3
votes
1answer
64 views

Evaluation of a class of continued fractions

Is there a closed-form way of writing the continued fraction: $$ 1 + \frac{2}{3+ \frac{4}{5 + \frac{6}{7 + ...}}} $$ EDIT: Since the above has been determined as $\frac{1}{\sqrt{e}-1}$, is there a ...
3
votes
0answers
27 views

Convergence of sum of antiderivative and derivative

This question is inspired by this question: Solutions for $ \frac{dy}{dx}=y $?. It makes me wonder if there are any function where the sum of all antiderivative and derivative converges. The ...
3
votes
1answer
89 views

$f_{n+1}(x)=f_n(x+1)-f_n(x)$ functional equation and “classification of functions”

Doing a quiz I found a question of this kind "given $a_0, a_1, a_2, ...,a_n$ find $a_{n+1}$" In order to find the $f$ such that $f(a_n)=a_{n+1}$ I tryed for a function like $f(x)=k+x$ ...
2
votes
1answer
41 views

The “trick” in the Herglotz trick

In How does the Herglotz trick work?, is explained as in "Proofs from THE BOOK" by Aigner and Ziegler, but the "trick" itself I found to be not so clear. The trick says: It follows from (4) ...
6
votes
3answers
267 views

How to solve this infinite set of equations?

If I can find a solution to the following set of equations then, with a bit of luck, I should be able to derive all sorts of nifty new results in non-equilibrium statistical mechanics. However, I'm ...
0
votes
2answers
45 views

Summarize my formula?

I would like to summarize my formula. $p$ and $y$ are constant value, $10000$ and $0.65$. When $n = 3$, my formula recalculate the result of $n = 2$. I don't want to recalculate. Is there way to ...
6
votes
2answers
95 views

Solution of Equation $ 1 + x + \frac{1\cdot3}{2!}x^2 + \frac{1\cdot3\cdot5}{3!}x^3 + \ldots = \sqrt{2} $

I have an equation whose left side is a infinite series . I can solve the equation if I am able to find a close form of the series . The equation is as follows : $$ 1 + x + \frac{1\cdot3}{2!}x^2 + ...
4
votes
2answers
92 views

Deriving the series representation of the digamma function from the functional equation

By repeatedly using the functional equation $ \displaystyle\psi(z+1) = \frac{1}{z} + \psi(z)$, I get that $$ \psi(z) = \psi(z+n) - \frac{1}{z+n-1} - \ldots - \frac{1}{z+1} - \frac{1}{z}$$ or ...
0
votes
2answers
978 views
2
votes
1answer
66 views

Solving the equation $\displaystyle \frac{e^x}{x}=\int_n^{n+1}f(t)\,dt$

Suppose the equation $\displaystyle \frac{e^x}{x}=\int_n^{n+1}f(t)\,dt$ as $f(t)=\frac{e^t}{t}$ and $n\in \mathbb{N} \setminus{0}$. How to prove that: The equation above has a unique solution $U_n$ ...
0
votes
2answers
74 views

Complex Constant and Convergent Power Series

Suppose that the function $f$ is defined by a convergent power series and suppose that $f (z + w) = f (z) f (w)$ for all complex $z$, $w$. (a) Prove directly from this assumption that there is a ...
6
votes
2answers
249 views

Solving the functional equation $f(x) = f(x / \phi) f(x / \phi^2 - 1)$

I'm trying to find a function $f(x)$ such that the spacing between consecutive roots looks like the infinite Fibonacci word: $$1, \phi^{-1}, 1, 1, \phi^{-1}, 1, \phi^{-1}, 1, 1, \phi^{-1}, 1, 1, ...
0
votes
1answer
329 views

Complexity of $T(n)=\sqrt{n}T(\sqrt{n})+n$

I tried to find the complexity of this recursion equation: $T(n)=\sqrt{n}T(\sqrt{n})+n$, by doing couple of iterations and getting a general idea, but I completely got lost. I'd really love your ...
1
vote
1answer
55 views

Transform the sample to make it more similar to a given

$X=\{x_{i}\}$ and $Y=\{y_{i}\}$ are numeric samples: $y_i \ge 0, x_i \ge 0, i \in [0..N]$. I need to find the mapping $F(X)=\{F(x_i)\}$ with fairly simple formula such that: Euclidean distance ...
5
votes
3answers
138 views

Looking for a function $f$ such that $f(i)=2(f(i-1)+f(\lceil i/2\rceil))$

I'm looking for a solution $f$ to the difference equation $$f(i)=2(f(i-1)+f(\lceil i/2\rceil))$$ with $f(2)=4$. Very grateful for any ideas. PS. I've tried plotting the the initial values into ...
7
votes
2answers
315 views

Evaluating $f(x) f(x/2) f(x/4) f(x/8) \cdots$

Let $f : \mathbb R \to \mathbb R$ be a given function with $\lvert f(x) \rvert \le 1$ and $f(0) = 1$. Is there a nice simplified expression for $$\begin{align}F(x) &= f(x) f(x/2) f(x/4) f(x/8) ...
0
votes
2answers
131 views

How to solve the following system?

I need to find the function c(k), knowing that $$\sum_{k=0}^{\infty} \frac{c(k)}{k!}=1$$ $$\sum_{k=0}^{\infty} \frac{c(2k)}{(2k)!}=0$$ $$\sum_{k=0}^{\infty} \frac{c(2k+1)}{(2k+1)!}=1$$ ...
1
vote
3answers
119 views

Solving Series of equations

I have the following series of equations (n+2 equations n+2 variables): $k_0q_0+\lambda q_0 + c_0 = 0$ $k_1q_1+\lambda q_1 + c_1 = 0$ $k_nq_n+\lambda q_n + c_n = 0$ $q_1+q_2+....+q_n = 1$ ...
0
votes
1answer
185 views

Solving this set of quadratic equations

I have a set of quadratic equations of the form.. $ 2S_0q_0(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_0 = 0$ $ 2S_1q_1(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_1 = 0$ . . . $ 2S_nq_n(q_0S_0 + q_1S_1 ...