1
vote
2answers
26 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
13 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
28 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
22 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
25 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
16 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
13 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
39 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
51 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
0answers
22 views

Superadditivity ; How to prove Superadditive function?

How can I prove this superadditive function? It's quite confused
0
votes
1answer
26 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
17 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
59 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
14 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
16 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
41 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
30 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
40 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 ?
0
votes
1answer
12 views

Distribution function and its concavity

Let $\gamma$ be a parameter which is $0<\gamma<4$. Is the following distribution function concave $$ F(x)=1-x^{-\gamma} $$
2
votes
2answers
34 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
91 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
43 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
57 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
102 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
56 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
97 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
45 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
37 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
88 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
371 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
45 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 ?
33
votes
0answers
643 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
527 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
58 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
102 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
137 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 ...
0
votes
1answer
133 views

How do I create a sigmoid-esque function with the following properties?

For a range of $x$ values between $A$ and $B$ I would like $f(x) \rightarrow x$. For values less than $A$ I would like $f(x)$ to exhibit a sigmoid-esque convergence to $A'$ where $A'$ is $A - ...
1
vote
2answers
42 views

Looking for function with specific properties

I need a function $f$ that is arbitrarly times differentiable and which has integral $$\int _a^b f(x) dx $$ strictly positive (where $a$ and $b$ are fixed), and for all derivatives, we have ...
2
votes
1answer
95 views

Construct a generating function for the components of a sum

Let $j \in Z_+$. Set $$ a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!} $$ and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$. Find generating function $\sum_{j}a_jx^j$ so that allows to ...
1
vote
2answers
713 views

Meaning of function with circle and cross

I've seen this function M2 = tmp ⊕ Pi. What does the circle with cross do?
1
vote
2answers
131 views

An injective map where each value is mapped to many others?

I want "something" ("something" because maybe it is not really a mathematical function, called F in the above image) that can describe what is shown on the image. A given value from a domain Xi can ...
2
votes
0answers
244 views

Determining the probability density function from an equation

I have the following (for me quite interesting) densities for which I am completely stuck. I only hope that you can provide me some help. Let me introduce my problem. I have two probability ...
2
votes
1answer
120 views

Upper bound for $\Gamma(x+y)$

Let $x, y \geq 1$ be two real numbers. I am wondering if one can get an upper bound for $\Gamma(x+y)$ in terms of $\Gamma(x)\Gamma(y)$? Any references or ideas are very appreciated. Thank you.
5
votes
3answers
365 views

Inverse function of $y=W(e^{ax+b})-W(e^{cx+d})+zx$

I have a simple question for which I am looking for a closed form expression (If there exits one). In other words, given: $$y=W(e^{ax+b})-W(e^{cx+d})+zx$$ where $W$ is the Lambert $W$ function and ...
1
vote
0answers
60 views

Functional equation for the given function

For instance, there is functional equation for Lambert W function $z=W(z) e^{W(z)}$ And moreover, there is differential one: $z(1+W)\frac{dW}{dz}=W$. At the same time, there is no known functional ...
0
votes
1answer
2k views

Calculation of bessel function versus matlab solution

I am looking to calculate the Bessel function of the first kind $J_o(\beta)$. I am using the formula (referenced from wikipedia) to accomplish this. $$J_\alpha (\beta) = ...