All Questions
Tagged with cv.complex-variables special-functions
82 questions
1
vote
1
answer
63
views
Need bound for absolute value of complex-valued special functions (Taylor coefficients of Faddeeva's w(z))
To guarantee accuracy for code [1] that computes Faddeeva's w(z) [2] using Taylor expansions around different centers, I would need upper bounds for the absolute values $|w_n(z)|$ of the coefficients
$...
- 111
3
votes
1
answer
136
views
What's the asymptotic behaviour of $_1F_1(a,b,az)$ when $a\to\infty$?
I'm working towards the solution to a problem about involving the Landau-Zener transition, but I'm finding some difficulties. I need to estimate $ \,_1 F_1\left(\frac{\mathrm i}{4\epsilon},\frac12;\...
- 87
13
votes
1
answer
251
views
Is $(n!^{-d})_{n\geq 0}$ a Pólya frequency sequence?
Fix a positive integer $d$. Is the sequence $(n!^{-d})_{n\geq 0}$ a
Pólya frequency (PF) sequence? Equivalently, is the Toeplitz matrix
$A=[a_{ij}]_{i,j\geq 0}$, where $a_{ij}=0$ if $i>j$ and
$a_{...
- 50.8k
6
votes
1
answer
406
views
Conjectured closed form of $\int_0^1 \frac{\text{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx$
After reading some meta posts, I've decided to post this question on MathOverflow since I didn't receive any comments or answers on MSE
Certainly, I apologize for any oversight. Here's a more refined ...
- 224
0
votes
1
answer
138
views
Is $\Gamma(z,1)\not=0$ for all $z$ with $\Re(z)<0$?
I found this paper online which appears to present zeros of the incomplete gamma function within the right half plane. It makes me think that there are no zeros in the left half plane. Not sure how to ...
- 402
7
votes
0
answers
218
views
Analytic continuation of Dixon's identity
Many well-known combinatorial identities has an analytic version. For example, the following identities
$$
2^n = \sum_{k=0}^n \binom{n}{k}
$$
$$
\binom{2n}{n} = \sum_{k=1}^n \binom{n}{k}^2
$$
can be ...
- 1,608
2
votes
0
answers
217
views
Zeta function associated with a function $f$
Let the function $f(t) = \cos(at)$, where ($0 < a < 1$). Let us define
$$\zeta(z, f) = \frac{1}{\Gamma(z)} \int_0^{+\infty} \frac{t^{z-1}\cos(at)}{e^t-1}\, dt.
$$
Is there a general formula that ...
- 463
2
votes
4
answers
739
views
Is the hypergeometric function ${}_1F_2(1;a,a+\frac12;-x^2)$ an elementary function? How about its positivity, monotonicity, and convexity in $x$?
Is the generalized hypergeometric function ${}_1F_2\bigl(1;a,a+\frac12;-x^2\bigr)$ for $a>-1$ and $x>0$ an elementary function?
How about the positivity, monotonicity, and convexity of the ...
- 1,091
2
votes
1
answer
215
views
An integral transform computation
In Erdelyi, Tables of Integral Transforms, p. 344 Section 7.2.
they note that
\begin{align}
\frac{1}{2 \pi i} \int_{c-i\infty}^{c+i\infty} s^{\nu} e^{\alpha s^2} x^{-s} \, ds
= 2^{-\nu/2} \pi^{-...
- 21
4
votes
0
answers
186
views
Asymptotic analysis for a double integral related to Airy functions
Let $Ai(x,y)$ be the Airy kernel which is given by
\begin{equation}\label{equ2.12}
Ai(x,y)=
\begin{cases}
\dfrac{Ai(x)Ai'(y)-Ai(y)Ai'(x)}{x-y}, & x\ne y, \\
Ai'(x)^2-xAi(x)^2 & x=y. \\
\end{...
- 879
0
votes
0
answers
111
views
The upper bound of hyperbolic cosine function in complex plane
I want to find the upper bound of the following function :
\begin{equation}
M(\lambda)=\max _{E \in \mathbb{C}}\left|L_{+}-L_{-}\right|
\end{equation}
where
\begin{equation}
\begin{aligned}
& 4 \...
0
votes
1
answer
163
views
Solutions of complex linear difference equations
I'm wondering what the solutions of complex linear difference equations like
\begin{equation}
f(z+\eta_1)+f(z+\eta_2)+\ldots+f(z+\eta_n)=0,\ \ \ \eta_1 \cdots\eta_n \in \mathbf{C}
\end{equation}
look ...
- 3
1
vote
1
answer
344
views
Where or what is the general formula for the $n$th derivative of the power-exponential function $x^x$?
It is well-known that the power-exponential function $x^x$ and its first few derivatives are often taught in calculus.
Does the general formula for the $n$th derivative of the power-exponential ...
- 1,091
2
votes
0
answers
129
views
Existence of analytic function in disk algebra [closed]
Does there exist an analytic function $f\in A(\mathbb{D})$, where $A(\mathbb{D})$ is the disk algebra, such that $f(0)=0$ and the real part of $f(z)$ is strictly positive?
- 149
5
votes
2
answers
449
views
Zeros of hypergeometric functions with complex variables
Let $z$ be a complex number and let $a,b,c > 0$. I would like to know the zeros of the following hypergeometric function:
$$_{2}F_{1} (a,b; c :z )=\sum_{k=0}^{+ \infty} \frac{(a)_{k}(b)_{k}}{(c)_{...
0
votes
0
answers
103
views
Computing $\int^{4b}_0 {e^{-tx}\biggl(\frac{\sqrt{4bx-x^2}}{(2b-2c)^2+4cx}\biggr) dx}$
I am a PhD student working on complex analysis. After integrating over a keyhole contour to obtain the inverse of a particular Laplace transform, I ended up with the following integral:
$$\frac{1}{\pi}...
- 9
2
votes
1
answer
387
views
Upper bound for the complex Beta function
The question is almost the same as here.
What is the upper bound for a complex Beta function
$$\DeclareMathOperator{\Im}{Im}\DeclareMathOperator{\Re}{Re}
\displaystyle B(s,z)=\frac{\Gamma(s) \Gamma(z)}...
- 29
0
votes
1
answer
513
views
When is a product of two two-parameter Mittag-Leffler functions a Mittag-Leffler function?
I am studying properties of the two-parameter Mittag-Leffler function.
$$ E_{\alpha,\beta}(z)=\sum_{k=0}^\infty \dfrac{z^k}{\Gamma(\alpha k+\beta)}.$$
I am particularly interested in recurrences and ...
- 123
3
votes
1
answer
464
views
Asymptotic behavior of infinite product of cosines
Consider the function
$$F(z) := \cos(z) \cos(z/3) \cos(z/5) \cos(z/7) \cdots$$
Note that $\cos(z/n) = 1 - \frac{z^2}{n^2} + \cdots$ so the product is absolutely convergent to an entire function.
I ...
- 1,243
7
votes
1
answer
524
views
continued fraction for logarithmic integral
Does the logarithmic integral function $\operatorname{li}(x)$ have the continued fraction expansion
$$\operatorname{li}(x) = \cfrac{x}{\log x -1 -{}} \ \cfrac{1}{\log x -3 -{}} \ \cfrac{4}{\log x -...
- 4,985
4
votes
1
answer
470
views
Beta function, harmonic numbers, and integral values
A problem of Mathematical Physics that I am working on involves the computation of a certain integral. Part of the result reads:
$$
I_k:= [\beta_x( -1 - k, 0) + H_{-2 - k}]x^k
$$
where $\beta_x( -1 - ...
- 255
1
vote
1
answer
110
views
Is trace of a slice of an elementary function of a matrix also elementary?
Let we have an elementary function $f(W)$, applicable to a matrix.
Now consider the function
$g(x)=\operatorname{tr} f(W+x),$
where $x$ is scalar. Is $g(x)$ necessarily an elementary function?
Simple ...
- 10.1k
5
votes
0
answers
225
views
Validity of $\ln z=\frac{\pi}{\operatorname{AGP}(\theta_2^2(1/z),\theta_3^2(1/z))}$
Definitions
For the definitions of $\operatorname{AGP}$ and $\operatorname{AGO}$, see here or here. $\theta_2(z)$ and $\theta_3(z)$ are defined as follows:
$$\theta_2(z)=\sum_{n=-\infty}^\infty z^{(n+...
- 83
3
votes
1
answer
179
views
Analytic or holomorphic extension of the ellipse perimeter function
Let ${\mathbb{R}^2}^+=\{(x,y)\in \mathbb{R}^2\mid x>0, y>0\}$.
Let $P:{\mathbb{R}^2}^+\to \mathbb{R}$ be the function with $P(a,b)=$ $\text{The perimeter of ellipse}\;\; \frac{x^2}{a^2}+\frac{y^...
- 356
9
votes
0
answers
321
views
When exactly is the principal AGM equal to the optimal AGM?
Definitions
Following the terminology of SageMath, let the principal arithmetic-geometric mean, $\operatorname{AGP}$, of $(a,b)\in\mathbb{C}^2$ for $a\ne 0$, $b\ne 0$, $a\ne\pm b$ be defined as ...
- 83
1
vote
2
answers
779
views
Inverse of exponential integral function
The exponential integral function $x \mapsto E_1(x)$ is strictly decreasing on the positive real
axis and, so, is globally real analytically invertible there. Where can I find information concerning
...
- 19
4
votes
0
answers
249
views
Two questions on asymptotic expansion of confluent hypergeometric functions for real variable $x, |x| \to \infty$
I'm looking into the asymptotic expansion for confluent hypergeometric function $_1F_1(a;b;z) \equiv M(a;b;z)$ and I've two quick questions regarding its asymptotic behavior for real values $x,$ i.e. ...
- 223
12
votes
2
answers
1k
views
Short research articles
I am a masters student. I am interested in short articles which have counter examples and very few references. I want to write a short and interesting article.
For example; One of the best known ...
2
votes
0
answers
65
views
Separating a Riemann-Hilbert problem
Consider a RHP on the real line a jump is piece-wise H\"older continuous(or $L^2$), say for example the jump is
$$g(x)=g_1(x)\chi_1+g_2(x)\chi_2,$$
where $g_j(x)$ are Holder continuous functions and $...
- 145
3
votes
0
answers
247
views
Limit of Hankel function for large complex order, fixed real argument
Consider the Hankel function $H_\nu(z)$ where $\nu=re^{i\theta}$ (real $z>0$, $r>0$, $0\leq\theta<\pi$) as $r\rightarrow\infty$. I am aware that the Bessel function $J_\nu(z)$ has the ...
- 573
1
vote
0
answers
97
views
Construct a doubly periodic function on $\mathbb{C}$ using Jacobi elliptic functions with anti-holomorphic involution
I would like to find explicit examples of non-constant meromorphic doubly periodic $f(z)$ on $\mathbb{C}$, i.e. a meromorphic function on $\mathbb{C} / \Gamma$ for some lattice $\Gamma$, such that ...
- 23
4
votes
1
answer
533
views
Real and imaginary parts of $\ln \Gamma(i b)$
The imaginary part of the digamma function when its argument is pure imaginary is known as
$$\Im\psi(\mathrm{i}b)=\frac{1}{2}b^{-1}+\frac{1}{2}\pi\coth{\pi b},$$ and its real part is much more ...
user147095
2
votes
0
answers
83
views
Positivity and zeros of Heun's function
I am interested in understanding where in the complex plane a Heun function might vanish, or where it (or its real part) is positive. Consider the case where the Frobenius indices at $0$ are $(0,1- \...
- 213
2
votes
1
answer
214
views
Ratio of hypergeometric function
Given $a>b>0$, is there any upper bound of the following ratio of hypergeometric function?
$$\frac{_2F_1(a,1-b;a+1;x)}{_2F_1(a,1-b;a+1;y)}$$
for $1>x>y>0$ ideally in the form like some ...
- 1,495
3
votes
1
answer
223
views
Ratio of Selberg integral
I'm considering a ratio of incomplete Selberg integral:
$$f_n(a,b)=\frac{\int_{\Delta_a}\prod_{i=1}^nx_i^{\alpha-\frac{n+1}{2}}\prod_{i=1}^n(1-x_i)^{-1/2}\prod_{i<j}|x_i-x_j|}{\int_{\Delta_b}\prod_{...
- 1,495
1
vote
0
answers
76
views
Identities for beta functions and twisted cohomology
This is a question about notation, I apologize if it is too basic. In the paper
Cho, Koji; Matsumoto, Keiji, Intersection theory for twisted cohomologies and twisted Riemann’s period relations. I, ...
- 8,992
0
votes
1
answer
375
views
Bringing a Heun equation into canonical form
It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly $4$ regular singular points may be suitably ...
- 213
3
votes
0
answers
237
views
best-possible inequalities for hypergeometric functions
In what follows, let $n$ be a positive integer and $0<a<1/2$. I am interested in the Gauss hypergeometric functions, $_{2}F_{1}( -n, -n-a; 1-a; z)$. Notice that these are polynomials, if that is ...
- 113
1
vote
0
answers
98
views
Bound for truncation error of continued fraction for $E_1(z)$
Let $z \in \mathbb C \setminus(-\infty,0)$. It is known that
$$E_1(z) = \cfrac{e^{-z}}{z+\cfrac{1}{1+\cfrac{1}{z+\cfrac{2}{1+\cfrac{2}{z+\cfrac{3}{1+\cdots}}}}}}.$$
For example, see http://functions....
- 4,985
8
votes
1
answer
693
views
lower bound for absolute value of a hypergeometric function
I am working with a certain Gauss hypergeometric function, $_{2}F_{1}(a,a-b;2a;1-z)$, where $a, b \in {\mathbb R}$ with $a, a-b > 0$ and $0<b<1$.
It appears that $\left| _{2}F_{1}(a,a-b;2a;1-...
- 113
3
votes
0
answers
171
views
Nekrasov Partition function and the leading term of Prepotential
I've got a pretty basic question from the paper SEIBERG-WITTEN THEORY
AND RANDOM PARTITIONS, https://arxiv.org/pdf/hep-th/0306238.pdf.
In (4.25) the author expressed the partition function ...
- 425
1
vote
1
answer
256
views
Seeking the derivation of the Fourier Sine Transform of $x^{2\nu}(x^2+a^2)^{-\mu-1}$
In this answer on math.stackexchange.com the Fourier Sine Transform of $x^{2\nu}(x^2+a^2)^{-\mu-1}$ is given in terms of the generalized hypergeometric function:
$$\frac{1}{2}a^{2\nu-2\mu}\frac{\Gamma(...
- 2,239
1
vote
1
answer
655
views
Series involving factorials
Playing around with this series for natural values of $a,b$, it appears that more generally for $c\in\mathbb N$, $$\sum_{k=0}^\infty \frac{ (a+k)! \ (b+k)!}{k!\ (a+b+c+ k+1)! }=\frac{a!\ b!\ (c-1)!}{...
- 13.4k
1
vote
0
answers
443
views
Multivariate solution to Lambert W / product-log function
Consider solving the following system for $x$
\begin{align*}
a - b e^{\theta x} - cx = 0
\end{align*}
According to your favorite computer algebra program, one possible (and the simplest) is
\begin{...
- 229
3
votes
0
answers
106
views
Does the Riemann characterization of the hypergeometric function have a q-analog?
This question is inspired by another recent one here, Characterization of the hypergeometric function. The latter is about the classical result of Riemann characterizing the hypergeometric functions ...
- 18.4k
8
votes
1
answer
1k
views
The inverse of the digamma function
The gamma function is increases on the interval $(x_0, \infty),$ where $x_0$ denotes the unique zero of the digamma function on the positive half line.
The inverse function of gamma function defined ...
- 551
5
votes
1
answer
690
views
Simple integral representation for a beta function with more than two variables
The beta function $B(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}$ can be written for $\Re x, \Re y > 0$ symmetrically as $$ B(x,y) = \int_{-\frac12}^{\frac12}(\frac12-t)^{x-1}(\frac12+t)^{y-1}\,...
- 13.4k
5
votes
0
answers
388
views
Is a basic hypergeometric function ${}_2\phi_1(a, b; c; q, z)$ a meromorphic function in $z$?
Here a basic hypergeometric function is the analytic continuation of the basic hypergeometric series (or called the $q$-hypergeometric series)
$$
{}_2\phi_1(a, b; c; q, z) = \sum^{\infty}_{n = 0} \...
- 123
3
votes
1
answer
205
views
Understand the properties of this function
We define a function
$f(t):=\sum_{n=0}^{\infty}e^{-nt}= \frac{1}{1-e^{-t}}= \frac{e^{\frac{t}{2}}}{e^{\frac{t}{2}}-e^{-\frac{t}{2}}}=\frac{2e^{\frac{t}{2}}}{\sinh\left(\frac{t}{2} \right)}$
observe ...
- 31
10
votes
1
answer
808
views
"unexpected" residue formula for $\Gamma^3(s)/(\Gamma(3s)(e^{2\pi is}-1)) $
There is a related problem in my current work: to find the residue of the following function at any negative integer $s=-n$:
$$f(s)=\frac{\Gamma^3(s)}{\Gamma(3s)(e^{2\pi is}-1)}$$
It seems to be a ...
- 101