0
votes
0answers
24 views

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 ...
1
vote
0answers
33 views

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 ...
2
votes
8answers
705 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 ...
2
votes
1answer
35 views

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 $f(x)$?
2
votes
0answers
38 views

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 $b$: $x (t)-λ\int_0^1 t^2s^2x(s) \, ds = 4t + bt^2$
1
vote
1answer
56 views

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 ...
1
vote
1answer
82 views

Bessel function and upper bound

I'm stuck on this following problem: Let $G$ a function such that $0\leq G(t)\leq 1$, and $G(t)=1$ if $B^2\leq t\leq 4B^2$, with $\operatorname{supp}G\subset [\frac{1}{4}B^2, 9B^2]$ and $G^{(j)}\ll ...
0
votes
0answers
47 views

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 ...
1
vote
1answer
35 views

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 ...
2
votes
1answer
48 views

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: $$z_x=\frac{1}{\tan \frac xy} \frac{1}{1+(\frac xy)^2} \frac 1y=\frac {y}{\tan \frac xy (x^2+y^2)}$$ ...
2
votes
1answer
22 views

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!
4
votes
3answers
62 views

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 ...
1
vote
2answers
36 views

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 : $f(0)=0$, ...
0
votes
0answers
29 views

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 ...
0
votes
2answers
41 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?
0
votes
1answer
27 views

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 ...
0
votes
1answer
41 views

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 ...
0
votes
0answers
33 views

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 ...
0
votes
0answers
35 views

Question about an exponential funtion

Now we have a function: $f(x)=e^x, x\in \mathbb R$ Question: 1) Assume that $x>0$, discuss the number of the intersects between $f(x)$ and $y=mx^2,m>0$ under different situation. 2)Assume ...
0
votes
0answers
15 views

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 ...
0
votes
1answer
60 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 $a<\infty$
0
votes
1answer
59 views

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 $m\geq0$
0
votes
1answer
40 views

Bessel function with shifted argument

Is there any standard practice which may represents $J_m(a\pm kx)$ in terms of $J_m(kx)$
1
vote
0answers
18 views

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 ...
3
votes
1answer
95 views

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 ...
0
votes
2answers
17 views

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 $0<x<1$ ...
0
votes
0answers
20 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, ...
1
vote
0answers
46 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 ...
0
votes
1answer
37 views

Differentiation of Bessel function

How can I represent the following differentiation in terms of $J_n$? The equation is: $\frac{\partial}{\partial x}[xJ_n'(x)]$
1
vote
1answer
41 views

Meaning of function $f(x) = [x]$

What does the function $f (x) = [x]$ mean? How is it different from least and greatest integer function ?
2
votes
2answers
35 views

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 $F(x,y)$ ...
2
votes
3answers
97 views

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?
2
votes
1answer
46 views

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 ...
0
votes
1answer
63 views

Representation of heaviside step functions

Can the heaviside step function, $u(t)$ be represented like so: $$u(t)=\frac{1}{2}\left(\frac{|x|}{x}+1\right)$$
0
votes
0answers
25 views

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 ...
2
votes
2answers
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 ...
0
votes
1answer
57 views

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 ...
3
votes
1answer
106 views

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
3
votes
2answers
47 views

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. ...
0
votes
2answers
43 views

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 $(1) f(x)=x-|x|$ ...
0
votes
1answer
106 views

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 ...
1
vote
2answers
555 views

How can a unit step function be differentiable??

Recently, I am taking a Signal & System course at my college. In all of the signal & system textbooks I have read, we see that it is written " When we differentiate a Unit Step Function, we ...
1
vote
2answers
46 views

Function by properties

I am looking for a function $f(x)$, which is vertical at the origin ($f'(0)\approx -\infty$) goes to zero further from the origin ($f(x) \to 0$ for $x\to \infty$ and for $x\to -\infty$) and is ...
2
votes
1answer
38 views

Simplifying confluient hypergeometric functions

I need to simplify the confulent hypergeometric function: $U(x>1,1/2,y>0)$. I don't know if someone knows a simpler form ?
43
votes
0answers
1k views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where ...
2
votes
2answers
36 views

Simplify this expression that came from integration

I was doing a calculation and arrived at a term $\left[P_{l-1}(\cos(\theta)) -P_{l+1}(\cos(\theta))\right]_{0}^{\pi}$(So this is the result of an integration). Does anybody of you know how to simplify ...
3
votes
2answers
696 views

When is the sum of two exponentials functions equal to another exponential function?

Fix real numbers $a_1$, $a_2$, $a_3$ and $b_1$, $b_2$, $b_3$ and $c_1$, $c_2$, $c_3$. Consider the equation $$ a_1\exp(b_1 (x-c_1)) + a_2\exp(b_2 (x-c_2)) = a_3\exp(b_3 (x-c_3)) $$ in $x$. My ...
0
votes
2answers
61 views

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 ...
3
votes
1answer
116 views

Addition formulas for Jacobi amplitude function

Are there any known summation formulas for the Jacobi amplitude function? I need a formula like $\mathrm{am}(t+x)=\mathrm{am}(t) + f(x)$. I have plotted some graphs and it seems that $f(x)$ is ...
1
vote
0answers
154 views

Graph for lower and upperside bound absolute value function

The relation to be represented in graph is as follows $$ y = k \text{ for } k-1 < |x| < k \text{ where } k\in \mathbb Z $$ Normally we plot the area where the relation holds good is where ...