# Tagged Questions

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### Intepolate from linear to step function, and one application for shading colors

I'm running after a particular function $f_\sigma : [−1,+1] \rightarrow [-1,+1]$ that could take three different forms depending on the value of its parameter $\sigma$. Could anyone help me ...
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### Generalized Legendre differential equation

In an application I encountered the ODE $$\left( x^2-1 \right) \frac {{\rm d}^{2}}{{\rm d} x^2} f ( x ) +x \left( \frac {\rm d}{{\rm d}x} f (x) \right) ( 8x^2-7 ) -4 (C+1) f( x ) =0.$$ which is ...
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### Series $\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$

Does anyone know the sums of the following two series? $$\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$$ $$\sum_{n=1}^\infty (-1)^{n+1} \frac{x^{4n-1}}{4n-1}$$ I encounter such series in my work.
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### bessel function maximizer

I try to find global maximum for $\frac{J_2(x)}{x^2}$ I suspect it happens at x=0 ( plotting the graph) where the value of the function is $\frac{1}{8}$ I know local maximizers are at zeros of ...
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### An example of a homeomorphism on $[0,1]^2$ with constant Jacobian determinant $\pm1$

Let $T(x,y):=(t_1(x,y),t_2(x,y))$ be a continuous bijection, namely a homeomorphism on $[0,1]^2$. I am trying to find a $T$ such that $\det(J_T)=1$. (*) The trivial cases are $T(x,y)=(x,y)$, ...
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### modern analysis: step functions with upper and lower sums [duplicate]

A function $f$ defined on $[a,b]$ is a step function if there is a partition $P$ such that $f$ is constant on each subinterval of $P$ a. Show that upper and lower sums are integrals of step ...
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### Integral in $n-$dimensional euclidean space

I want to calculate this integral in $n$-dimensional euclidean space. $$I(x)=\int_{\mathbb{R}^n}\frac{d^n k}{(2\pi)^n}\frac{e^{i(k\cdot x)}}{k^2+a^2},$$ where $k^2=(k\cdot k)$, ...
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### An example of almost periodic function

"I need a continuous almost periodic function $f(x)$ such that $f(x)$ exists as $x$ tends to infinity. But this function should not be constant, which is a trivial example." Definition of almost ...
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### Bernoulli Map properties

I am referring to the function stated here http://en.wikipedia.org/wiki/Dyadic_transformation This map is defined on $[0,1]$ by $f_n(x)=nx [mod 1]$ There are three things I do not quite understand, ...
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### Evaluating $\int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx$

I am trying to prove that $$\int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx = -\frac{\pi^3}{48}-\frac{\pi}{8}\log^2 2 +G\log 2$$ where $G$ is the Catalan's Constant. Numerically, it's ...
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### A Challenging Logarithmic Integral $\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx$

How can we prove that: $$\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx=\frac{7\pi^2}{48}\zeta(3)-\frac{25}{16}\zeta(5)$$ where $\zeta(z)$ is the Riemann Zeta Function. The best I could do was ...
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### function defined as an integral involving Bessel functions

i need to analyze a function of the form $$F(x,y) = \int_0^{1} e^{-(1+s)\alpha x}\sinh((1-s)\beta y) I_0(\sqrt{(x^2-y^2)s}) ds$$ Where $I_0$ is the modified Bessel function. $x>y$ always. ...
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### Fourier transform of the indicator of the unit ball

What is the Fourier transform of the indicator of the unit ball in $\mathbb R^n$? I think it is known as one of special functions, so I would be happy to know which one.
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### Looking for an analytical expression of this horror-integral

I have given a function $$a(m,n,\mu,\nu,p):=\frac{2p+1}{2}\frac{(p-m-\mu)!}{(p+m+\mu)!}\int_{-1}^1 P^m_n(x)P^\mu_{\nu}(x)P^{m+\mu}_p(x)dx.$$ (of course all parameters are appropriate integers, so that ...
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### This limit of the hypergeometric function makes me stunning…

I am currently reading this paper Physics paper please have a look at the definition of (20) and then (36). In (36) they investigate the limit of the hypergeometric function ...
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### Smooth step between $-1$ and $1$

I am currently interpolating the step between two functions from $-1$ and $1$ smoothly therefore I used $\tanh$. Since I am quite confident with the result, but interested in further ways to do this, ...
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### On the absolute integrability of radially symmetric functions

Let $\phi:\mathbb R\to\mathbb R$ be an smooth, even function and $\int_\mathbb R|\phi(t)|^p\,\mathrm dt<\infty$, that is, $\phi$ is pth-power integrable in $\mathbb R$ iff $p\geq p_0$ for ...
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### Express complex Bessel function in terms of functions taking real arguements

I want to use the Bessel function in C++. Since this one is not implemented there for complex arguments, I am looking for a way to express the bessel function(first and second kind) as: ...
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### Difference between Rician distribution and Gaussian distribution

could any one please tell me the difference between Rician and Gaussian Distribution and the advantages of using one over other please.With some mathematical proof would be truly appreciated Thank ...
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### Find local maxima of this quadratic function

How can I find local maxima of this quadratic function? $$f(x) = \sum _{i=1}^n -\frac{(z_i - x)_+^2}{2} - \left\{((\frac{(z_i - x)_+^2}{2})-(\frac{(y_i - x)_+^2}{2}) ) * c_i\right\}$$ which ...
### A problematic integral: $\int_0^{2\pi} e^{-2\pi i\lambda\cos(t)}\,dt$
Is there a special trick to calculate this integral? $$\int_0^{2\pi} e^{-2\pi i\lambda\cos(t)}\,dt$$ for $\lambda>0$.