### Dirichlet problem in the disk: behavior of conjugate function, and the effect of discontinuities

Dirichlet's problem in the unit disk is to construct the harmonic function from the given continuous function on the boundary circle. It is solved by the convolution with the Poisson kernel, and we ...
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### Showing that Gaussians are eigenfunctions of the Fourier transform

I'm having a bit of trouble on this problem: I've tried to evaluate the integral directly (using the trick from multivariable calculus where you "square" the integral and convert to polar ...
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### asymptotical behavior of integral

I'm interest in the asymptotical of $$\int_{-\pi}^{\pi}\exp\Big((\cos z+i\alpha\sin z-1)t\Big)dz\hspace{3mm}\text{as}\hspace{2mm}t\to\infty$$ for $-1<\alpha<1$. Numberical result suggest that ...
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### Proving that two functions involving integrals with Legendre polynomials are equal

I have two functions that I expect to be equal (where $P_{2l}$ are the even Legendre Polynomials): $$F_{2l}(x)=x\, \tanh(\pi x/2)\left|\int_0^1 u^{i x-1}P_{2l}(u)\,du\right|^2$$ ...
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### Solution of definite integral of product of bessel function and exponential

I have an integral $I=\int_{\theta} \int_r J_m(k_1r)e^{-j[P_x r \cos(\theta)+P_y r \sin(\theta)]} r dr d\theta$ $0\leq\theta\leq2\pi; r<\infty$ is there any method to solve this?
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### Complex Integration: theorem of Residues

I am trying to prove a theorem that is doing my head in a bit. I have tried to simplify the problem as much as possible and leave out the details, even though it might look a bit too big. The ...
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### Fourier series without Fourier analysis techniques

It is known that one can sometimes derive certain Fourier series without alluding to the methods of Fourier analysis. It is often done using complex analysis. There is a way of deriving the formula ...
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### an “alternate derivation” of Poisson summation formula and discrete Fourier transformation

Inspired by this post, I am trying to do a derivation of a Poisson summation formula. My starting point is this: $$\frac{1}{2\pi} \int^{\infty}_{-\infty} e^{i k x} dx=\delta(k)$$ I simply wish ...
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### A naive example of discrete Fourier transformation

We know a discrete Fourier transformation with discrete $n$ and continuous $x_1,x_2$: $$\sum_{n\in\mathbb{Z}} e^{-in(x_1-x_2)\frac{2\pi}{L}}=L\delta(x_1-x_2)$$ with Dirac delta function $\delta$. ...
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### a generalization of normal distribution to the complex case: complex integral over the real line

How to prove $\int_{\mathbb{R}} e^{-\frac{(x+it)^2}{2}}dx=\sqrt{2\pi}$ for any $t\in \mathbb{R}$? I only obtained the case that $t=0$, $\int_{\mathbb{R}} e^{-\frac{x^2}{2}}dx=\sqrt{2\pi}$. Thanks.
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### Solve $\mathscr{F}^{-1} [ \cot{a \omega} \times \mathscr{F} \{ U(t) \sin{\omega_0 t} \} ]$ using contour integration

I wish to evaluate $y(t) = \mathscr{F}^{-1} [ \cot{a \omega} \times \mathscr{F} \{ U(t) \sin{\omega_0 t} \} ]$, where $\mathscr{F}$ represents the Fourier transform, and U(t) represents the ...
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### compute the derivative of a function defined by an integration over the whole real line

Let $h(t)=\int _{\mathbb{R}}e^{-\frac{(x+it)^2}{2}}d\lambda(x)$, where $\lambda$ is the Lebesgue measure. I want to prove that $h$ is differentiable and compute the derivative of $h$: ...
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### Real part of a holomorphic function is bounded by polynomial then the holomorphic function is a polynomial [duplicate]

Let $u$ be a harmonic function on $\mathbb{R}^2\cong \mathbb{C}$ such that $Ref= u$ where $f$ is an entire function. If $|u(z)|\leq |z|^n$ for any $z\in\mathbb{C}$, then $f$ is a polynomial of degree ...
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### Multiplying and Dividing Series

For example, how do you compute the taylor series for $$e^x \sin x=\sum_{n=0}^{\infty} \frac {x^n}{n!} \sum_{n=0}^{\infty} (-1)^n\frac {x^{2n+1}}{(2n+1)!}$$ Of course I want the result to contain ...
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### conflictions of analytic functions to the boundary and Schwarz reflection principle

Let $\Omega$ be an open subset of $\mathbb{C}$ and $f:\Omega\longrightarrow \mathbb{C}$ be a holomorphic function. Then for any $z\in \Omega$ and any $r>0$ such that $D(z,r)\subseteq \Omega$, $f$ ...
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### On $\int_{-\infty}^{+\infty} {\frac{\tan(t-t_0)}{\cosh^2(t-t_0)} \cos(\omega t) \,\mathrm{d}t}$

How to count this? $$\int_{-\infty}^{+\infty} {\frac{\tan(t-t_0)}{\cosh^2(t-t_0)} \cos(\omega t) \,\mathrm{d}t}$$ Can we use residue formula?
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### Integral $\int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx$

Hi I'm trying to solve this integral Fourier Transform $$\int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx=\sqrt{2\pi|k|}(1+i) (-1+\text{sgn}(k))$$ where sgn(k)$=1$ for k>1 and $-1$ for k<1. I am ...
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### Require help with the convolution of two complex conjugates

I need to find the convolution of the following two functions: When rationalizing the denominator, the numerators become complex conjugates of each other. I have tried obtaining the Fourier ...
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### Need help with the convolution of two complex functions

Could someone start me off with how to find the convolution of these two functions? Using the normal equation for convolution seems impossible as a common overlap interval is required for ...
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### Uniform Boundedness in N of $\sum\nolimits_{0<\mid n \mid \leq N} \frac{\mathrm{e}^{inx}}{n}$

Show that $\int_0^\infty \frac{\sin(x)}{x}\,\mathrm{d}x = \frac{\pi}{2}$, and using that show that $\sum\nolimits_{0<\mid n \mid \leq N} \frac{\mathrm{e}^{inx}}{n}$ is uniformly bounded in N and ...
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I'm reading about Fourier analysis and there is one equality, which I don't understand. Why does: $$\left\|\sum_{i=1}^{N}\langle f,\phi_i\rangle\phi_i\right\|^2 = \sum_{i=1}^{N}|\langle ... 1answer 78 views ### Laplace transform of g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases} Find Laplace transform for this function "g"$$g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$$Then Take advantage of it to calculate the following ... 1answer 69 views ### Inhomogeneous diffusion equation and initial conditions inversion While working on a physical diffusion process, I encountered the following Fokker-Planck equation$$ \frac{\partial F}{\partial t} = D (x) \frac{\partial^2 F}{\partial x^2} \tag1$$where D(x) > ... 1answer 45 views ### Fourier Transform Identity I'm trying to verify the following:$$ \int_{\mathbb{R}} e^{-z\xi^2} \hat{f}(\xi) \; d\xi = \sqrt{\frac{\pi}{z}} \int_{\mathbb{R}} e^{-\pi^2x^2/z} f(x) dx, $$for z = \alpha i purely imaginary and ... 1answer 61 views ### Fourier transform of \frac{d}{dt}\ln\frac{1}{it} I'd like to proove the identity$$\mathcal{F}\left(\frac{d}{dt}\ln\frac{1}{it}\right)=2i\pi H$$with H=\mathbb{I}_{\mathbb{R}^+} ie the Heaviside step function, \mathcal{F} denote the Fourier ... 0answers 20 views ### Does there exists f\in L^{1}(\mathbb R)\cap FL^{1}(\mathbb R)(=Fourier algebra) but |f|\not \in A(\mathbb R)? For f\in L^{1}(\mathbb R); We define the Fourier transform of f as follows:$$\hat{f}(\xi)= \int_{\mathbb R}f(x) e^{-2\pi i \xi x}dx; \ \text {for} \ \xi \in \mathbb R.$$Consider a Fourier ... 1answer 44 views ### Complex Fourier integral Why is the \omega in the solution for this integral written in absolute value?$$\int_{-\infty}^{\infty} \frac{x e^{i\omega x}}{(x^2+1)^2}dx = \frac{\pi \omega}{2}e^{-|\omega|}$$0answers 59 views ### Is the Fourier series also a Laurent series? For a holomorphic function f(z) = \sum^{\infty}_{n=-\infty}{a_nz^n}, the substitution z = e^{j\pi\omega} yields the Fourier series f(z) = \sum^{\infty}_{n=-\infty}{a_ne^{j\pi\omega n}}. Would it ... 2answers 122 views ### A contour integral for Fourier transform How does one show the following, preferably with contour integral on the complex plane?$$\frac{\Gamma(\alpha)}{2\pi}\int_{-\infty}^\infty (ik)^{-\alpha}e^{-ikx}dk = (-x)_+^{\alpha-1},$$where x is ... 1answer 120 views ### Is Fourier transform of a L^{1} integrable function is L^{1} integrable? Let f:\mathbb R \to \mathbb R such that$$f(x)= \frac{\sin \pi x}{x (x^{2}-1)}$$for x\in \mathbb R - \{ 0, -1, 1 \} and f(x):= \pi  for x=0 and f(x)=-\frac{\pi}{2} for x= -1, 1. Let ... 1answer 46 views ### Complex Fourier Transform Quick question, say I have a complex series of data f(t), so that at each data point t_i have a real and imaginary number, is it correct to calculate the power spectrum of that series (so I want ... 2answers 62 views ### How to estimate (compute) Fourier transform? Let f:\mathbb R \to \mathbb R such that$$f(x)= \frac{\sin \pi x}{x (x^{2}-1)}$$for x\in \mathbb R - \{ 0, -1, 1 \} and f(x):= \pi  for x=0 and f(x)=-\frac{\pi}{2} for x= -1, 1. ... 1answer 152 views ### Contour Integral of Exponential I want to show the following for a > 0:$$e^{-a} = \int_{0}^{\infty}{\frac{e^{-x}}{\sqrt{x}}e^{-a^{2}/(4x)}dx}.
I am really struggling with this one. Any help is welcome! For equation $f''(z) + p(z) f'(z) + q(z) f(z) = 0$, where $p(z)$ and $q(z)$ are fixed polynomials. Given $f(0)=f_0$, $f'(0)=f_1$, prove that ...
My question is if it is possible to find a compactly supported smooth function $\varphi:\mathbf{R}\to \mathbf{R}$ s.t. the following integration $\int_{\mathbf{R}}\varphi(t)e^{itx}e^{tx}dt$ stays ...