6
votes
3answers
256 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
44 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
91 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 + ...
3
votes
1answer
78 views

Deriving the series formula for the digamma function using the functional equation

By repeatedly applying 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
609 views
7
votes
1answer
131 views

The value of the trilogarithm ($\text{Li}_{3} (z)$) at $\frac{1}{2}$

From the functional equation $$\text{Li}_{3}(z) + \text{Li}_{3}(1-z)+ \text{Li}_{3} \Big( 1 - \frac{1}{z} \Big) = \zeta(3) + \frac{\ln^{3} (z)}{6}+ \frac{\pi^{2} \ln (z) }{6}- \frac{\ln^{2} (z) ...
2
votes
1answer
65 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
0answers
43 views

What is the complete general solution to $s(z)-s(z-1)=f(z)$?

$$ \displaystyle s(z)=\sum _{k=-\infty }^{\infty } c_h(k) e^{i 2 \pi k z}+\sum _{k=-\infty }^{\infty } c_p(k) \left(\zeta \left(-k,a_0-z_0\right)-\zeta \left(-k,z-z_0+1\right)\right) $$ I developed ...
0
votes
2answers
67 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
233 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
318 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
54 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
290 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
130 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
118 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
183 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 ...