Skip to main content

All Questions

Filter by
Sorted by
Tagged with
2 votes
4 answers
740 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 ...
qifeng618's user avatar
  • 1,091
1 vote
1 answer
345 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 ...
qifeng618's user avatar
  • 1,091
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 -...
Jesse Elliott's user avatar
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^...
Ali Taghavi's user avatar
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 ...
Max's user avatar
  • 213
3 votes
0 answers
172 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 ...
user113988's user avatar
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 ...
Kreimer's user avatar
  • 31