# Tagged Questions

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### Regularity of Daubechies wavelet

I am reading the book Wavelets: Theory and applications by A. K. Louis, D. Maass, A. Rieder ...
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### Distribution function and its concavity

Let $\gamma$ be a parameter which is $0<\gamma<4$. Is the following distribution function concave $$F(x)=1-x^{-\gamma}$$
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### comparison of two integrals

Let $n \in N$. How to compare two integrals: $$I_1=\int_0^{\infty}\left(\frac{\sin t}{t}\right)^n dt \quad \text{and} \quad I_2=\int_0^{\pi}\left(\frac{\sin t}{t}\right)^n dt\,\, ?$$ I've beet ...
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### Calculation of integral with Bessel function

I have a trouble with to calculating (or bounding from above) the following integral: $$\int_{-\infty}^{\infty}\left(\frac{J_2(x)}{x^2}\right)^p\, dx, \quad p\geq 1,$$ where $J_2(x)$ is a Bessel ...
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### How to find support of functions

$\textbf{Support}$:$f$ is real valued function with domain $E^n$ the support of $f$ is the smallest closed set $K$ such that $f(x)=0$ for all $x$ is not in $K$ Find the support $(1) f(x)=x-|x|$ ...
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### Integral $\int_0^{\pi}\frac{dx}{a + \cos{x}}$

For $a > 1$ show that $$J = \int\limits_0^{\pi}\frac{dx}{a + \cos{x}}=\frac{\pi}{\sqrt{a^2 -1}}$$ Neither of trivial methods like integration by parts works. ...
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### What is the family / equation of this function?

I'm making up a function and I want to figure out the equation for it so that I can define it continuously. Right now I'm using ExcelGoogle Spreadsheets to define it on a point-by-point basis. I have ...
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### How to solve for $z$ in $\dfrac{xy}{1-x}=(1-z)(x-x^{1/z})$

How to solve the following for $z$: $$\frac{xy}{1-x}=(1-z)(x-x^{1/z})$$ where $0 < x < 1$, $\;0 < y < 1$, $\;0 < z \leq 1$.
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### On the Hurwitz Zeta Function

In my mathematics course in Uni. (I'm a physics student) my prof. gave us the following exercise: to express the Hurwitz Zeta function $\zeta(2k+1,\frac{1}{4})$ with $k=1,2,3,\dots$ in terms of the ...
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### Realization of Bessel functions

As I know when whe trying to execute Bessel functions when $z < 8$ we using this formula $$J_\nu(z) = (\frac{1}{2}z)^\nu \sum_{k = 0}^{\infty} \frac{(-\frac{1}{4}z^2)^k}{k!\Gamma(\nu+k+1)}$$ And ...
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### Dirichlet L-series and Gamma function question

Could someone help me, please, with this exercise? Consider a sequence of complex numbers $\{a_n\}$ such that $a_n=a_m$ iff $n\cong m$ mod $q$ for some positive integer $q$. Define the ...
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### Calculate an integral involving Hermite polynomials

I have to calculate the integral $$\frac{1}{\sqrt{2^nn!}\sqrt{2^ll!}}\frac{1}{\sqrt{\pi}}\int_{-\infty}^{+\infty}H_n(x)e^{-x^2+kx}H_l(x)\;\mathrm{d}x$$ where $H_n(x)$ is the $n^{th}$ Hermite ...
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### How to solve $(m_{(t)} x')' + kx = 0$ Sturm Liouville equation with bessel functions

I have been working on this problem for a while now and think I need assistance. I am trying to solve with respect to $x_{(t)}$ over the interval $t = [0, \infty]$: $$(m_{(t)} x')' + kx = 0$$ ...
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### Electric Potential of an off axis charge (Legendre Generating Function)

An insulated disk, uniform surface charge density $\sigma$, of radius R is laid on the xy plane. Deduce the electric potential $V(z)$ along the z-axis. Next ...
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### Evaluating $\int_0^1 \! C(x) \, \mathrm dx$ through integration by parts

$$\int_0^1 \! C(x) \, \mathrm{d} x.$$ where $C(x) = \int_0^x \cos(t^2) \, \mathrm{d} t$. I am really not quite sure how to go about this one, especially given that it needs to be calculated ...
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### Condition for frame of $L_2$

Let $f$ be continuous, real valued and compactly supported with exactly one maximum function in $L_2$. Form the functions $$f_{m,k}=f^m(x-2^k)$$ Under which conditions $\{f_{m,k}\}$ would be a ...
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### Beta integral transformation

It's a homework task and I can't get past the last step. Task is to prove that $$B(x,y)=\int\limits_0^1 \frac{\tau^{x-1}+\tau^{y-1}}{(1+\tau)^{x+y}} \mathrm{d}\tau$$ By substituting ...
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