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Special Functions - Science topic
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Questions related to Special Functions
What is missing is an exact definition of probability that would contain time as a dimensionless quantity woven into a 3D geometric physical space.
It should be mentioned that the current definition of probability as the relative frequency of successful trials is primitive and contains no time.
On the other hand, the quantum mechanical definition of the probability density as,
p(r,t)=ψ(r,t)*.ψ(r,t),
which introduces time via the system's destination time and not from its start time is of limited usefulness and leads to unnecessary complications.
It's just a sarcastic definition.
It should be mentioned that a preliminary definition of the probability function of space and time proposed in the Cairo technique led to revolutionary solutions of time-dependent partial differential equations, integration and differentiation, special functions such as the Gamma function, etc. without the use of mathematics.
It is well known that Painlevé transcendent have their origin in the study of special functions, which often arise as solutions of differential equations, as well as in the study of isomonodromic deformations of linear differential equations. So, What are the current research directions in the relation between Nonlinear Special Functions and Painleve equations ?
Practical applications of special functions of mathematics in the oil and gas industry and related fields, thank you
Is the reciprocal of the inverse tangent $\frac{1}{\arctan x}$ a (logarithmically) completely monotonic function on the right-half line?
If $\frac{1}{\arctan x}$ is a (logarithmically) completely monotonic function on $(0,\infty)$, can one give an explicit expression of the measure $\mu(t)$ in the integral representation in the Bernstein--Widder theorem for $f(x)=\frac{1}{\arctan x}$?
These questions have been stated in details at the website https://math.stackexchange.com/questions/4247090
Dear Colleagues
I need an inequality for the ratio of two Bernoulli numbers, see attached picture. Could you please help me to find it? Thank you very much.
Best regards
Feng Qi (F. Qi)
Bell polynomials of the second kind Bn,k(x1,x2,...,xn-k+1) are also called the partial Bell polynomials, where n and k are positive integers. It is known that Bn,k(1,1,...,1) equals Stirling numbers of the second kind S(n,k).
What are the values of the special Bell polynomials of the second kind Bn,k(0,1,0,1,0,1,0,...) and Bn,k(1,0,1,0,1,0,...)? Where can I find answers to Bn,k(0,1,0,1,0,1,0,...) and Bn,k(1,0,1,0,1,0,...)? Do they exist somewhere?
The Riemann zeta function or Euler–Riemann zeta hypothesis is the more challenging and unsolved problem in mathematics. What's the applications in physics and science engineering ? Some research advances to solve it ?
I am looking for compact formula for approximation of Mittag Lefller function.
Please give me a hint on how to do that. By which tool? By which website?
Thanks a million
The representation of the special functions in series form is more useful.
Greeting to all
I have one integration, I tried with it many times, so I need your help and feedback, please.
f(x)= integration from 0 to infinity {[x3 cos(ax)]\[expx-1]}
- the first one, I used the integration by part
but the problem with the 0-> infinity .
- the second one, I use the some Special function like ψ(x), function
Regards
What are the specialized functions of the right and left hemispheres of the human brain?
Dear researchers , I'm a student in Master 1 (EDP) , and am a beginner in research , I have one international paper entitled " A new special function and it's application in probability " , I want people here to Give me comments to improve that research for the futur contribution in mathematics ? , Now I want theorist in probability and numerical analyis to give us any constrictive opinion about that research in all needed sides , For checking that paper via the journal webpage , just to check this link , Thanks som much for any comments or any kind of help.
In my profile are my co-authors Anastasiia Garanina and Anastasiia S. Garanina. But this is the same person. I know that there are twins of my profile (R. Uzbekov, R. E. Uzbekov, Rustem E. Uzbekov, Rustem Uzbekov) because the names of the authors are written differently in different journals. In a similar database of the Moscow University “ISTINA” for this there is a special function “profile merging”. Is it possible to make a similar function for ReceachGate?
Suppose I take a simple exponential function of decrease with respect to velocity. i.e. exp -(alpha* mod(velocity)). How can I find alpha? Is there a way to find it numerically?
I need to make a relatively quick calculation of confluent Heun function (It is HeunC in Maplesoft Maple notation) values with different parameters in different points. I understand that it is possible to solve a Heun equation numerically, but that way seems not to be effective.
Of course, it is possible to use, e.g. Maple, for numerical procedure realization I need this in my study, but applying some series expansion or hypergeometric functions is more preferable.
So, I would appreciate simple (because I'm not a pure mathematician) information about series expansions of the Heun confluent function, its expression by hypergeometrics or any other known functions which are available to realize the numerical calculation.
Can Special Function approach be used in painleve analysis for dicrete and utra-discrete cases??
Hi,
Similar to Matlab xlswrite(filename,A), what is the function name in Scilab to write on a new excel file ? I tried searching, but couldn't find out. There was only write_csv available.
Thanks,
Praveen
Is it a variant of Vandermonde convolution formula for falling factorials? What is the answer? See the picture.
See the picture uploaded here.
DOC in water has special function and role in metal polluants cycle. I will need to sample DOC concentration and also need to analysis DOC fluorescence.
My question is, is it enough to just take one sample for both analysis? for the DOC concentration analysis, I usually put acid to conditionnante the sample, does this action (add acid) will affect DOC fluorescence analysis ?
Thanks
Can one explicitly compute the Laplace transform of a function involving the reciprocal of the gamma function? See the picture for the Laplace transform. Thank you very much.
HI everybody! Looking for an expert who is specialized in the functionalization of cellulose nanocrystals with proteins and/or polysaccharides. Is there anyone who would like to share his/her experience? I would be very grateful if you could help (article, drawback, etc.)
Is the series in the picture convergent? If it is convergent, what is the sum of the series?
Dear All
Happy New Year.
Can one verify an identity involving factorials? For details, please see the picture uploaded with this message.
Best regards
Feng Qi (F. Qi)
What is the name of this continued fraction? How can I express it? How can I compute it? Could you please recommend me some related references? Thank a lot in advance.
I remember that there exists a list of papers for the gamma function on internet, but now I can not find it. In my memory, the list was collected by a mathematician from a country, such as Romania and Croatia, in the East Europe. As you know, there are so many papers for the gamma function that no one can collect all of them. Therefore, the list I am finding is very dear for me. Can you help me find such a list? Anyway, appreciate your kind help!
What are completely monotonic functions on an interval $I$? See the picture 1.png
What is the Bernstein-Widder theorem for completely monotonic functions on the infinite interval $I=(0,\infty)$? See the picture 1.png
My question is: is there an anology on the finite interval $I=(a,b)$ of the Bernstein-Widder theorem for completely monotonic functions on the infinite interval $I=(0,\infty)$? In other words, if $f(x)$ is a completely monotonic function on the finite interval $I=(a,b)$, is there an integral representation like (1.2) in the picture 1.png for the completely monotonic function $f(x)$ on the finite interval $(a,b)$?
The answer to this question is very important for me. Anyway, thank everybody who would provide answers and who would pay attention on this question.
Dear all
What and where is the formula for higher derivatives of a product of many functions? In other words, how to compute higher derivatives of a product of $n$ functions? Concretely speaking, what is the answer (a general formula) to
$$
\frac{d^m}{d x^m}\left[\prod_{k=1}^n f_k(x)\right]=?
$$
where $m,n\in\mathbb{N}$. Could you please show me a reference containing the answer? Thank a lot!
Best regards
Feng Qi (F. Qi)
Dear all, please does anyone know this products of hypergeomtric functions 2F1(-n,c;d;a*z)*2F1(-m,e;f;b*z) ; with n and m non-negative integers. How to get the simple form ?
For compiling molecular information that all a plant cell needs to trigger its growth differently then the surrounding cells. e.g. Megaspore Mother Cell (MMC) and trichome are the cells which shows enlargement over other cells.
So please indicate about the types of cells in a plant body and the molecular triggers behind their enlargement to perform specialized function.
Say for a given a function f, there are functions h and g satisfying h leq f leq g. What do we call h and g? I've used upper and lower bounds, which is incorrect. I reckon g could be called a dominant. What is h?
Related with new area of research.
Lambert function has been proposed by Jean-Henri Lambert, some times called Omegat function (W).
the equation can be written as following:
xexp(x) = z this means that x= W0(z)
So the question is , how can I compute W0?
Since last 15 years, I am working on Special Functions and their applications. Recently, number of applications of special functions found in many fields. Now, I am interested in extensions of special functions? Those are interested please contact to me at <goyal.praveen2011@gmail.com>
Hello,
My question is how can I evaluate a discrete metaheuristic using a benchmack mathematical functions. in other words, the concerned metaheuristic handel a discrete solutions xi with D dimensions where each dimension can have discrete values (or either binary 0 or 1).
How can this metaheuritics be evaluated using the existing mathematical functions that have a continous inputs ?
In the case of a binary discrete values is their a special functions for that ?
Thank you very much.
Best regards.
I am working on pulsatile flow of blood in an artery with slightly elliptic cross section and landed up with Mathieu Functions. The computational complexity of dealing with continued fractions and recurrence relations poses a great threat. There are some classical work on these lines way back in 1950's but due to poor computational facility at that time, the authors have just provided mathematical solutions without carrying out any computational work to understand the flow characteristics. Any comments on this? References?
I would like to prove that the following real integral
$$I_t(x)=\int_{-\infty}^{\infty}e^{-\frac{\cosh^2(u)}{2x}}\,e^{-\frac{u^2}{2 t}}\,\cos\left(\frac{\pi\,u}{2t }\right)\,\cosh(u)\,du$$
(with x,t>0 ) is non negative, that is, $I_t(x)\geq 0$. The main problem appear with the oscillatory term
\cos\left(\frac{\pi\,u}{2t }\right)
It will be used where normal overcurrent relay cannot be set properly.
With the help of physical applications, we will develop more and more properties of these useful functions.
It is well known that the secant $\sec z$ may be expanded at $z=0$ into the power series
\begin{equation}\label{secant-Series}
\sec z=\sum_{n=0}^\infty(-1)^nE_{2n}\frac{z^{2n}}{(2n)!}
\end{equation}
for $|z|<\frac\pi2$, where $E_n$ for $n\ge0$ stand for the Euler numbers which are integers and may be defined by
\begin{equation}
\frac{2e^z}{e^{2z}+1}=\sum_{n=0}^\infty\frac{E_n}{n!}z^n =\sum_{n=0}^\infty E_{2n}\frac{z^{2n}}{(2n)!}, \quad |z|<\pi.
\end{equation}
What is the power series expansion at $0$ of the secant to the power of $3$? In other words, what are coefficients in the following power series?
\begin{equation}
\sec^3z=\sum_{n=0}^\infty A_{2n}\frac{z^{2n}}{(2n)!}, \quad |z|<\frac\pi2.
\end{equation}
It is clear that the secant to the third power $\sec^3z$ is even on the interval $\bigl(-\frac\pi2,\frac\pi2\bigr)$.
The logarithmic concavity must be used in the required result.
I am asking for a combinatorial interpretation of a formula for Bell numbers in terms of Kummer confluent hypergeometric functions and Stirling numbers of the second kind. See the formula (8) and Theorem 1 in the attached PDF file or http://arxiv.org/abs/1402.2361. Could you please help me? Thank a lot.
Modified Bessel function of first kind and second kind of order n with complex argument a+ib are denoted by I[n, a+ib] and K[n, a+ib] respectively. How can these functions I[n, a+ib] and K[n, a+ib] be expressed in real and imaginary parts separately ?
Just as that we make use of the Gaussian function for infinite dimensionals.
Approximation of t distribution is essential for finding the p-value in a computer program (while testing the hypothesis about the means). Is it enough to have three decimal point accuracy of the approximation? How many decimals should be correctly approximated by a function? Finally, how to approximate the cdf of t-distribution efficiently? Please suggest some thoughts on the same. Thank you.
_2F_1(a,b;c;z) is the Gauss hypergeometric function.
It is known that Legendre and Chebyshev have contributions in the field.
Li_2 is the dilogarithm function of complex argument.
I tried everything I could by using Sec. 5 of Lewin's book, but failed to find an answer.
I need to find the Fourier transform of displaced airy function.
The function is ψn(ξ) = Nn Ai (ξ − ξn), where ξ=x/x0 x0=(1/2)^1/3, and ξn = 3π/2*(n − 1/4) 2/3. Nn is normalization constant.
I need to find its Fourier transform in momentum basis. I need to plot that out on mathematica, but didn't get the correct results. Do I need to confirm the Fourier transform?
I used the Fourier shift theorem and found the answer as
ψn(p) = Nn*x0/sqrt(2Pi) Exp(i((p*x0)^3-(p*x0*ξn))
Is it a correct Fourier?
Can someone could help me with following question? I need to prove the following bound:
- 2 \int_{-1}^1 |p_{n}(s)|^2 T_n(s) ds \le c \it_{-1}^1 |p_n(s)|^2 ds
with c<1 independent of $n$. (Note the "-" sign in the integral). Here, T_n is Chebyshev polynomial of first kind with p_n either
a) polynomials of degree n-2,
b) polynomials of degree n vanishing at -1,1
c) p_n=(1-s^2)^{1/2}q_{n-2} with q_{n-2} a polynomial of degree n-2
I've checked this bound numerically using a simple code in Matlab (it can be reduced to compute the algebraically largest eigenvalue of a symmetric matrix), up to degree 2500 and that I've observed:
a) c=0.71
b) c=0.83
c) c=0
I've sought for more information in the literature, but I'm not an expert in special functions. Can anyone help?
