All Questions
17
questions
6
votes
3
answers
167
views
Find $f(x)$ such that $\sum_{k=0}^{\infty}f^k\left(x\right)=\sum_{k=0}^{\infty}f^{\left(k\right)}\left(x\right)$
I wonder how we can find all $f(x)$ that satisfies$$\sum_{k=0}^{\infty}f^k\left(x\right)=\sum_{k=0}^{\infty}f^{\left(k\right)}\left(x\right)\tag{1}$$Here $f^{(k)}(x)$ means the $k$th derivative of $f(...
- 6,549
1
vote
0
answers
43
views
Computing $\frac{d^{n-1}}{dz^{n-1}}\frac{z}{\ln\left(z!\right)}^{n}$
I am trying to compute
$$\frac{d^{n-1}}{dz^{n-1}}\left(\frac{z}{\ln\left(z!\right)}\right)^{n}$$
The problem arises when dealing with inversion formulae. My question is, can this expression be ...
- 11
1
vote
1
answer
62
views
Finding the $n$th derivative of $1/(1-a(1-x)(1+x))$ with respect to $x$.
I need to find the $n$th derivative of
$$\frac{1}{1-a(1-x)(1+x)}$$
with respect to $x$. Wolfram says that when $a=1$ we have the $n$th derivative as
$$\frac{1}{1-(1-x)(1+x)}^{(n)} = (-1)^n \frac{(n+1)!...
- 2,691
1
vote
0
answers
79
views
Find an expression whose derivative is multiplicatively separable?
Is there an example of an expression using $f(x)$ and $g(x)$ (perhaps composed with other elementary functions) whose derivative can be factorized into multiplicatively separable expressions, one ...
- 11
9
votes
2
answers
4k
views
Derivative of $\operatorname{arctan2}$
I'm currently working on some navigation equations and I would like to write down the derivative with respect to $x$ of something like $$f(x) = \operatorname{arctan2}(c(x), d(x))$$
I've searched ...
- 117
1
vote
3
answers
200
views
What is the $n$ th derivative of $\ln(x)/(1+x^2)\:?$
I'm into something but I came across the problem of finding a closed form for
$$\left( \frac{\ln x}{1+x^2} \right)^{(n)}$$
where the little $(n)$ denotes the $n$ th derivative of the function. After ...
- 2,691
0
votes
2
answers
189
views
The $n$ th derivative of $\ln(xy)/(1-xy)$ with respect to $x$.
I'm looking for a closed form of
\begin{align}
\frac{d^n}{dx^n}\left( \frac{\ln(xy)}{1-xy} \right)
\end{align}
I tried using the Taylor series:
\begin{align}
\frac{\ln(xy)}{1-xy} = \frac{\sum_{...
- 2,691
7
votes
1
answer
309
views
Is there a closed-form for $\frac{d^ny}{dx^n}$?
I am dealing with this...
#Question#
Given $y$ is a function of $x$, $x^n+y^n=1$, where $n$ is a positive integer.
Find $\displaystyle\frac{d^ny}{dx^n}$ in terms of $x$, $y$ and $n$.
###Example 1###...
- 5,217
4
votes
1
answer
437
views
What's the $n$-th derivative of $\ln(\sin(x))$?
I want to find the $n$-th derivative of $\ln(\sin x)$, i.e.
$$
\frac{d^n\ln(\sin x)}{dx^n}
$$
where $x\in (0,\pi/2)$ such that $\sin x>0$. To make the problem definitely, $x=\pi/4$ is assumed. In ...
- 881
2
votes
1
answer
109
views
Computing the integral of $-1/f''$
I think this is a very silly question but I have some problems nonetheless.
If I know that $g'=-\frac{1}{f''}$, is then
$$
g=(f')^{-1}?
$$
- 2,049
3
votes
1
answer
173
views
Finding the $n^{th}$ derivative of $\frac{x^n}{(1+x)}$
Find the $n^{th}$ derivative of $\frac{x^n}{(1+x)}$ .
I think we have to use Leibnitz's Formula to evaluate this, but I haven't succeeded in it as well. I have already received an answer of $\frac {x^...
12
votes
1
answer
512
views
Derivative of a generalized hypergeometric function
Let $$f(a)={_2F_3}\left(\begin{array}c1,\ 1\\\tfrac32,\ 1-a,\ 2+a\end{array}\middle|-\pi^2\right).$$
How to find $f'(0)$ in a closed form?
- 1,475
8
votes
1
answer
252
views
Closed form for derivative $\frac{d}{d\beta}\,{_2F_1}\left(\frac13,\,\beta;\,\frac43;\,\frac89\right)\Big|_{\beta=\frac56}$
As far as I know, there is no general way to evaluate derivatives of hypergeometric functions with respect to their parameters in a closed form, but for some particular cases it may be possible. I am ...
- 47.3k
24
votes
2
answers
2k
views
Derivative of the Meijer G-function with respect to one of its parameters
Are there any approaches that allow to find a derivative of the Meijer G-function with respect to one of its parameters in a closed form (or at least numerically with a high precision and in ...
- 47.3k
12
votes
1
answer
258
views
Derivatives of the Struve functions $H_\nu(x)$, $L_\nu(x)$ and other related functions w.r.t. their index $\nu$
There are some known formulae for derivatives of the Bessel functions $J_\nu(x),\,$$Y_\nu(x),\,$$K_\nu(x),\,$$I_\nu(x)\,$with respect to their index $\nu$ for certain values of $\nu$, e.g.
$$\left[\...
- 47.3k