1
vote
0answers
11 views

Derivative of a generalized hypergeometric function

Let $$f(a)={_2F_3}\left(\begin{array}c1,\ 1\\\tfrac32,\ 1-a,\ 2+a\end{array}\middle|-\pi^2\right).$$ How to find $f'(0)$ in a closed form?
1
vote
2answers
74 views

Lyapunov function for non-autonomous non-linear differential equations

I have read some lecture notes about Lyapunov’s Second Method for autonomous system. Now, I want to deal with the stability of a non-autonomous system. Suppose there is a non-autonomous non-linear ...
2
votes
1answer
48 views

Let $z=\ln \tan\frac xy.$ What is $z_x$ and what is $z_y$?

Let $$z=\ln \tan\frac xy.$$ What is $z_x$ and what is $z_y$? Thanks ahead:) What I have tried: $$z_x=\frac{1}{\tan \frac xy} \frac{1}{1+(\frac xy)^2} \frac 1y=\frac {y}{\tan \frac xy (x^2+y^2)}$$ ...
0
votes
0answers
29 views

How to evaluate the derivate of a hypergeometric function w.r.t. one of its parameters?

I have to numerically evaluate the derivative of the hypergeometric function w.r.t. its first and second parameters $\large\frac{\partial}{\partial a}{_2F_1}\left(a , b ,c;z\right)$ and ...
6
votes
1answer
55 views

Closed form for derivative $\frac{d}{d\beta}\,{_2F_1}\left(\frac13,\,\beta;\,\frac43;\,\frac89\right)\Big|_{\beta=\frac56}$

As far as I know, there is no general way to evaluate derivatives of hypergeometric functions with respect to their parameters in a closed form, but for some particular cases it may be possible. I am ...
19
votes
1answer
356 views

Derivative of the Meijer G-function with respect to one of its parameters

Are there any approaches that allow to find a derivative of the Meijer G-function with respect to one of its parameters in a closed form (or at least numerically with a high precision and in ...
8
votes
0answers
81 views

Derivatives of the Struve functions $H_\nu(x)$, $L_\nu(x)$ and other related functions w.r.t. their index $\nu$

There are some known formulae for derivatives of the Bessel functions $J_\nu(x),\,$$Y_\nu(x),\,$$K_\nu(x),\,$$I_\nu(x)\,$with respect to their index $\nu$ for certain values of $\nu$, e.g. ...
13
votes
1answer
167 views

How to calclulate a derivate of a hypergeometric function w.r.t. one of its parameters?

Is it possible to take a derivative of a hypergeometric function w.r.t. one of its parameters and express it in a closed form? I am particularly interested in this case: ...
1
vote
2answers
82 views

Why do we have $\psi^{(m)} (z)=(-1)^{m+1}\int_{0}^{\infty}\frac{t^me^{-zt}}{1-e^{-t}}dt$?

Why is the following representation true? $$\psi^{(m)} (z)=(-1)^{m+1}\int_{0}^{\infty}\frac{t^me^{-zt}}{1-e^{-t}}dt,$$ where $\psi^{(m)} (z)$ denotes the Polygamma function.
1
vote
0answers
136 views

$n$-th derivative of exponential integral

I would like to adapt Abramowitz and Stegun 5.1.27 formula to my problem. Equation 5.1.27 states: ...
1
vote
1answer
85 views

Reasoning about the gamma function using the digamma function

I am working on evaluating the following equation: $\log\Gamma(\frac{1}{2}x) - \log\Gamma(\frac{1}{3}x)$ If I'm understanding correctly, the above is an increasing function which can be demonstrated ...
2
votes
1answer
104 views

Polylogarithm - derivative with respect to order

Does anybody know where I could find the expression for $$\frac{\partial}{\partial s}\mathrm{Li}_s(z)\bigg|_{s=0}$$ or something similar?
11
votes
0answers
318 views

Divergence of the Derivative of the Prime Counting Function

On the one hand, the Prime Counting Function $\pi_0(x)$ maybe be written $$ \pi_0(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1} $$ with $ \operatorname{R}(z) = ...
2
votes
0answers
170 views

Problem with understanding first (and second) derivative of a two-sided infinite series

For the function $$f(x)=b^x-1 = x_1 \qquad g(x)=\log(1+x)/\log(b) $$ and its iterative notation $$ x_0=x \qquad x_h=f(x_{h-1})=g(x_{h+1}) \qquad x_{-1}=g(x_0) $$ with b from the interval $1 \lt b \lt ...
1
vote
0answers
70 views

$n$-th derivative of the prolate spheroidal function

For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined $$ ...
1
vote
0answers
263 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 ...
5
votes
3answers
331 views

Proof that $Γ'(1) = -γ$?

I know that $Γ'(1) = -γ$, but how does one prove this? Starting from the basics, we have that: $$Γ(x) = \int_0^\infty e^{-t} t^{x-1} dt$$ How do we differentiate this? How do we then find that ...