# Tagged Questions

51 views

### Terminology for $1/(e^x+1)$?

$\frac{1}{e^x+1}$ and $\frac{e^x}{e^x+1}$ Just wonder if either of the above function has a term/name associated with it? Or they are just functions that look beautiful without names? Maybe they ...
16 views

### function undefined at odd inputs

I am a high-school student in pre-calculus. My teacher told me today that it is impossible to define a function using only multiplication, division, exponents, addition, subtraction such that it ...
71 views

### List of functions $f(cx) = C\cdot f(x)$

I was looking for some complex functions f(x), which satisfies the condition: $$\exists (c, C) \in \Bbb C^2 \backslash\{(1,1)\}, \forall x \in \Bbb C, f(cx) = C\cdot f(x)$$ Till now I have got ...
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### Proving concavity for complicated function

I have a rather complicated function, $f$, that I am trying to demonstrate is log-log-concave, i.e., $$\frac{d^2\log f}{(d\log x)^2}\leq 0.$$ The reason I think it is concave is purely heuristic. ...
107 views

### Why is it hold for types of operators?

We ‎let ‎the ‎state ‎space ‎be‎ ‏‎‎‎$\mathcal{H} =‎ ‎‎H_{E}^{2}(0 , 1) \times L^2(0 , 1)$‎ equipped with the norm ‎ \begin{align} \| (f , g) \| = \int_{0}^{1} [ |f''(x)|^2 + |g(x)|^2] ‎\mathrm{d}x‎ ...
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### Hyperbolic sinc function

Cardinal sine function or sinc function is defined by: $$\mathrm{sinc}x=\begin{cases}\frac{\sin x}{x}, & x \neq 0,\\ 1, & x = 0,\end{cases}$$ Is there any ...
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### Differentiation of the Beta function

I suppose that \begin{align*} \frac{\partial}{\partial x}\left[B\left(x,y\right)\right]=&\frac{\partial}{\partial x}\left[\int_0^1t^{x-1}(1-t)^{y-1}dt\right]\\ ...
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### Simplify $L_{-1}(x) + I_1(x)$

Is there a simple solution for x << 0 of the following equation: $$Y(x) = L_{-1}(x) + I_1(x)$$ Where $L_{-1}(x)$ is modified Struve function and $I_1(x)$ is modified bessel function. For ...
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### Even function about a point over a restricted range

Why is $f(x)=(x-1)^2$sin$(n\pi x)$ even about $x=1$ for $0\leq x \leq2$? I understand that $(x-1)^2$ is even about $x=1$ and I can plot the graph for various values of $n$ on wolfram alpha, but how ...
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### Help with operator $f(x^q)=\frac{1}{q+1}x^q$.

This question is somewhat related to this. I am looking for an operator $f:\mathbb{R}[x]\to\mathbb{R}[x]$, that is, $f$ is an operator that maps polynomials in one variable to polynomials in one ...
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### Help with function $f_r(x^q)=q^rx^{q-1}$

Let $r,q$ be a positive integers. I am looking for a function $f_r(x^q)$ such that it is satisfied $$f_r (x^q)=q^r x^{q-1}$$ (without explicit dependence on $q$ of course, and for $r>1$). I ...
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### Keep a Function Positive via Mod

I have a function $F$ and I want it to remain positive--i.e., $$- F=F,\quad F=F$$ Would sticking a $\mod 2$ in front of $F$ do this? That is, because $$-1\mod 2=1\mod 2=1$$ Then let ...
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### Is there a name for this property of a real function?

Let $M=\sup_{x \in [0,1]^n} f(x)$ where $f:[0,1]^n \rightarrow \mathbb{R}$ is differentiable twice, and write $x=(x_1, \dots, x_n)$. Let $M_{x_i=0}=\sup_{x \in [0,1]^n:x_i=0} f(x)$ and ...
721 views

### What are the most important functions every mathematician should know? [closed]

I am an undergrad in math and was wondering, what are for you the most important functions every mathematician should know? At the moment I think ...
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### composite function problem

If I have the following expression: $$g(f(x))-g(x)=1,$$ it is possible to express $f(x)$ in terms of the $g(x)$: $$f(x)=g^{-1}(1+g(x)).$$ Is it possible to express $g(x)$ in terms of $f(x)$?
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### In the space $L^2 [0,1]$ to solve for all values ​​of the complex parameters $\lambda$ and $b:$ [closed]

In the space $L^2 [0,1]$ to find a solution of the integral equation for all values ​​of the complex parameters $\lambda$ and $b$: $x (t)-λ\int_0^1 t^2s^2x(s) \, ds = 4t + bt^2$
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### Continuous non differentiable functions :)

I was searching for functions like Weierstrass (continuous but differentiable nowhere), but I haven't found any. If you could tell me some that would be great. Also, I would like to find some ...
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### Is $2\delta(x) \neq \delta(x)$?

$2\delta(x) \neq \delta(x)$ since, by definition, Can this been seen graphically though? If so, how? If not, why is it that they are mathematically different but graphically the same? Btw, I ...
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### Characterization of nowhere differentiable functions

Let $N:=\{f\in C([0,1])\vert \text{ f is nowhere differentiable } \}$ and $A_n = \{f\in C([0,1]) \vert \exists x\in [0,1]s.t. \forall y\in[0,1]: |f(x)-f(y)|\leq n |x-y|\}$. Now I have already ...
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### Let $z=\ln \tan\frac xy.$ What is $z_x$ and what is $z_y$?

Let $$z=\ln \tan\frac xy.$$ What is $z_x$ and what is $z_y$? Thanks ahead:) What I have tried: $$z_x=\frac{1}{\tan \frac xy} \frac{1}{1+(\frac xy)^2} \frac 1y=\frac {y}{\tan \frac xy (x^2+y^2)}$$ ...
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### How would I go about converting $U(n)= 4^n+U(n-1)$ into an explicit form? [closed]

I have the recursive function $U(n)= 4^n+U(n-1)$, and I'd like to convert it into an explicit form. If you could also walk me through the process that would be great. Thanks!
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### Game With 21 Squares, How Many Possible Answers? Function Building

We played this game in our math class, okay, I'll explain how it's played. There are 21 squares in a straight line across, the first person shades in 2 adjacent squares. The next player shades in 2 ...
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### How to find a function from an infinite sequence of derivatives at $x=0$

I need an odd function $f(x)$ which converges to $\pm \infty$ at $\pm a$ for some positive $a$. At $x=0$, the even derivatives must be $0$, and the odd derivatives must be factorials : $f(0)=0$, ...
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### Function that defines a skew bell shaped curve

The following formula describes a normal bell shaped curve: $$f(x,a,b,c) = \frac{1}{1+|\frac{x-c}{a}|^{2b}}$$ I am trying to model data that exhibits has skewed bell shaped behaviour (please see the ...
55 views

### How do you shift a sigmoidal curve to the right?

How do you shift the function $1$ $/ ( 1 + e ^ {-x} )$ to the right without altering the shape of the curve?
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### Discontinuity of the indicator function

Consider the function $q(x,\theta)=1\{ x \in \{x \text{ s.t. } \theta+x_i>0 \text{ }\forall i \}\}$ where 1 is the indicator function taking value 1 if the condition inside $\{ \}$ is satisfied and ...
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### Continuous superposition of bump functions

I am trying to "model" Fig 2 with a superposition of a bump function. I understand that bump functions are bounded and can be often differentiated. The bump function I have used is shown in Fig 1. My ...
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### Move Hill equation curve horizontally without changing its shape?

I have a normal Hill function of: $y = \dfrac{x^\lambda}{h^\lambda + x^\lambda}$; where $\lambda$ is Hill coefficient, and $h$ indicates the infection point. I am concerning if we could add another ...
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### Finding the Bessel function from its derivative

I have a situation: $A_k\frac{\partial J_m(k\rho)}{\partial \rho}=0$. where $k=k_1$ for $0\leq\rho\leq a$ and $k=k_2$ for $a \leq \rho \leq \Lambda-a$ with $a,\Lambda\leq \infty$. Can I proceed with ...
82 views

### Definite integral involving bessel functions of first and second kind

Is there any standard solution of the integral: $\lim_{\epsilon \to 0} \int_{\epsilon}^{a} J_m(k_1\rho)Y_m(k_2\rho)\rho \, d\rho$. where the integer $m\geq0$ and $a<\infty$
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### Definite integral of product of two bessel functions of different order and different argument

What is the solution of the integral: $\int_0^a J_m(k_2\rho)J_{m+1}(k_1\rho)d\rho$ where the integer $m\geq0$
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### Bessel function with shifted argument

Is there any standard practice which may represents $J_m(a\pm kx)$ in terms of $J_m(kx)$
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### An additive function with capability to retrieve individual components

Assume a set of points in a path $A,B,C,D$ and $E$. Starting from point $A$ assuming single direction there are 4 possible paths as $AB, ABC, ABCD, ABCDE$ with identifies 1 to 4 representing each ...
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### Solution of definite integrals involving incomplete Gamma function

The solution of the integral $$\int_0^{\infty}e^{-\beta x}\gamma(\nu,\alpha \sqrt x)dx$$ is given as ...
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### Simple function with a couple of properties

Please supply any simple function $f(x | p)$ which the following properties: $f(0 | p) = 0$ and $f(1 | p)=0$ $f'(0 | p) = 0$ and $f'(1 | p)=0$ $f(x | p)>0$ for $0<x<1$ For $0<x<1$ ...
22 views

### How to solve this Fourier-Bessel integral

I want to solve this integral: $\{\int_0^{a\cos{\phi}}\rho e^{-bk\rho}J_q(k\rho)d\rho\}$ where $b=\frac{jn\kappa \cos \phi}{k}$ and $q$ is integer. Since it has a form like Fourier-Bessel transform, ...
47 views

### Solution of some Bessel integrals

The solution of the integration $\int_0^\infty e^{-\alpha x}J_v(\beta x)x^{\mu-1}dx$ is given in a standard form. Can I use the same result when the upper limit of the integration is finite? The ...
45 views

### Differentiation of Bessel function

How can I represent the following differentiation in terms of $J_n$? The equation is: $\frac{\partial}{\partial x}[xJ_n'(x)]$
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### Meaning of function $f(x) = [x]$

What does the function $f (x) = [x]$ mean? How is it different from least and greatest integer function ?
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### Function F(x,y) which is high when (both x and y are high) and (x is close to y)

Given two positive real numbers $x$ and $y$, I'm looking for a function $F(x,y)$ that has the following behaviour: the higher $x$, $y$ values and the smaller $|x-y|$, the higher the number $F(x,y)$ ...
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### Mathematical function to convert two numbers into one? [closed]

Is there a mathematical function that converts two numbers into one so that the two numbers can always be extracted again?
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### different results for the solution of bessel function with exponential

I have this integral $$\int_0^\infty e^{-\alpha x}K_1(\beta \sqrt{x}) \, dx.$$ for $\Re[\alpha] >0$, and $\Re[\beta]>0$ According to the (Table of Integrals, Series, and Products, Seventh ...
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### Representation of heaviside step functions

Can the heaviside step function, $u(t)$ be represented like so: $$u(t)=\frac{1}{2}\left(\frac{|x|}{x}+1\right)$$
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### higher order hash functions

I am trying to find a functions which can efficiently transform (input) N number to and Integer K (and uniformly distribute it over some range - will use this as hash function). It is kind of a hash ...
105 views

### Prove cardinality

Let $V = \{x \in \mathbb{R} | 2 < x < 5\}$. Prove that $S$ and $V$ have the same cardinality, where $S$ denotes the set of real numbers between $0$ and $1$. The part I don't get is where my ...
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### Autocorrelation functions of 2 correlated stationairy processes

I have some trouble solving the following problem: Given are the stationairy processes $X_t$ and $Y_t$: $X_t = Z_t*\sqrt{7+0.5X_{t-1}^2}$ $Y_t = 2+(2/3)*Y_{t-1}+X_t$ Where $Z_t$ is distributed IID ...
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### is this expansion possible (f+g)(t)= f(t)+g(t)?

Hi if the functions were polynomials is (f+g)(t)= f(t)+g(t) possible? I am trying to integrate a function of that form
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### Approximate a piecewise function that is 0 for a while and then has constant slope after a certain point.

I'm interested in finding a non-piecewise approximation to a simple piecewise function. $$f(x) = \begin{cases} 0 & : x < T \\ x-T & : x \geq T \end{cases}$$ i.e. ...
$\textbf{Support}$:$f$ 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 $(1) f(x)=x-|x|$ ...
### Is there a name for the normal CDF function $\Phi(\cdot)$?
I can't seem to find a plain English name for the CDF of the normal distribution $\Phi(x)$. However, I am aware of several other related functions that have a name, so I feel like this one should as ...