Tagged Questions
1
vote
2answers
37 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
84 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
193 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
91 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
213 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
97 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
278 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
55 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
1k 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.
$$J_\alpha (\beta) = ...
2
votes
1answer
189 views
To find the closed form of $ f^{-1}(x)$ if $3f(x)=e^{x}+e^{\alpha x}+e^{\alpha^2 x}$
$3f(x)=e^{x}+e^{\alpha x}+e^{\alpha^2 x}$ where $\alpha=e^{\frac{2\pi i}{3} }$
I would like to find a closed form of $ f^{-1}(x)$
$f(x)=\sum \limits_{k=0}^\infty \frac{x^{3k}}{(3k)!}$
We can see ...
2
votes
1answer
49 views
A certain family of continuous functions on $[0,1]^2$ the closure of which linear span is $\tilde{\mathcal{C}}([0,1]^2,\mathbb{R}))$
First of all I must apologize for the vague title and am open to suggestions.
This is not a Homework Assignment but something I once again encountered while reading a very compactly written paper.
...
3
votes
2answers
153 views
Does anyone recognize this function?
I am looking for a function $f(n)$ that satisfies the following two conditions at the same time $$ \frac{f(n-1)}{f(n)}=(-1)^n\quad ,\quad \frac{f(n+1)}{f(n)}=(+1)^n\equiv 1,\quad \forall ...
1
vote
0answers
55 views
How to find a function with the following properties?
I want to find a function $f(s,x)$ such that
$f(s,x)$ is analytic
for any $s \in Z^+ $,
$f(s,x)=B_s(x)$,
where $B_s(x)$ are the Bernoulli polynomials
$f(a, x)$ is elementary against $x$ at any ...
3
votes
1answer
101 views
Hypergeometric functions inequality
Let $_2F_1(a,b;c,z)$ be the (Gauss) hypergeometric function, and $m$ and $n$ positive integers.
From a simple plot it looks like
$_2F_1(m+n,1,m+1,\frac{m}{m+n})>\frac{m}{n} ...
2
votes
1answer
258 views
What can we say about functions satisfying $f(a + b) = f(a)f(b) $ for all $a,b\in \mathbb{R}$? [duplicate]
Possible Duplicate:
Is there a name for such kind of function?
I am investigating functions satisfying the exponentiation identity $f(a + b) = f(a)f(b)$ for all $a,b\in \Bbb R$. This is ...
1
vote
0answers
209 views
Using Rouche's theorem
Let $p>1$.
Consider $\phi(p)=\int_0^{\infty}\left|\frac{\sin t}{t}\right|^pdt$.
Function $\phi(p)$ is analytic on its domain.
It's derivative, $\phi'(p)=\int_0^{\infty}\left|\frac{\sin ...
2
votes
0answers
209 views
$L_2$-norm representation of the function
Let
$$
f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+,
$$
where $\alpha > -\frac 12$(see for reference ...
2
votes
0answers
74 views
Iterated Root Mean Square-Arithmetic Mean
Can I find iterated Root Mean Square-Arithmetic Mean as a function of Arithmetic-geometric mean (AGM) with some transformations if it is possible?
if not possible, what is the closed form of it as ...
0
votes
1answer
146 views
The ratio of two strictly increasing polynomial functions
I have the following question:
Given,
$f_1(a), f_2(a),\ldots, f_n(a)$ and $g_1(a), g_2(a),\ldots, g_n(a)$ are strictly increasing positive "polynomial" functions of $a$.
It is also known that
...
1
vote
1answer
120 views
Does division of polynomials give an increasing function?
How can I show that
\begin{equation} f(a)=\frac{\sum_{i=1}^{k^*-1} \left(\begin{array}{c} K \\ i \\ \end{array} \right) \left(-1-\frac{1}{ar}\right)^i+1}{\sum_{i=1}^{k^*-1} \left(\begin{array}{c} K \\ ...
1
vote
2answers
2k views
Definition of Sinc function
I just want to make clear of the definition of sinc(x). I know there is a normalized and unnormalized definition for the sinc function. If we have unnormalized sinc then we have: ...
0
votes
2answers
287 views
Looking for function of bell-like curve that peaks quickly.
I'm writing a little Sage/Python script that would graph the cumulative effects of taking a particular medication at different time intervals / doses.
Right now, I'm using the following equation:
...
1
vote
1answer
346 views
How to show that functions of this type are strictly decreasing
Let $f:[0,\infty)\to \mathbf{R}$ be defined by $$ f(x) = \frac{1}{x+1} \int_x^\infty g(r,x) dr,$$ where $g(r,x)$ is a "nice" function and all of this makes sense.
Suppose that I want to show that ...
0
votes
1answer
65 views
Is the hypergeometric function $F(5/4,3/4; 2, z)$ bounded on $(0,1]$
Consider the classical hypergeometric function $F(5/4,3/4; 2, z)$ for $z\in (0,1]$. Is this bounded by some real number (independent of $z$)?
I'm aware of Euler's formula:
$$F(5/4,3/4; 2, z) = ...
4
votes
2answers
146 views
Bound for the Legendre function of the second kind of degree $1/2$
Let $Q_{1/2}(u)$ be the Legendre function of the second kind of degree $1/2$.
One can show that $Q_{1/2}(u) = O(u^{-3/2})$ as $u\to \infty$; see Equation 21 in Section 3.9.2 of Higher transcendental ...
1
vote
1answer
114 views
Is Bessel function $J_0(n)$ absolutely summable?
Is the Bessel function $J_0(n)$ absolutely summable i.e $\sum_{n=0}^{\infty}|J_0(n)| < \rm C$? Since $\lim\limits_{n \to\infty} J_0(n) = 0$, I'd assume the absolute sum converges to a constant ...
7
votes
1answer
304 views
Higher Order Trigonometric Function
Once in a time, I had to work with functions that have the following Taylor series expansion:
$$
t_m(x)=1-\frac{x^m}{m!}+\frac{x^{2m}}{(2m)!}+\cdots =\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}.
$$
...
0
votes
0answers
138 views
integral of Bessel function
How to compute the following integral of Bessel function $J_1$
$$
\int_0^{\infty}|x^{-s+1}J_1^s|dx, s\geq 2
$$
Can it be computed the same way as in An integral about Bessel function
Or there are ...
2
votes
0answers
43 views
Deriving functions for empiracal distributions -very applied mathatics
First I am a new user of this site.
Second my math background is very limited, although I do have a lot of experience in applied statistics.
Component or piece part failures on high value parts($1000 ...
5
votes
1answer
198 views
Generalization of cos: is this function known?
Consider a function $f_1$ defined by $f_1(x)=1-x+o(x)$ and $f_1(2x)=f_1(x)^2 + 0$. It's simple to find that $f_1(x)=e^{-x}$ (for example by writing series near $x=0$).
Consider a function $f_2$ ...
