2
votes
0answers
37 views

Functional Equation involving derivatives and time-steps [duplicate]

I am attempting to solve the equation $$f(x + 1) = f'(x)$$ for distributions $C \rightarrow C: f(x)$ My first guess to exploit the fact that this seems similar to identity $$\sin\left( ...
1
vote
0answers
61 views

Are even Dirichlet series constants?

I consider a Dirichlet series with an absolute abscissa of convergence $\sigma$ that can be meromorphically extend to $\mathbb{C}$: $$\phi(s)=\sum_{n=1}^{+\infty}{\frac{a_n}{n^s}}$$ and the analytic ...
0
votes
0answers
24 views

Question on 2 functional equations.

Let $z,x$ be complex numbers. Im looking for analytic functions $f(z)$ such that : $$1) \exp(\ln^{5} (f(x))=\sum_i a_i f(b_ix)$$ $$2)f(x)^5=\sum_j c_j f(d_jx)$$ holds for all $x$ and where both ...
3
votes
3answers
107 views

Solving equation system of complex funtions

Does there exist two complex functions $f$ and $g$ satisfy below equation system? $$ \begin{cases} f=e^g\\ g=e^f \end{cases} $$ What about analytic funtions?
8
votes
1answer
438 views

Different methods to prove $\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) \Gamma (1-s) \zeta (1-s)$.

I've recently encountered this strangely attractive equation (Riemann's functional equation), along with Riemann's original proof. $$\displaystyle\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) ...
5
votes
2answers
254 views

Iterative roots of sine

Is there an analytical function $f(z)$ such that $f(f(z)) = \sin(z)$? More generally, an analytical function such that f applied $n$ times to $z$ gives $\sin(z)$? Is there a general theory for ...
1
vote
2answers
82 views

Solve this equation: $f(s)=P(s)\exp(Q(s))$

Let $f,P,Q$ three analytic functions. Here $P$ is a polynomial. I want to solve this equation: $$f(s)=P(s)\exp(Q(s)).$$ The unknown here are $P, Q$ and $f$ is known.
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 ...
3
votes
2answers
80 views

Solution to functional equation

I have the following functional equation: $$f(a+b)=f^{n-1}(a)f(b)+f^{n-2}(a)f^{1}(b)+...+f(a)f^{n-1}(b)=\sum_{k=0}^{n-1}f^{n-1-k}(a)f^{k}(b)$$ where $a,b$ are complex and the function $f$ is an ...
0
votes
0answers
72 views

Interpretation of functional equation of dedekind eta function

It's well known that the Dedekind eta function defined by $\eta(z) = \displaystyle e^{\frac{\pi i z}{12}} \prod_{n=1}^{\infty} (1 - e^{2 \pi i n z}) $ converges for $z$ in the upper half plane to a ...
6
votes
1answer
96 views

Can we give a definition of the cotangent based on a functional equation?

I've recently learned that the cotangent satisfies the following functional equation: $$\dfrac1{f(z)}=f(z)-2f(2z)$$ (true for $f(z)\neq 0$). Can we solve this equation for real or complex ...
1
vote
3answers
189 views

Solution of functional equation $f(x)=-f(x-a)$

I have a problem with finding solution. I suppose it will be something like $f(x) =G(x)\Re(e^{\frac{x\pi}{a}})$, where $\Re$ is real part of a complex number, $G(x)$ periodic function whith period ...
1
vote
1answer
167 views

Analytic solution of a certain functional equation

Here's my question. Let $b_2$, ..., $b_d \in \mathbb{C}$ ($d$ is an integer greater than 2), and consider the functional equation $$V(z^d)=dz^{d-1} V(z)+(b_2 z^{d-2} + b_3 z^{d-3}+\ldots + b_d)$$ ...
7
votes
1answer
400 views

Entire function with $f(z)=\sin(f(z))$ must be constant?

I'm trying to show why an entire function with the property $f(z)= \sin(f(z))$ everywhere must be constant. Is it sufficient to say that when taking the derivatives, we will get $f'(z)=f'(z) ...
3
votes
0answers
144 views

Analytic function that provide $f^2(z)=z$

I am trying to solve this problem: Does there exist a function $f(z)$, that is analytic at $E=\{x+iy :x>y\}$ and provides $f^2(z)=z$ for every $z \in \mathbb C$. I have seen a solution that ...
0
votes
1answer
223 views

functional equation uniqueness $f(x^{2})= f(x)^{2} $ in $\mathbb{C}$ and $f(x)+f''(x) = 0 $ in $\mathbb{C}$

This is an exercise from a book I tried: One would like to find all holomorphic equations that satisfy:$$i) \ f(z)+f''(z) = 0 \text{ in } \mathbb{C} $$$$ii)\ f(z^{2})=f(z)^{2} \text{ in } ...
10
votes
2answers
542 views

Entire functions such that $f(z^{2})=f(z)^{2}$

I'm having trouble solving this one. Could you help me? Characterize the entire functions such that $f(z^{2})=f(z)^{2}$ for all $z\in \mathbb{C}$. Hint: Divide in the cases $f(0)=1$ and $f(0)=0$. ...
6
votes
3answers
276 views

Solving $\forall x \in \mathbb{R}_+^*, f'(x) = f\left(\frac1{x}\right)$

I recently came across this equation : $$\forall x \in \mathbb{R}_+^*, f'(x) = f\left(\frac1{x}\right)$$where $f \in \mathcal{C}^1(\mathbb{R}, \mathbb{R})$. I've done the following, but I'm stuck at ...